Relative energy functionals in fluid
dynamics
Eduard Feireisl
Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague
AICES Seminar, Aachen, 9 November 2015
The research leading to these results has received funding from the European Research Council
under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement 320078
Eduard Feireisl
Relative energy/entropy
Compressible barotropic fluid flow
Equation of continuity - mass conservation
∂t % + divx (%u) = 0
Momentum equation - Newton’s second law
∂t (%u) + divx (%u ⊗ u) + ∇x p(%) = divx S(∇x u)
2
S(∇x u) = µ ∇x u + ∇tx u − divx uI + ηdivx uI
3
u|∂Ω = 0
Energy inequality
Z
τ
Z
S(∇x u) : ∇x u dx dt = E (0)
E (τ ) +
0
Z
E=
Ω
Ω
1
%|u|2 + P(%) dx, P(%) = %
2
Z
1
%
p(z)
dz
z2
Weak formulation
Field equations
Z
t=t2 Z
%ϕ dx
=
Ω
t2
Z
[%∂t ϕ + %u · ∇x ϕ] dx dt
t1
t=t1
Ω
t=t2
Z
%u · ϕ dx
Ω
Z
t2
t=t1
Z
[%u · ∂t ϕ + %u ⊗ u : ∇x ϕ + p(%)divx ϕ] dx dt
=
t1
Ω
Z
t2
Z
S(∇x u) : ∇x ϕ dx dt
−
t1
Ω
Energy inequality
Z
Ω
1
%|u|2 + P(%) dx
2
t=τ
Z
τ
Z
S(∇x u) : ∇x u dx dt ≤ 0
+
t=0
0
Ω
Relative energy/entropy
Lyapunov function
Z 1
%|u − 0|2 + P(%) − P 0 (%)(% − %) − P(%) dx
E=
Ω 2
Coercivity of the pressure potential
% 7→ p(%) non-decreasing ⇒ % 7→ P(%) convex
Relative energy (relative entropy Dafermos [1979] )
E %, u r , U
Z 1
=
%|u − U|2 + P(%) − P 0 (r )(% − r ) − P(r ) dx
Ω 2
Relative energy inequality
Weak formulation
h it=τ
E %, u r , U
t=0
t=τ Z
t=τ
1
2
%|u| + P(%) dx
−
%u · U dx
=
Ω
Ω 2
t=0
t=0
Z
t=τ
1
+
%|U|2 dx
Ω 2
t=0
Z
t=τ Z
t=τ
−
P 0 (r )% dx
+
[P 0 (r )r − P(r )] dx
Z Ω
Ω
t=0
t=0
Constraints imposed on the test functions
r >0
U|∂Ω = 0
Eduard Feireisl
Relative energy/entropy
Dissipative solutions
Relative energy inequality
h it=τ
E %, u r , U
t=0
Z τZ
+
(S(∇x u) − S(∇x U)) : (∇x u − ∇x U) dx dt
0
Ω
Z
≤
0
τ
R %, u r , U dt
Test functions
r > 0, U|∂Ω = 0 (or other relevant b.c.)
Eduard Feireisl
Relative energy/entropy
Remainder
Z
0
τ
R %, u r , U dt
Z
% (∂t U + u · ∇x U) · (U − u) dx
Ω
Z
Z
S(∇x U) : (∇x U − ∇x u) dx +
+
Ω
(p(r ) − p(%)) divx U dx
Ω
Z
+
Ω
[(r − %)∂t P 0 (r ) + ∇x P 0 (r ) · (r U − %u)] dx
Results for dissipative solutions
Positive results
Weak solutions exist for p(%) ≈ a%γ , γ > N2 (P.L.Lions [1998]
certain γ, EF, A. Novotný, H. Petzeltová [2000])
Any weak solution is a dissipative solution (EF, B.J. Jin, A.
