The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
The Heat Equation
Vipul Naik
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
April 9, 2007
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Outline
The Heat Equation
Vipul Naik
Basic properties of the heat equation
Partial derivatives
The heat equation in one dimension
Heat equation in more than one dimension
The obvious properties of the heat equation
Physical intuition behind the heat equation
Heat and its flow
Temperature and two laws relating temperature to heat
Properties of the heat equation
Recapitulation
Obvious structural properties
What improves and what is unchanged
The general concept of flow and fixed points
Flow and time-dependent vector field
Picture of the flow
The heat equation as a flow
Basic properties of
the heat equation
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
Solving the heat equation in one variable
Separation of variables
The general
concept of flow
and fixed points
Variations on the heat equation
The heat equation as a diffusion equation
Gradient and reaction terms
Supersolutions and subsolutions
Solving the heat
equation in one
variable
Maximum principles
Maximum principle for the heat equation
Effect of a reaction term
Why the heat equation matters
In the physical world
In the mathematical world
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
A quick recall of partial derivatives
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Suppose u = u(x1 , x2 , . . . , xn ) is a function Rn → R. Then
we define:
∂u
:= (x1 , x2 , . . . , xn ) 7→ (d/dxi )(xi 7→ u(x1 , x2 , . . . , xn ))
∂xi
In other words, the partial derivative in xi equals the
derivative when viewed as a function of xi keeping the other
variables constant.
∂u
is also a map from Rn to R, viz it
Note that each ∂x
i
evaluates at any point in Rn to give a real number.
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Higher order partial derivatives
In addition to making sense of the first partials, we can also
make sense of higher partials. To do this, observe that each
partial is also a function from Rn to R, and can hence be
differentiated in its own right.
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Higher order partial derivatives
In addition to making sense of the first partials, we can also
make sense of higher partials. To do this, observe that each
partial is also a function from Rn to R, and can hence be
differentiated in its own right.
Interestingly, we have the result:
∂u
∂ ∂x
i
=
∂u
∂ ∂x
j
∂xj
∂xi
provided both sides are continuous functions. This is (a
weak form of) Fubini’s theorem.
We can thus simplify the notation and write:
∂2u
∂xi ∂xj
In the particular case where i = j, we can write as:
∂2u
(∂xi )2
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The actual equation
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The heat equation is a differential equation involving three
variables – two independent variables x and t, and one
dependent variable u = u(t, x).
The equation states:
∂2u
∂u
=k
∂t
(∂x)2
k ∈ R is a real number.
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The actual equation
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The heat equation is a differential equation involving three
variables – two independent variables x and t, and one
dependent variable u = u(t, x).
The equation states:
∂2u
∂u
=k
∂t
(∂x)2
k ∈ R is a real number.
Here the symbol ∂u/∂t means the derivative of u with
respect to t, keeping x constant, while the symbol
∂ 2 u/(∂x)2 denotes the second partial derivative in x,
keeping t constant.
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Terminology associated with the heat equation
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
For the heat equation, we use the following terminology
(which shall be clear once we get to the physical motivation):
1. The variable t is termed the time parameter, or the
time variable.
2. The variable x is termed the spatial parameter, or the
spatial variable.
3. The function u is termed the heat function
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The n-dimensional heat equation
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The heat equation in n dimensions is defined as follows:
There are n + 1 independent variables, namely t (the time
parameter) and x1 , x2 , . . . , xn (the space parameters), and
one dependent variable u = u(t, x1 , x2 , . . . , xn ), subject to:
∂u
=k
∂t
n
X
i=1
∂2u
(∂xi )2
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The Heat Equation
The Laplacian operator
Vipul Naik
Basic properties of
the heat equation
The Laplacian operator(defined) is a second-order differential
operator that takes as input a function f : Rn → R, and
outputs another function, ∆f : Rn → R, where:
n
X
∂2f
∆f =
(∂xi )2
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
i=1
Properties of the
heat equation
Note that the Laplacian operator, as expressed in this form,
appears to be heavily coordinate-dependent – a change of
basis would change the Laplacian.
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The Heat Equation
The Laplacian operator
Vipul Naik
Basic properties of
the heat equation
The Laplacian operator(defined) is a second-order differential
operator that takes as input a function f : Rn → R, and
outputs another function, ∆f : Rn → R, where:
n
X
∂2f
∆f =
(∂xi )2
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
i=1
Properties of the
heat equation
Note that the Laplacian operator, as expressed in this form,
appears to be heavily coordinate-dependent – a change of
basis would change the Laplacian.
However, it turns out that any change of basis by an
orthogonal matrix does not alter the value of the Laplacian.
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The heat equation using the Laplacian operator
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
We now think of u as a function R × Rn → R, where the
first R is the time coordinate and the remaining Rn is the
space coordinate. We then have:
∂u
= k∆u
∂t
where the Laplacian on the right side is taken only in terms
of the space coordinates (for a fixed time).
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Autonomous nature
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The heat equation is an autonomous differential equation. In
other words, the only way the dependent variables are
invoked is through differentiation – none of them appear
explicitly in the differential equation.
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Autonomous nature
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The heat equation is an autonomous differential equation. In
other words, the only way the dependent variables are
invoked is through differentiation – none of them appear
explicitly in the differential equation.
This means, in particular, that the heat equation is invariant
under both spatial translation and temporal translation.
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Invariance under orthogonal transformations
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The heat equation states:
∂u
= ∆u
∂t
Now, since ∆ (the Laplacian) is invariant under orthogonal
transformations, the overall heat equation is also invariant
under orthogonal transformations.
