Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague joint work with Bum Ja Jin (Muan), A.Novotný (Toulon), and Yongzhong Sun (Nanjing) Benasque, August-September 2011
Incompressible Navier-Stokes system Caffarelli, Kohn, and Nirenberg [1982] div x u = 0 t u + div x (u u) + x p = u Energy inequality p L 3/2 (0, T ) Ω t u 2 + x u 2 u + 2div x (pu) + 2 x u 2 u 2
Compressible Navier-Stokes system t ϱ + div x (ϱu) = 0 t (ϱu) + div x (ϱu u) + x p(ϱ) = div x S( x u) u Ω = 0 S = µ( x u + x u t 2 3 div xui) + ηdiv x ui
Germain [2010]: weak-strong results in the framework of better weak solutions suitable conditions formulated in terms of hypothetical smooth solutions periodic boundary conditions
Finite-energy weak solutions Equation of continuity t ϱ + div x (ϱu) = 0 Momentum equation t (ϱu) + div x (ϱu u) + x p(ϱ) = div x S( x u)
Energy inequality Ω ( 1 τ 2 ϱ u 2 + H(ϱ)) (τ, ) dx + S( x u) : x u dx dt 0 Ω ( ) 1 2 ϱ 0 u 0 2 + H(ϱ 0 ) dx Ω ϱ p(z) H(ϱ) = ϱ 1 z 2 dz
Relative entropy Ω Germain [2010], Berthelin and Vasseur [2005] E(ϱ, r) = H(ϱ) H (r)(ϱ r) H(r) ϱ p(z) H(ϱ) ϱ 1 z 2 dz, P H ( ) 1 2 ϱ u U 2 + H(ϱ) H (r)(ϱ r) H(r) dx
Suitable weak solutions Renormalized equation of continuity ( ) t b(ϱ) + div x (b(ϱ)u) + b (ϱ)ϱ b(ϱ) div x u = 0 Momentum equation t (ϱu) + div x (ϱu u) + x p(ϱ) = div x S( x u)
Generalized energy inequality τ + 0 + Ω Ω Ω τ ( 1 2 ϱ u U 2 + E(ϱ, r)) (τ, ) dx [S( x u) S( x U)] : x (u U) dx dt ( ) 1 2 ϱ 0 u 0 U(0, ) 2 + E(ϱ 0, r(0, )) dx 0 R (ϱ, u, r, U) dt for a.a. τ (0, T ), for any smooth r > 0, U Ω = 0
Remainder term = Ω R (ϱ, u, r, U) ( ( ) ) ϱ t U + u x U (U u) + div x S( x U)(u U) ( ( ) + (r ϱ) t P(r) + x P(r) ru ϱu Ω ( ( ) )) div x U ϱ P(ϱ) P(r) E(ϱ, r) dx dx
Global existence [E.F., A. Novotný, Y. Sun, Indiana Univ. Math. J., to appear] Theorem Let Ω R 3 be a bounded smooth domain. Let the pressure p be a continuously differentiable function satisfying p(0) = 0, p p (ϱ) (ϱ) > 0 for all ϱ > 0, lim ϱ ϱ γ 1 = a > 0 for a certain γ > 3/2. Assume that the initial data satisfy ϱ 0 0, ϱ 0 0, ϱ 0 L γ (Ω), ϱ 0 u 0 2 L 1 (Ω). Then the compressible Navier-Stokes system possesses a suitable weak solution on (0, T ) Ω.
Strong solutions: 0 < ϱ ϱ(t, x) ϱ, ũ(t, x) u (1) x ϱ L 2 (0, T ; L q (Ω; R 3 )), 2 xũ L2 (0, T ; L q (Ω; R 3 3 3 )) (2) q > max{3, 3 γ 1 }.
Theorem Let Ω R 3 be a bounded domain of class C 2+ν, ν > 0. In addition to hypotheses of existence theorem, suppose that p is twice continuously differentiable on the open interval (0, ). Assume that the Navier-Stokes system admits a weak solution ϱ, ũ in (0, T ) Ω belonging to the regularity class specified through (1), (2). Then ϱ ϱ, ũ u, where ϱ, u is the suitable weak solution of the Navier-Stokes system emanating from the same initial data.
Using the result of Sun, Wang and Zhang [2010] we have: Theorem Let Ω R 3 be a bounded domain of class C 2+ν, ν > 0. Let ϱ 0, u 0 be given such that ϱ 0 W 1,6 (Ω), 0 < ϱ ϱ 0 (x) ϱ for all x Ω, u 0 W 2,2 (Ω; R 3 ) W 1,2 0 (Ω; R 3 ). Suppose that the pressure p satisfies the hypotheses of existence theorem, and that µ > 0, η = 0.
Let ϱ, u be a suitable weak solution of the Navier-Stokes system in (0, T ) Ω. If, in addition, ess sup ϱ <, (0,T ) Ω then ϱ, u is the unique (strong) solution of the Navier-Stokes.
Corollary Let Ω and the initial data ϱ 0, u 0 be the same as in the previous theorem. Assume that ϱ, u is a suitable weak solution of the Navier-Stokes system such that ess inf ϱ(τ, x) = 0 for a certain τ (0, T ). x Ω The there exists 0 < τ 0 τ such that lim sup t τ 0 [ess sup ϱ(t, x)] =. x Ω
Theorem Let Ω R 3 be a bounded domain of class C 2+ν, ν > 0. In addition to hypotheses of existence theorem, suppose that p is twice continuously differentiable on the open interval (0, ). Assume that the Navier-Stokes system admits a (strong) solution ϱ, ũ. In addition, let ϱ 0,ε ϱ 0 in L γ (Ω), ϱ 0,ε 0, ϱ 0,ε u 0,ε ũ 0 2 dx 0 as ε 0. Ω
Then and sup ϱ ε (τ, ) ϱ(τ, ) L γ (Ω) 0 τ [0,T ] sup ϱ ε u ε (τ, ) ϱũ(τ, ) L 1 (Ω;R 3 )) 0, τ [0,T ] u ε ũ in L 2 (0, T ; W 1,2 0 (Ω; R 3 )), where ϱ ε, u ε is a suitable weak solution of the Navier-Stokes system emanating from the initial data ϱ 0,ε, u 0,ε.
Finite energy solutions are suitable [E.F., A.Novotný, Bum Ja Jin, Preprint 2011] Finite energy weak solutions are suitable weak solutions general domains (bounded, unbounded, irregular) general boundary conditions (no-slip, complete slip, Navier s slip) general driving force