Introduction
A general theory
Existence theory for non linear transport
equations
P-E Jabin
In collaboration with : F. Ben Belgacem
The new idea
Introduction
A general theory
The new idea
What are non linear transport equations?
Classical, linear transport equation reads
∂t n(t, x) + div (a(t, x) n(t, x)) = 0,
where the velocity field a is either given or is related to n through
another equation.
Recently new models were introduced in various settings (traffic
flow for cars or pedestrian, movement of bacteria/cells...) taking
local non linear effects into account
∂t n(t, x) + div (a(t, x) f (n(t, x))) = 0,
t ∈ R+ , x ∈ Rd
The function f is given and typically decreases as the density
increases.
Introduction
A general theory
The new idea
What are non linear transport equations?
Classical, linear transport equation reads
∂t n(t, x) + div (a(t, x) n(t, x)) = 0,
Recently new models were introduced in various settings (traffic
flow for cars or pedestrian, movement of bacteria/cells...) taking
local non linear effects into account
∂t n(t, x) + div (a(t, x) f (n(t, x))) = 0,
t ∈ R+ , x ∈ Rd
As before the field a is given or related to n, for instance
u = −∇φ,
−∆φ = g (n),
again for a non linear function g of n. See Topaz-Bertozzi,
Burger, Dolak, Schmeiser and Dalibard-Perthame.
Introduction
A general theory
The new idea
What are non linear transport equations?
Classical, linear transport equation reads
∂t n(t, x) + div (a(t, x) n(t, x)) = 0,
Recently new models were introduced in various settings (traffic
flow for cars or pedestrian, movement of bacteria/cells...) taking
local non linear effects into account
∂t n(t, x) + div (a(t, x) f (n(t, x))) = 0,
t ∈ R+ , x ∈ Rd
As before the field a is given or related to n, for instance
u = −∇φ,
|φ|2 − ∆φ = g (n),
still for a non linear function g of n. See DiFrancesco, Markowich,
Pietschmann.
Introduction
A general theory
The new idea
What are non linear transport equations?
Classical, linear transport equation reads
∂t n(t, x) + div (a(t, x) n(t, x)) = 0,
Recently new models were introduced in various settings (traffic
flow for cars or pedestrian, movement of bacteria/cells...) taking
local non linear effects into account
∂t n(t, x) + div (a(t, x) f (n(t, x))) = 0,
t ∈ R+ , x ∈ Rd
Many other models of the same type have been derived, see
Goatin, Maury, Rascle...
Introduction
A general theory
The new idea
Existence theory
Take for example a vanishing viscosity approximation
∂t nε (t, x) + div (aε (t, x) f (nε (t, x))) − ε2 ∆x nε = 0,
nε (t = 0, x) = nε0 (x),
x ∈ Rd .
With aε given or coupled to nε , can we pass to the limit in this
equation to obtain a weak solution?
Introduction
A general theory
The new idea
Existence theory
Take for example a vanishing viscosity approximation
∂t nε (t, x) + div (aε (t, x) f (nε (t, x))) − ε2 ∆x nε = 0,
nε (t = 0, x) = nε0 (x),
x ∈ Rd .
Easy a priori estimates
• By the maximum principle, in general all Lp bounds are
propagated
knε (t, .)kLp (Rd ) ≤ Ct knε0 (.)kLp (Rd ) ,
nε (t, .) ≥ 0 if n0 ≥ 0.
Introduction
A general theory
The new idea
Existence theory
Take for example a vanishing viscosity approximation
∂t nε (t, x) + div (aε (t, x) f (nε (t, x))) − ε2 ∆x nε = 0,
nε (t = 0, x) = nε0 (x),
x ∈ Rd .
Easy a priori estimates
• By the maximum principle, in general all Lp bounds are
propagated
knε (t, .)kLp (Rd ) ≤ Ct knε0 (.)kLp (Rd ) ,
nε (t, .) ≥ 0 if n0 ≥ 0.
• Taking for example aε = −∇φε with −δφε = g (n), one has
compactness on aε
kdiv aε kL∞ ≤ Cg knε kL∞ ,
kaε kW 1,p ≤ Cg knε kLp
1 < p < ∞.