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The heat equation could thus encode a physical
law
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
In the classical picture of a physical law, we expect the
following:
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The heat equation could thus encode a physical
law
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
In the classical picture of a physical law, we expect the
following:
I
The physical law should be invariant under
time-translation
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The heat equation could thus encode a physical
law
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
In the classical picture of a physical law, we expect the
following:
I
I
The physical law should be invariant under
time-translation
The physical law should be invariant under spatial
translation
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The heat equation could thus encode a physical
law
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
In the classical picture of a physical law, we expect the
following:
I
The physical law should be invariant under
time-translation
I
The physical law should be invariant under spatial
translation
I
The physical law should be invariant under any
orthogonal transformation (that is, any
distance-preserving transformation)
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The heat equation could thus encode a physical
law
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
In the classical picture of a physical law, we expect the
following:
I
The physical law should be invariant under
time-translation
I
The physical law should be invariant under spatial
translation
I
The physical law should be invariant under any
orthogonal transformation (that is, any
distance-preserving transformation)
We have seen that the heat equation encodes all these
properties.
Hence, it may well encode a physical law.
Partial derivatives
The heat equation in
one dimension
Heat equation in more
than one dimension
The obvious
properties of the heat
equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Outline
The Heat Equation
Vipul Naik
Basic properties of the heat equation
Partial derivatives
The heat equation in one dimension
Heat equation in more than one dimension
The obvious properties of the heat equation
Physical intuition behind the heat equation
Heat and its flow
Temperature and two laws relating temperature to heat
Properties of the heat equation
Recapitulation
Obvious structural properties
What improves and what is unchanged
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
The general concept of flow and fixed points
Flow and time-dependent vector field
Picture of the flow
The heat equation as a flow
The general
concept of flow
and fixed points
Solving the heat equation in one variable
Separation of variables
Solving the heat
equation in one
variable
Variations on the heat equation
The heat equation as a diffusion equation
Gradient and reaction terms
Supersolutions and subsolutions
Variations on the
heat equation
Maximum principles
Maximum principle for the heat equation
Effect of a reaction term
Why the heat equation matters
In the physical world
In the mathematical world
Maximum
principles
Why the heat
equation matters
The Heat Equation
Heat and heat change
Vipul Naik
The heat of a body (or object) is a kind of total measure of
some energy in it. Since heat is a kind of measure totalled
across the volume, we can talk of the thermal density or the
heat density at a point, and the total heat of the body is:
Z
h(x)
V
where h(x) denotes the thermal density at the point x.
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The Heat Equation
Heat and heat change
Vipul Naik
The heat of a body (or object) is a kind of total measure of
some energy in it. Since heat is a kind of measure totalled
across the volume, we can talk of the thermal density or the
heat density at a point, and the total heat of the body is:
Z
h(x)
V
where h(x) denotes the thermal density at the point x.
Note that the total heat at a body is not something of direct
relevance; what is of relevance, though, is the difference in
heat across time. That is, we given times t and t 0 , we are
interested in:
0
δtt h(x)
that is, the difference between the heat density of spatial
point x at times t 0 and t.
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Instantaneous picture of heat change
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
At any given instant of time and at any point in the body,
heat is flowing in some direction, and with some magnitude.
Thus, the instantaneous picture of heat flow is a vector field
on the body that associates to each point the heat flow
vector.
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Heat change in a region of finite volume
The Heat Equation
Vipul Naik
Heat flowing through a body does not necessarily mean that
heat will accumulate in it. This is analogous to the fact that
water flowing through a pipe does not cause any water
accumulation in it.
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Heat change in a region of finite volume
The Heat Equation
Vipul Naik
Heat flowing through a body does not necessarily mean that
heat will accumulate in it. This is analogous to the fact that
water flowing through a pipe does not cause any water
accumulation in it.
The heat change in a body over a period of time is given by:
Heat change = Total heat flown in − Total heat flown out
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Heat change in a region of finite volume
The Heat Equation
Vipul Naik
Heat flowing through a body does not necessarily mean that
heat will accumulate in it. This is analogous to the fact that
water flowing through a pipe does not cause any water
accumulation in it.
The heat change in a body over a period of time is given by:
Heat change = Total heat flown in − Total heat flown out
Differentiating with respect to time:
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Heat change per unit time =
Rate of heat inflow per unit timeMaximum
principles
−Rate of heat outflow per unit time
Why the heat
equation matters
This is with respect to a region (portion) in the body of
positive volume.
The Heat Equation
Change in heat density
Vipul Naik
To measure the heat density, we need to differentiate the
formula:
Heat change per unit time =
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Rate of heat inflow per unit timeHeat and its flow
Temperature and two
laws relating
temperature to heat
−Rate of heat outflow per unit time
For this, take a point x, and look at every line through x.
We want to measure the outward heat flow through x, minus
the inward heat flow through x. In other words, we want to
measure at which the rate of heat flow changes in a very
small neighbourhood of x. This corresponds to the physical
notion of divergence, viz:
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Rate of heat density change per unit time
= ∇.Heat flow vector at the point
Why the heat
equation matters
The Heat Equation
Temperature controls heat flow
Vipul Naik
The temperature at a point is an intrinsic value that governs
the way heat flows through the point. Namely, the
temperature is the potential function whose gradient defines
the heat flow, or in other words, if u denotes the
temperature at position x and time t:
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
Heat flow vector through x at time t = k(∇u)(t, x)
(1)
where k is some suitable constant that measures thermal
conductivity and ∇ is the gradient function, defined as:
∂u
∂u
∂u
(x),
(x), . . . ,
(x))
∇u = x 7→ (
∂x1
∂x2
∂xn
In other words, heat flows in order to equalize temperature.
The k is a conductivity constant that depends on the nature
of the material.