Novotný, Y. Sun [2014])
Eduard Feireisl
Relative energy/entropy
Relative energy for complete fluids
Relative energy
E %, ϑ, u r , Θ, U
Z =
Ω
∂HΘ (r , Θ)
1
%|u − U|2 + HΘ (%, ϑ) −
(% − r ) − HΘ (r , Θ) dx
2
∂%
Ballistic free energy
HΘ (%, ϑ) = % (e(%, ϑ) − Θs(%, ϑ))
Hypothesis of thermodynamic stability
∂p(%, ϑ)
∂e(%, ϑ)
> 0,
>0
∂%
∂ϑ
Weak-strong uniqueness, conditional regularity
Results (EF, B.J. Jin, A.Novotný, Y.Sun [2014])
Weak and strong solutions of the barotropic Navier-Stokes
system emanating from the same initial data coincide as long as
the latter exists
Weak solution with bounded density component emanating
from smooth initial data is smooth
Eduard Feireisl
Relative energy/entropy
Singular limits
Rotating fluids
∂t % + divx (%u) = 0
1
1
∂t (%u) + divx (%u ⊗ u) + %b × u + 2M ∇x p(%)
ε
ε
1
= εR divx S(∇x u) + 2F ∇x G
ε
Path dependent singular limit
ε → 0, certain relation between M, R, F > 0
low Mach ⇒ compressible → incompressible
high Rossby ⇒ 3D → 2D
high Reynolds ⇒ viscous → inviscid
Convergence to singular limit system
Target problem - Euler system
∂t v + v · ∇x v + ∇x Π = 0, divx v = 0, x ∈ R 2
Convergence results (joint work with A.Novotný, Y.Lu [2014])
Spatial geometry - infinite strip:
Ω = R 2 × (0, π)
Complete slip (Navier) boundary conditions:
u · n|∂Ω = 0, (S(∇x u) · n) × n|∂Ω = 0
Limits on domains with variable geometry
Channel like domains
n
o
2
Ωε = (x, z) z ∈ (0, 1), |x − εX(z)| < ε2 R 2 (z) , |X(z)| < R(z)
Boundary conditions
u · n|Σ = 0, (S(∇x u) · n) × n|Σ = 0
Σ = ∂Ω ∩ {z ∈ (0, 1)}
u|z=0,1 = 0
Target systems
Inviscid limit
∂t (%E A) + ∂z (%E uE A) = 0
∂t (%E uE A) + ∂z (%E uE2 A) + A∂z p(%E ) = 0
Viscous limit
∂t (%NS A) + ∂z (%NS uNS A) = 0
2
∂t (%NS uNS A) + ∂z (%NS uNS
A) + A∂z p(%NS )
= Aν∂z2 uNS + ν∂z (∂z log(A)uNS ) , ν =
A = R2
4
µ+η >0
3
Convergence
Korn-Poincaré inequality
Z
Z
2
|v| dx ≤ cKP
Ωε
∇x v + ∇tx v2 dx
Ωε
Convergence (joint work with P.Bella, M.Lewicka, A.Novotný
[2015])
Convergence to the target Euler system with geometric terms in
the inviscid limit
Convergence to the Navier-Stokes system in the viscous limit
provided the bulk viscosity in the primitive system is positive
Equations driven by stochastic forces
Navier–Stokes system with stochastic forcing
d% + divx (%u) dt = 0
d(%u)+[divx (%u ⊗ u) + ∇x p(%)] dt = divx S(∇x u) dt+G(%, %u)dW ,
White–noise forcing
G(%, %u) dW =
X
Gk (%, %u) dWk .
k≥1
Eduard Feireisl
Relative energy/entropy
Relative energy inequality
Relative energy inequality - joint result with D.Breit and
M.Hofmanová [2015]
h it=τ
E %, ur , U
t=0
Z τZ
+
(S(∇u) − S(∇x U) : (∇x u − ∇x U) dx dt
0
Ω
≤
t=τ
[MRE ]t=0
Z
+
0
τ
R %, ur , U dt
Test functions
d r = Dtd r dt + Dts r dW , dU = Dtd U dt + Dts U dW
Stochastic remainder
Remainder
R %, ur , U
Z
Z =
S(∇x U) : (∇x U − ∇x u) dx + % Dtd U + u · ∇x U (U − u) dx
Ω
Ω
+
Z (r − %)H 00 (r )Dtd r + ∇x H 0 (r )(r U − %u) dx
Ω
Z
−
Z
G (%, %u)
2
k
%
− Dts Uk dx
%
Ω
Ω
k≥1
Z
Z
X
1
1X
000
s
2
+
%H (r )|Dt rk | dx +
p 00 (r )|Dts rk |2 dx
2
2
Ω
Ω
divx U(p(%) − p(r )) dx +
k≥1
1X
2
k≥1
Eduard Feireisl
Relative energy/entropy
Results for stochastic Navier-Stokes system
Weak–strong uniqueness (with D.Breit, M.Hofmanová [2015])
Pathwise weak-strong uniqueness
Weak-strong uniqueness in law
Inviscid–incompressible limit in the stochastic setting (with
D.Breit, M.Hofmanová [2015])
Convergence to the limit stochastic Euler system for vanishing
viscosity and the Mach number. Results for well-prepared data.
Eduard Feireisl
Relative energy/entropy
Possible extensions
Numerical analysis (T. Gallouet, R. Herbin, D. Maltese,
A.Novotný [2014])
Relative energy inequality for the numerical scheme proposed by
K.Karlsen and T. Karper. Error estimates.
Measure–valued solutions
Weak–strong uniqueness for measure-valued solutions - work in
progress with P. Gwiazda, A. Swierczewska-Gwiazda, E.Wiedemann