Introduction
A general theory
The new idea
Existence theory
Take for example a vanishing viscosity approximation
∂t nε (t, x) + div (aε (t, x) f (nε (t, x))) − ε2 ∆x nε = 0,
nε (t = 0, x) = nε0 (x),
x ∈ Rd .
Easy a priori estimates
• By the maximum principle, in general all Lp bounds are
propagated
knε (t, .)kLp (Rd ) ≤ Ct knε0 (.)kLp (Rd ) ,
How to obtain compactness on nε ?
nε (t, .) ≥ 0 if n0 ≥ 0.
Introduction
A general theory
The new idea
State of the art
Several existence results are already available in restricted settings.
• Existence (even of strong solutions) for short times is
relatively easy. It is only valid up to the first shock.
• The 1-d case can usually be solved with compensated
compactness or other regularizing effects.
• For some very precise couplings like
∂t n(t, x) − div (∇φ(t, x) f (n(t, x))) = 0,
−∆φ = n,
gradient flows techniques can be used (see Dolbeaut, Maury,
Santambroggio).
• For the case
∂t n(t, x) − div (∇φ(t, x) f (n(t, x))) = 0,
−∆φ = g (n),
compactness has been proved by Dalibard-Perthame using
the rigidity in the kinetic formulation.
Introduction
A general theory
The new idea
General assumptions
Let us summarize the general framework we would like to work in.
Consider for nε0 compact and uniformly bounded in L1 ∩, L∞ a
sequence of solutions to
∂t nε (t, x) + div (aε (t, x) f (nε (t, x))) − ε2 ∆x nε = 0,
nε (t = 0, x) = nε0 (x),
x ∈ Rd .
Assume
∃ p > 1,
sup sup kaε (t, .)kW 1,p (Rd ) < ∞,
ε t∈[0, T ]
sup kdivx aε kL∞ ([0,
ε
And
T ]×Rd )
< ∞.


 divx aε = dε + rε with dε compact and
∃C > 0, s.t. ∀ε > 0, ∀x, y ,


|rε (x) − rε (y )| ≤ C |nε (t, x) − nε (t, y )|.
Introduction
A general theory
The new idea
The main result
Theorem
Under the previous assumptions, the sequence nε is compact in
L1loc and if aε → a, it converges to the entropy solution to
∂t n(t, x) + div (a(t, x) f (n(t, x))) = 0,
n(t = 0, x) = lim nε0 (x),
x ∈ Rd .
Entropy solution means the usual: For any χ convex and smooth,
there exists η s.t.
∂t χ(n(t, x)) + div (a(t, x) η(n(t, x))) ≤ 0.
Introduction
A general theory
The new idea
A transparent example
The case where a(t, x) = a(x) with div a = 0 is straightforward.
Define the flow
∂t X (t, x) = a(X (t, x)),
X (0, x) = x,
and introduce
g (t, s, x) = n(t, X (s, x)).
Then g solves the 1 − d scalar conservation law
∂t g + ∂s (f (g )) = 0.
Compactness follows from the usual theory for linear transport
equations and scalar conservation laws (with some complication
with the Laplacian).
Introduction
A general theory
The new idea
The difficulty in the general case
The problem is that the usual methods to obtain compactness
either for linear transport equations or SCL are not compatible.
For instance, for scalar conservation laws,
• Compensated compactness or other regularizing effects require
a non degeneracy assumption.
Introduction
A general theory
The new idea
The difficulty in the general case
The problem is that the usual methods to obtain compactness
either for linear transport equations or SCL are not compatible.
For instance, for scalar conservation laws,
• Compensated compactness or other regularizing effects require
a non degeneracy assumption. Here, except for d = 1, the
equation is always degenerate as the derivative is taken in the
direction of a.
Introduction
A general theory
The new idea
The difficulty in the general case
The problem is that the usual methods to obtain compactness
either for linear transport equations or SCL are not compatible.
For instance, for scalar conservation laws,
• Compensated compactness or other regularizing effects require
a non degeneracy assumption.
• The propagation of BV bounds is false for linear transport
equations.
Introduction
A general theory
The new idea
The difficulty in the general case
The problem is that the usual methods to obtain compactness
either for linear transport equations or SCL are not compatible.
For instance, for scalar conservation laws,
• Compensated compactness or other regularizing effects require
a non degeneracy assumption.