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Temperature depends on heat
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The temperature at a point is related to the heat density at
the point as follows:
Physical intuition
behind the heat
equation
Heat density = Mass density × Specific heat × Temperature
Properties of the
heat equation
Differentiating with respect to time:
The general
concept of flow
and fixed points
Rate of heat density change =
Mass density × Specific heat
×Rate of temperature change
Heat and its flow
Temperature and two
laws relating
temperature to heat
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Putting the things together
The Heat Equation
Vipul Naik
We have the following:
I
The temperature is a scalar function on the body that is
proportional to the heat density at the point
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Putting the things together
The Heat Equation
Vipul Naik
We have the following:
I
The temperature is a scalar function on the body that is
proportional to the heat density at the point
I
The heat flow vector is a vector function on the body
that is proportional to the gradient of the temperature
function at the point
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Putting the things together
The Heat Equation
Vipul Naik
We have the following:
I
The temperature is a scalar function on the body that is
proportional to the heat density at the point
I
The heat flow vector is a vector function on the body
that is proportional to the gradient of the temperature
function at the point
I
The rate of heat change at a point is the divergence of
the heat flow vector
Putting all these together, we get the heat equation:
∂u
= k∆u
∂t
where ∆ is the Laplacian function, viz the composite of the
gradient function and the divergence function.
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Notational simplification
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
To simplify notation, we adopt the convention of using
subscripts for derivatives. That is, we denote ∂u/∂x as ux ,
and ∂ 2 u/∂x∂y as uxy .
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Notational simplification
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
To simplify notation, we adopt the convention of using
subscripts for derivatives. That is, we denote ∂u/∂x as ux ,
and ∂ 2 u/∂x∂y as uxy .
With this notation, the heat equation becomes:
ut = k∆u
Heat and its flow
Temperature and two
laws relating
temperature to heat
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Outline
The Heat Equation
Vipul Naik
Basic properties of the heat equation
Partial derivatives
The heat equation in one dimension
Heat equation in more than one dimension
The obvious properties of the heat equation
Physical intuition behind the heat equation
Heat and its flow
Temperature and two laws relating temperature to heat
Properties of the heat equation
Recapitulation
Obvious structural properties
What improves and what is unchanged
The general concept of flow and fixed points
Flow and time-dependent vector field
Picture of the flow
The heat equation as a flow
Solving the heat equation in one variable
Separation of variables
Variations on the heat equation
The heat equation as a diffusion equation
Gradient and reaction terms
Supersolutions and subsolutions
Maximum principles
Maximum principle for the heat equation
Effect of a reaction term
Why the heat equation matters
In the physical world
In the mathematical world
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
Recapitulation
Obvious structural
properties
What improves and
what is unchanged
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Properties that we already saw
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
We saw the following about the heat equation:
I
It is autonomous in t and x. That is, it is invariant
under time-translation and spatial translation
Physical intuition
behind the heat
equation
Properties of the
heat equation
Recapitulation
Obvious structural
properties
What improves and
what is unchanged
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Properties that we already saw
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
We saw the following about the heat equation:
I
I
It is autonomous in t and x. That is, it is invariant
under time-translation and spatial translation
It is invariant under orthogonal transformations in the
spatial variables. This essentially follows from the fact
that the Laplacian is invariant under orthogonal
transformations
We now look at some other properties that arise from the
mathematical structure and the physical interpretation.
Physical intuition
behind the heat
equation
Properties of the
heat equation
Recapitulation
Obvious structural
properties
What improves and
what is unchanged
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The Heat Equation
Linearity
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
One thing we can say about the equation:
ut = k∆u
is that it is linear, that is:
I
If u and v are two solutions, so is u + v
I
If u is a solution and λ ∈ R, λu is also a solution
Properties of the
heat equation
Recapitulation
Obvious structural
properties
What improves and
what is unchanged
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Dependence on k
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
We would like to know whether solutions to the heat
equation for one value of k are related to solutions for
another value of k.
Physical intuition
behind the heat
equation
Properties of the
heat equation
Recapitulation
Obvious structural
properties
What improves and
what is unchanged
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The Heat Equation
Dependence on k
Vipul Naik
Basic properties of
the heat equation
We would like to know whether solutions to the heat
equation for one value of k are related to solutions for
another value of k.
In fact here is an obvious relation for k 6= 0:
u is a solution to ut = k∆u if and only if (t, x) 7→ u(kt, x) is
a solution to ut = ∆u.
Thus, we can restrict ourselves to a study of the equation:
ut = ∆u
Physical intuition
behind the heat
equation
Properties of the
heat equation
Recapitulation
Obvious structural
properties
What improves and
what is unchanged
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Linear functions are solutions
The Heat Equation
Vipul Naik
The solution to the equation:
∆y = 0
is of the form
y (x) = l(x) + b
where l is a linear functional and b ∈ R.
Thus, the map:
u(t, x) = l(x) + b
gives a solution to the heat equation.
This also tells us that the maps of the form:
u 7→ ((t, x) 7→ u(t, x) + l(x) + b)
are symmetries of the solutions to the heat equation.
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
Recapitulation
Obvious structural
properties
What improves and
what is unchanged
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The total heat is unchanged
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The intuition should tell us that the total heat change of the
system equals the amount of heat that flows out through the
boundary.
Physical intuition
behind the heat
equation
Properties of the
heat equation
Recapitulation
Obvious structural
properties
What improves and
what is unchanged
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The total heat is unchanged
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The intuition should tell us that the total heat change of the
system equals the amount of heat that flows out through the
boundary.
The mathematical justification for this is as follows: the rate
of heat change is the divergence of the heat flow vector field.
Hence, its integral over the whole volume equals the integral
over the boundary area of the heat flow vector field. This is
the area flowing through the boundary.
In particular, if the object doesn’t have boundary, or if it is
insulated at the boundary, then the total heat in the system
doesn’t change.
Physical intuition
behind the heat
equation
Properties of the
heat equation
Recapitulation
Obvious structural
properties
What improves and
what is unchanged
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The extent of heat variation keeps reducing
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
We want to say something like: there is some quantity that
measures the extent to which the heat deviates from its
standard value, such that that quantity keeps reducing as
heat flows within the body.
Properties of the
heat equation
Recapitulation
Obvious structural
properties
What improves and
what is unchanged
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The extent of heat variation keeps reducing
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
We want to say something like: there is some quantity that
measures the extent to which the heat deviates from its
standard value, such that that quantity keeps reducing as
heat flows within the body.
Coming up with an exact description of this quantity is a
hard task. The rough idea would be that this should be the
negative of a physical notion of entropy.
Properties of the
heat equation
Recapitulation
Obvious structural
properties
What improves and
what is unchanged
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Outline
The Heat Equation
Vipul Naik
Basic properties of the heat equation
Partial derivatives
The heat equation in one dimension
Heat equation in more than one dimension
The obvious properties of the heat equation
Physical intuition behind the heat equation
Heat and its flow
Temperature and two laws relating temperature to heat
Properties of the heat equation
Recapitulation
Obvious structural properties
What improves and what is unchanged
The general concept of flow and fixed points
Flow and time-dependent vector field
Picture of the flow
The heat equation as a flow
Solving the heat equation in one variable
Separation of variables
Variations on the heat equation
The heat equation as a diffusion equation
Gradient and reaction terms
Supersolutions and subsolutions
Maximum principles
Maximum principle for the heat equation
Effect of a reaction term
Why the heat equation matters
In the physical world
In the mathematical world
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Differential equation without time
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Suppose we have a differential operator F taking as input
functions y1 , y2 , . . . , ym of independent variables
x1 , x2 , . . . , xn . F could depend on the values yi as well as
their partial derivatives in the xj s. Consider the differential
equation:
F ≡0
That is, we want to determine choices of the functions yi
such that F (y1 , y2 , . . .) = 0 for all i.
For the moment, we concentrate on situations with only one
dependent variable y .
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Moving towards a solution in discrete time
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
To find a solution function y , we can try finding a map G
which takes as input a function y : Rn → R and outputs
another function G (y ) : Rn → R, such that the fixed points
of G are precisely the solution functions y .
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Moving towards a solution in discrete time
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
To find a solution function y , we can try finding a map G
which takes as input a function y : Rn → R and outputs
another function G (y ) : Rn → R, such that the fixed points
of G are precisely the solution functions y .
The idea is to then start off with any arbitrary function y ,
and compute the iterated sequence y , G (y ), G 2 (y ), . . ..
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
In more fancy language
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
In more fancy language what we have done is constructed a
map u : N × Rn → R satisfying the condition:
un+1 (x) − un (x) = F (un , x)
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Moving towards a solution in continuous time
The Heat Equation
Vipul Naik
Recall the setup: we want to find a function y such that
F (y , x1 , x2 , . . . , xn ) = 0 where F may also involve partial
derivatives.
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Moving towards a solution in continuous time
The Heat Equation
Vipul Naik
Recall the setup: we want to find a function y such that
F (y , x1 , x2 , . . . , xn ) = 0 where F may also involve partial
derivatives.
Consider a function u : R × Rn → Rn , written as u = ut (x)
where t is the time parameter, satisfying the following
differential equation:
∂u
= F (u, x1 , x2 , . . . , xn )
∂t
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Moving towards a solution in continuous time
The Heat Equation
Vipul Naik
Recall the setup: we want to find a function y such that
F (y , x1 , x2 , . . . , xn ) = 0 where F may also involve partial
derivatives.
Consider a function u : R × Rn → Rn , written as u = ut (x)
where t is the time parameter, satisfying the following
differential equation:
∂u
= F (u, x1 , x2 , . . . , xn )
∂t
Now suppose y is a solution to the differential equation
associated with F . Then the function u : (t, x) 7→ y (x) (that
is, a function independent of t) is clearly a solution to this
differential equation.
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
How this parallels the discrete case
The Heat Equation
Vipul Naik
In the discrete case, the left-hand side was:
un+1 − un
And we hope to reach some stage n (either finite or infinity)
where this left-hand side vanishes (and hence we get a
solution)
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
How this parallels the discrete case
The Heat Equation
Vipul Naik
In the discrete case, the left-hand side was:
un+1 − un
And we hope to reach some stage n (either finite or infinity)
where this left-hand side vanishes (and hence we get a
solution)
And in the continuous case, the left-hand side was:
∂u
∂t
And we hope to reach some time t (either finite or infinity)
where this left-hand side vanishes (and hence we get a
solution)
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The general picture
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
The picture is this:
We have the set of all possible functions y : Rn → R.
Now, suppose we start off with a function y0 . We want to
investigate the conditions under which we can find a solution
u to the differential equation such that y0 is the function
u(0, ). Such a solution can be viewed as follows: we start
off at y0 at time 0, and then flow along in the space of
possible functions with time.
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Some immediate questions
The Heat Equation
Vipul Naik
Here are natural properties we would seek for the flow
equation:
I
Short-time existence: This means that the solution u
is defined for all x and for t in some neighbourhood of 0
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Some immediate questions
The Heat Equation
Vipul Naik
Here are natural properties we would seek for the flow
equation:
I
I
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Short-time existence: This means that the solution u
is defined for all x and for t in some neighbourhood of 0
Properties of the
heat equation
Global existence: This means that the solution u is
defined for all x and all t
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Some immediate questions
The Heat Equation
Vipul Naik
Here are natural properties we would seek for the flow
equation:
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Short-time existence: This means that the solution u
is defined for all x and for t in some neighbourhood of 0
Properties of the
heat equation
I
Global existence: This means that the solution u is
defined for all x and all t
The general
concept of flow
and fixed points
I
Uniqueness: This means that any two solutions
(defined in suitable time neighbourhoods) must be
equal wherever they are both defined
I
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Some immediate questions
The Heat Equation
Vipul Naik
Here are natural properties we would seek for the flow
equation:
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Short-time existence: This means that the solution u
is defined for all x and for t in some neighbourhood of 0
Properties of the
heat equation
I
Global existence: This means that the solution u is
defined for all x and all t
The general
concept of flow
and fixed points
I
Uniqueness: This means that any two solutions
(defined in suitable time neighbourhoods) must be
equal wherever they are both defined
I
Short-time existence and uniqueness are typically shown
using the general theory of existence and uniqueness of
solutions to differential equations, while global existence may
require further exploitation of the particular structure of the
differential equation.