• The propagation of BV bounds is false for linear transport
equations.
• In general using L1 contraction does not work because
n(t, x + h) is not a solution if n is.
Introduction
A general theory
The new idea
The difficulty in the general case
The problem is that the usual methods to obtain compactness
either for linear transport equations or SCL are not compatible.
For linear transport equations
• DiPerna-Lions renormalized solutions require to pass to the
limit and use the uniqueness of the limit. Here we do not
know how to pass to the limit...
Introduction
A general theory
The new idea
The difficulty in the general case
The problem is that the usual methods to obtain compactness
either for linear transport equations or SCL are not compatible.
For linear transport equations
• DiPerna-Lions renormalized solutions require to pass to the
limit and use the uniqueness of the limit. Here we do not
know how to pass to the limit...
• Methods based on the characteristics (Crippa-DeLellis,
Hauray-LeBris-Lions) do not work well with shocks.
Introduction
A general theory
The new idea
A new norm
Define
knkph,p =
Z
R2d
|n(x) − n(y )|p
I
dx dy .
(h + |x − y |)d |x−y |≤1
This is a sort of extension of usual Sobolev or Besov norms to the
case of 0 derivative. Recall for example that for s > 0
Z
|n(x) − n(y )|2
2
knkḢ s =
dx dy .
d+2s
R2d |x − y |
Introduction
A general theory
The new idea
A new norm
Define
knkph,p
Z
=
R2d
|n(x) − n(y )|p
I
dx dy .
(h + |x − y |)d |x−y |≤1
This is a sort of extension of usual Sobolev or Besov norms to the
case of 0 derivative. Recall for example that for s > 0
Z
|n(x) − n(y )|2
knk2Ḣ s =
dx dy .
d+2s
R2d |x − y |
However the previous equality fails for s = 0 and the
corresponding norm is stronger than L2 and is enough to control
the high frequencies.
Introduction
A general theory
The new idea
A new norm
Define
knkph,p
Z
=
R2d
|n(x) − n(y )|p
I
dx dy .
(h + |x − y |)d |x−y |≤1
Proposition
Assume that for a sequence nk uniformly bounded in Lp
lim sup | log h|−1 knk kph,p −→ 0
k
then the sequence nk is compact in Lploc .
as h → 0,
Introduction
A general theory
The new idea
A quantitative estimate
It is possible to obtain explicit bounds on knε kh,1 for the problem
we consider.
Proposition
Under all previous assumptions and in particular aε ∈ W 1,p
uniformly, one has for a constant C independent of ε and h
ε2
h2
0
+ C knε kh,1 + C kdε kh,1 ,
knε kh,1 ≤C | log h|1/p̄ + C
with p̄ = min(2, p).
Introduction
A general theory
The new idea
Conclusion and perspectives
A norm was introduced, critical for transport equations and scalar
conservation laws, implying the propagation of compactness.
Introduction
A general theory
The new idea
Conclusion and perspectives
A norm was introduced, critical for transport equations and scalar
conservation laws, implying the propagation of compactness.
Many unsolved questions remain
• Technical improvements are probably possible: the rate is
likely not optimal, the case where aε ∈ BV (W 1,1 can be
obtained by interpolation).
Introduction
A general theory
The new idea
Conclusion and perspectives
A norm was introduced, critical for transport equations and scalar
conservation laws, implying the propagation of compactness.
Many unsolved questions remain
• Technical improvements are probably possible: the rate is
likely not optimal, the case where aε ∈ BV (W 1,1 can be
obtained by interpolation).
• Developing good numerical schemes for those models is
crucial.
Introduction
A general theory
The new idea
Conclusion and perspectives
A norm was introduced, critical for transport equations and scalar
conservation laws, implying the propagation of compactness.
Many unsolved questions remain
• Technical improvements are probably possible: the rate is
likely not optimal, the case where aε ∈ BV (W 1,1 can be
obtained by interpolation).
• Developing good numerical schemes for those models is
crucial.
• The condition div aε ∈ L∞ is in principle not necessary for
some models. The non linearity lets us keep L∞ bounds on
the density nε .
Introduction
A general theory
The new idea
Conclusion and perspectives
A norm was introduced, critical for transport equations and scalar
conservation laws, implying the propagation of compactness.