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Bidirectionality of the flow
One fundamental way in which the continuous flow differs
from the discrete flow is that it is bidirectional. For the
discrete flow, we had defined:
G (x) = x + F (x)
Now if x and G (x) were fairly close, we could possibly think
of F (x) as being equal to F (G (x)), and get:
G (x) = x + F (G (x))
which would allow us to write:
G −1 (x) = x − F (x)
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Bidirectionality of the flow
One fundamental way in which the continuous flow differs
from the discrete flow is that it is bidirectional. For the
discrete flow, we had defined:
G (x) = x + F (x)
Now if x and G (x) were fairly close, we could possibly think
of F (x) as being equal to F (G (x)), and get:
G (x) = x + F (G (x))
which would allow us to write:
G −1 (x) = x − F (x)
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Making the flow continuous actually helps us rigorize this,
namely, the flow for −F is the reverse of the flow for F .
(this is essentially because in the continuous thing, adjacent
things are close enough).
Maximum
principles
Why the heat
equation matters
The fixed points of the flow
The Heat Equation
Vipul Naik
We know that the fixed points of the flow are precisely the y
that are solutions to F (y ) = 0. We thus have the picture:
I
There will be some fixed points. These are points that
don’t move under the flow at all.
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The fixed points of the flow
The Heat Equation
Vipul Naik
We know that the fixed points of the flow are precisely the y
that are solutions to F (y ) = 0. We thus have the picture:
I
There will be some fixed points. These are points that
don’t move under the flow at all.
I
There will be some paths defined on finite time
intervals, and some paths defined on infinite time
intervals
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The fixed points of the flow
The Heat Equation
Vipul Naik
We know that the fixed points of the flow are precisely the y
that are solutions to F (y ) = 0. We thus have the picture:
I
There will be some fixed points. These are points that
don’t move under the flow at all.
I
There will be some paths defined on finite time
intervals, and some paths defined on infinite time
intervals
I
If a path converges at the limit, then the point of
convergence must be a fixed point under the flow. For
paths defined globally, the limits at ∞ and at −∞ may
both give (possibly distinct) fixed points
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The fixed points of the flow
The Heat Equation
Vipul Naik
We know that the fixed points of the flow are precisely the y
that are solutions to F (y ) = 0. We thus have the picture:
I
There will be some fixed points. These are points that
don’t move under the flow at all.
I
There will be some paths defined on finite time
intervals, and some paths defined on infinite time
intervals
I
I
If a path converges at the limit, then the point of
convergence must be a fixed point under the flow. For
paths defined globally, the limits at ∞ and at −∞ may
both give (possibly distinct) fixed points
There may be some circular flows as well – flows which
keep going round and round
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The flow for the Laplacian
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The heat equation is the flow equation corresponding to the
Laplacian, viz F = ∆ here.
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The flow for the Laplacian
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The heat equation is the flow equation corresponding to the
Laplacian, viz F = ∆ here.
Thus, we can apply all the ideas of flows to studying the
heat equation.
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
A definite directionality
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
In the case of the heat equation, there is a definite
directionality to things. That is, if k = 1, then the heat
equation evolves towards a fixed point as t → ∞, rather
than as t → −∞.
In fact, there are a number of quantities we can associate
with the heat equation that:
I
Are minimal (respectively maximal) when we are at a
fixed point
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
A definite directionality
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
In the case of the heat equation, there is a definite
directionality to things. That is, if k = 1, then the heat
equation evolves towards a fixed point as t → ∞, rather
than as t → −∞.
In fact, there are a number of quantities we can associate
with the heat equation that:
I
Are minimal (respectively maximal) when we are at a
fixed point
I
Decrease (respectively increase) monotonically at a
general point, with the decrease (respectively increase)
becoming steadily zero only once we reach a fixed point
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Possibilities for decreasing/increasing quantities
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
I
A bounding range for the heat function at any given
time. This is related to so-called maximum principles
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Possibilities for decreasing/increasing quantities
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
I
I
A bounding range for the heat function at any given
time. This is related to so-called maximum principles
Functions that measure the average deviation of the
heat function from its mean value. These include things
like entropy functions
Properties of the
heat equation
The general
concept of flow
and fixed points
Flow and
time-dependent vector
field
Picture of the flow
The heat equation as
a flow
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Outline
The Heat Equation
Vipul Naik
Basic properties of the heat equation
Partial derivatives
The heat equation in one dimension
Heat equation in more than one dimension
The obvious properties of the heat equation
Physical intuition behind the heat equation
Heat and its flow
Temperature and two laws relating temperature to heat
Properties of the heat equation
Recapitulation
Obvious structural properties
What improves and what is unchanged
The general concept of flow and fixed points
Flow and time-dependent vector field
Picture of the flow
The heat equation as a flow
Solving the heat equation in one variable
Separation of variables
Variations on the heat equation
The heat equation as a diffusion equation
Gradient and reaction terms
Supersolutions and subsolutions
Maximum principles
Maximum principle for the heat equation
Effect of a reaction term
Why the heat equation matters
In the physical world
In the mathematical world
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Separation of
variables
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
In the one-dimensional case
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
For the function u = u(t, x)
∂2u
∂u
=
∂t
∂x 2
We try to hunt for solutions of the form:
u(t, x) = f (t)g (x)
Such solutions are termed multiplicatively separable.
This technique is termed separation of variables.
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Separation of
variables
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
ODEs formed after we separate variables
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
After we separate the variable, we get the following:
f 0 (t)
g 00 (x)
=
f (t)
g (x)
Since this is true for every t and every x, we can set this to
a value λ, and we obtain:
f 0 (t) = λf (t)
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Separation of
variables
Variations on the
heat equation
And parallelly obtain:
g 00 (t) = λg (t)
Maximum
principles
Why the heat
equation matters
Linear combinations of multiplicatively separable
solutions
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Since the heat equation is linear, any linear combination of
multiplicatively separable solutions is also a solution.
Further, any solution that is the convergent sum of an
infinite series whose terms are all multiplicatively separable
solution.
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Separation of
variables
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Linear combinations of multiplicatively separable
solutions
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Since the heat equation is linear, any linear combination of
multiplicatively separable solutions is also a solution.
Further, any solution that is the convergent sum of an
infinite series whose terms are all multiplicatively separable
solution.
The question then would be: can every solution be obtained
in this form? That is, can the flow corresponding to every
point be obtained as an “infinite linear combination” of
multiplicatively separable solutions?
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Separation of
variables
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Linear combinations of multiplicatively separable
solutions
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Since the heat equation is linear, any linear combination of
multiplicatively separable solutions is also a solution.
Further, any solution that is the convergent sum of an
infinite series whose terms are all multiplicatively separable
solution.
The question then would be: can every solution be obtained
in this form? That is, can the flow corresponding to every
point be obtained as an “infinite linear combination” of
multiplicatively separable solutions?
For the heat equation, the answer in fact turns out to be yes.
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Separation of
variables
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
The more general situation of a flow
The same idea of separation of variables that we used for the
heat equation works in the greater generality of a flow. In
fact, it works well when the F for which we are considering
the differential equation, is a linear differential operator.
Consider:
∂u
= F (u)
∂t
Expressing u(t, x) as f (t)g (x) we obtain the system of
differential equations:
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Separation of
variables
f 0 (t) = λf (t)
F (g )(t) = λg (t)
Thus, g must be an eigensolution for F with eigenvalue λ.
We again have that under suitable assumptions, every
solution is an infinite linear combination of such separable
solutions.
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
Outline
The Heat Equation
Vipul Naik
Basic properties of the heat equation
Partial derivatives
The heat equation in one dimension
Heat equation in more than one dimension
The obvious properties of the heat equation
Physical intuition behind the heat equation
Heat and its flow
Temperature and two laws relating temperature to heat
Properties of the heat equation
Recapitulation
Obvious structural properties
What improves and what is unchanged
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
The general concept of flow and fixed points
Flow and time-dependent vector field
Picture of the flow
The heat equation as a flow
Solving the heat
equation in one
variable
Solving the heat equation in one variable
Separation of variables
Variations on the
heat equation
Variations on the heat equation
The heat equation as a diffusion equation
Gradient and reaction terms
Supersolutions and subsolutions
Maximum principles
Maximum principle for the heat equation
Effect of a reaction term
Why the heat equation matters
In the physical world
In the mathematical world
The heat equation as
a diffusion equation
Gradient and reaction
terms
Supersolutions and
subsolutions
Maximum
principles
Why the heat
equation matters
The more general idea of diffusion
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Instead of thinking of the heat equation in terms of heat
content and temperature, we can view it in terms of an
uneven mass density of a material in a body. If the mass
density is not equal everywhere, there is a tendency for mass
to flow from higher density to lower density, resulting in
greater mass equalization.
We can deduce, using similar reasoning to that for the heat
equation, that if u denotes the mass density at a point, u
satisfies the heat equation.
Hence, the heat equation is also termed the diffusion
equation.
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
The heat equation as
a diffusion equation
Gradient and reaction
terms
Supersolutions and
subsolutions
Maximum
principles
Why the heat
equation matters
Gradient term and reaction term
The Heat Equation
Vipul Naik
The heat equation with gradient term is given by the
following general equation:
ut = ∆u + hX , ∇ui
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
The heat equation as
a diffusion equation
Gradient and reaction
terms
Supersolutions and
subsolutions
Maximum
principles
Why the heat
equation matters
Gradient term and reaction term
The Heat Equation
Vipul Naik
The heat equation with gradient term is given by the
following general equation:
ut = ∆u + hX , ∇ui
The heat equation with both gradient term and scalar term
(also called reaction term) is given by the following general
equation:
∂u
= ∆u + hX , ∇ui + H(u)
∂t
If H(u) = βu we say that we have a heat equation with
linear reaction term.
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
The heat equation as
a diffusion equation
Gradient and reaction
terms
Supersolutions and
subsolutions
Maximum
principles
Why the heat
equation matters
Gradient term and reaction term
The Heat Equation
Vipul Naik
The heat equation with gradient term is given by the
following general equation:
ut = ∆u + hX , ∇ui
The heat equation with both gradient term and scalar term
(also called reaction term) is given by the following general
equation:
∂u
= ∆u + hX , ∇ui + H(u)
∂t
If H(u) = βu we say that we have a heat equation with
linear reaction term.
A general equation of the above setup is termed a
reaction-diffusion equation.
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
The heat equation as
a diffusion equation
Gradient and reaction
terms
Supersolutions and
subsolutions
Maximum
principles
Why the heat
equation matters
Physical significance of the reaction term
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Reaction terms arise in the physical situation as terms that
alter the heat at a point without heat flowing to or from the
neighbouring points. The word “reaction” stems, for
instance, from the fact that when heat is flowing through a
material, too much of it may concentrate at a point, leading
to some chemical change at that point.
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
The heat equation as
a diffusion equation
Gradient and reaction
terms
Supersolutions and
subsolutions
Maximum
principles
Why the heat
equation matters
Physical significance of the reaction term
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Reaction terms arise in the physical situation as terms that
alter the heat at a point without heat flowing to or from the
neighbouring points. The word “reaction” stems, for
instance, from the fact that when heat is flowing through a
material, too much of it may concentrate at a point, leading
to some chemical change at that point.
Note that once there are reaction terms, the total heat of
the system is no longer kept constant.
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
The heat equation as
a diffusion equation
Gradient and reaction
terms
Supersolutions and
subsolutions
Maximum
principles
Why the heat
equation matters
The four kinds of reaction terms
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
I
Positive and positively related reaction terms: These are
reaction terms that are positive in sign and are
positively related to the heat function. This means that
the greater the heat density at a point, the more it rises.
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
The heat equation as
a diffusion equation
Gradient and reaction
terms
Supersolutions and
subsolutions
Maximum
principles
Why the heat
equation matters
The four kinds of reaction terms
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
I
I
Positive and positively related reaction terms: These are
reaction terms that are positive in sign and are
positively related to the heat function. This means that
the greater the heat density at a point, the more it rises.
Positive and negatively related reaction terms: These
are reaction terms that are positive in sign and are
negatively related to the heat function. This means that
the greater the heat density at a point, the less it rises.
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
The heat equation as
a diffusion equation
Gradient and reaction
terms
Supersolutions and
subsolutions
Maximum
principles
Why the heat
equation matters
The four kinds of reaction terms
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
I
I
Positive and positively related reaction terms: These are
reaction terms that are positive in sign and are
positively related to the heat function. This means that
the greater the heat density at a point, the more it rises.
Positive and negatively related reaction terms: These
are reaction terms that are positive in sign and are
negatively related to the heat function. This means that
the greater the heat density at a point, the less it rises.
I
Negative and positively related reaction terms
I
Negative and negatively related reaction terms
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
The heat equation as
a diffusion equation
Gradient and reaction
terms
Supersolutions and
subsolutions
Maximum
principles
Why the heat
equation matters
All these heat equations satisfy the flow model
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The general form of heat equation with both gradient and
reaction term is also a flow equation, corresponding to F
being:
y 7→ ∆u + hX , ∇ui + H(u)
Hence, we can view them in the same way (doing the same
kind of analysis of flows) as we did for the heat equation.
However, if the reaction term has a time-dependence (that
is, it is dependent on t), then the heat equation cannot be
viewed as a flow equation corresponding to a differential
operator.
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
The heat equation as
a diffusion equation
Gradient and reaction
terms
Supersolutions and
subsolutions
Maximum
principles
Why the heat
equation matters
Supersolutions and subsolutions
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
u is termed a supersolution at (t, x) if:
∂u
≥ ∆(u) + hX , ∇ui + F (u)
∂t
u is termed a supersolution if it is a supersolution for all x
and all t
Similarly, u is a subsolution at (t, x) if:
∂u
≤ ∆(u) + hX , ∇ui + F (u)
∂t
u is termed a subsolution if it is a subsolution for all x and
all t
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
The heat equation as
a diffusion equation
Gradient and reaction
terms
Supersolutions and
subsolutions
Maximum
principles
Why the heat
equation matters
Outline
The Heat Equation
Vipul Naik
Basic properties of the heat equation
Partial derivatives
The heat equation in one dimension
Heat equation in more than one dimension
The obvious properties of the heat equation
Physical intuition behind the heat equation
Heat and its flow
Temperature and two laws relating temperature to heat
Properties of the heat equation
Recapitulation
Obvious structural properties
What improves and what is unchanged
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
The general concept of flow and fixed points
Flow and time-dependent vector field
Picture of the flow
The heat equation as a flow
Solving the heat
equation in one
variable
Solving the heat equation in one variable
Separation of variables
Variations on the
heat equation
Variations on the heat equation
The heat equation as a diffusion equation
Gradient and reaction terms
Supersolutions and subsolutions
Maximum
principles
Maximum principles
Maximum principle for the heat equation
Effect of a reaction term
Why the heat equation matters
In the physical world
In the mathematical world
Maximum principle
for the heat equation
Effect of a reaction
term
Why the heat
equation matters
The statement for the heat equation
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The following is the statement for the heat equation:
Let u be a solution to the heat equation with the property
that u(0, x) ∈ [T1 , T2 ] for all x ∈ Rn . Then, the maximum
principle states that:
u(t, x) ∈ [T1 , T2 ] ∀x ∈ Rn , ∀t ≥ 0
In other words, whatever bounds/limits control the heat
function at the initial time, control the heat function at all
later times.
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Maximum principle
for the heat equation
Effect of a reaction
term
Why the heat
equation matters
The statement for the heat equation
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The following is the statement for the heat equation:
Let u be a solution to the heat equation with the property
that u(0, x) ∈ [T1 , T2 ] for all x ∈ Rn . Then, the maximum
principle states that:
u(t, x) ∈ [T1 , T2 ] ∀x ∈ Rn , ∀t ≥ 0
In other words, whatever bounds/limits control the heat
function at the initial time, control the heat function at all
later times.
By the invariance under time-translation, we can say that the
range of u(t, x) for given t, keeps reducing as t increases.
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Maximum principle
for the heat equation
Effect of a reaction
term
Why the heat
equation matters
For a more general reaction-diffusion equation
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
A general reaction-diffusion equation is said to satisfy the
weak maximum principle if for any solution u of that
equation:
u(t, x) ∈ [T1 , T2 ] ∀x ∈ Rn , ∀t ≥ 0
In other words, whatever bounds/limits control the heat
function at the initial time, control the heat function at all
later times.
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Maximum principle
for the heat equation
Effect of a reaction
term
Why the heat
equation matters
A second version of the maximum principle
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Here’s another version of the maximum principle. Consider
the equation:
∂u
= ∆u + hX , ∇ui
∂t
Then suppose we have an α ∈ R satisfying the following two
conditions:
I
u(x, 0) ≥ α
I
u is a supersolution of the heat equation at any
(t, x) ∈ [0, T ) × Rn for which u(t, x) < α.
Then u(t, x) ≥ α for all (t, x) ∈ [0, T ) ×
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Rn .
Maximum
principles
Maximum principle
for the heat equation
Effect of a reaction
term
Why the heat
equation matters
Reaction terms can be disruptive
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Consider a positive and positively related reaction term.
What this does is to push up the heat at various points.
Moreover, those points where the heat density is higher,
tend to get pushed up more. Thus, in this case, the reaction
term actually increases disparity and obstructs the process of
attaining equilibrium.
We want to prove that a maximum principle persists even in
the presence of reaction terms.
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Maximum principle
for the heat equation
Effect of a reaction
term
Why the heat
equation matters
Outline
The Heat Equation
Vipul Naik
Basic properties of the heat equation
Partial derivatives
The heat equation in one dimension
Heat equation in more than one dimension
The obvious properties of the heat equation
Physical intuition behind the heat equation
Heat and its flow
Temperature and two laws relating temperature to heat
Properties of the heat equation
Recapitulation
Obvious structural properties
What improves and what is unchanged
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
The general concept of flow and fixed points
Flow and time-dependent vector field
Picture of the flow
The heat equation as a flow
Solving the heat
equation in one
variable
Solving the heat equation in one variable
Separation of variables
Variations on the
heat equation
Variations on the heat equation
The heat equation as a diffusion equation
Gradient and reaction terms
Supersolutions and subsolutions
Maximum
principles
Maximum principles
Maximum principle for the heat equation
Effect of a reaction term
Why the heat equation matters
In the physical world
In the mathematical world
Why the heat
equation matters
In the physical world
In the mathematical
world
For physical situations
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The heat equation that we studied originally is a prototype
for general reaction-diffusion equations, which are used to
model many physical and chemical situations. Further, many
of the concepts such as energy, entropy, maximum principle
and so on, that we develop for the heat equation, motivate
corresponding notions for more general reaction-diffusion
equations.
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
In the physical world
In the mathematical
world
For physical situations
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
The heat equation that we studied originally is a prototype
for general reaction-diffusion equations, which are used to
model many physical and chemical situations. Further, many
of the concepts such as energy, entropy, maximum principle
and so on, that we develop for the heat equation, motivate
corresponding notions for more general reaction-diffusion
equations.
The heat equation also inspires ideas in more general flow
equations governed by laws qualitatively very different from
those for the heat equation.
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
In the physical world
In the mathematical
world
The leading ideas
The heat equation, which originally started out from the
physical motivation, has helped in the development of a
whole lot of mathematics. To see how, let’s single out the
important aspects of flow equation:
I
We have a differential operator F and are looking at
functions annihilated by F .
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
In the physical world
In the mathematical
world
The leading ideas
The heat equation, which originally started out from the
physical motivation, has helped in the development of a
whole lot of mathematics. To see how, let’s single out the
important aspects of flow equation:
I
I
We have a differential operator F and are looking at
functions annihilated by F .
We consider the flow equation ut = F (u) and consider
the way arbitrary initial solutions evolve under this flow
equation. If we can find some initial solutions for which
the flow converges at ∞, we get functions annihilated
by F .
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
In the physical world
In the mathematical
world
The leading ideas
The heat equation, which originally started out from the
physical motivation, has helped in the development of a
whole lot of mathematics. To see how, let’s single out the
important aspects of flow equation:
I
I
I
We have a differential operator F and are looking at
functions annihilated by F .
We consider the flow equation ut = F (u) and consider
the way arbitrary initial solutions evolve under this flow
equation. If we can find some initial solutions for which
the flow converges at ∞, we get functions annihilated
by F .
We develop certain special techniques, such as the use
of energy, entropy, and maximum principles, to show
that the flow actually comes closer and closer to
something, viz it converges.
The above gives a concrete recipe for trying to locate
functions annihilated by F .
The Heat Equation
Vipul Naik
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
In the physical world
In the mathematical
world
Ricci flows and the Hamilton program
The Heat Equation
Vipul Naik
Hamilton formulated a program to solve the famous
Poincare conjecture using a technique called Ricci flows.
The idea (rough sketch) was:
I
We consider a differential operator on the space of
Riemannian metrics on a differential manifold (the Ricci
curvature tensor), such that we are interested in those
metrics which are annihilated by this operator
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
In the physical world
In the mathematical
world
Ricci flows and the Hamilton program
The Heat Equation
Vipul Naik
Hamilton formulated a program to solve the famous
Poincare conjecture using a technique called Ricci flows.
The idea (rough sketch) was:
I
I
We consider a differential operator on the space of
Riemannian metrics on a differential manifold (the Ricci
curvature tensor), such that we are interested in those
metrics which are annihilated by this operator
We consider the flow equation associated with this
operator – the so-called Ricci flow
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
In the physical world
In the mathematical
world
Ricci flows and the Hamilton program
The Heat Equation
Vipul Naik
Hamilton formulated a program to solve the famous
Poincare conjecture using a technique called Ricci flows.
The idea (rough sketch) was:
I
We consider a differential operator on the space of
Riemannian metrics on a differential manifold (the Ricci
curvature tensor), such that we are interested in those
metrics which are annihilated by this operator
I
We consider the flow equation associated with this
operator – the so-called Ricci flow
I
We then try to find some flows which actually converge
and hence find metrics annihilated by the Ricci
curvature tensor
Basic properties of
the heat equation
Physical intuition
behind the heat
equation
Properties of the
heat equation
The general
concept of flow
and fixed points
Solving the heat
equation in one
variable
Variations on the
heat equation
Maximum
principles
Why the heat
equation matters
In the physical world
In the mathematical
world