Boundary layers for Navier-Stokes equations with slip boundary conditions

Transcription

1 Boundary layers for Navier-Stokes equations with slip boundary conditions Matthew Paddick Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie (Paris 6) Séminaire EDP, Université Paris-Est Créteil 16 th April / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

2 1 Stability and instability in the 2D incompressible model 2 2 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

3 1 Stability and instability in the 2D incompressible model 2 3 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

4 2D incompressible Navier-Stokes on the half-space (x, y) Ω = R ]0, + [: t u ε + u ε u ε ε u ε + p ε = 0 div u ε = 0 (1) u ε t=0 = u0 ε, modelling the motion of a viscous, homogeneous fluid. Usual boundary conditions (BC) for macroscopic flows: impermeability of the boundary u 2 y=0 = 0, and the fluid does not slip along the boundary - Dirichlet BC u y=0 = 0. 4 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

5 Nondimensional viscosity ε = 1/Re, Re the Reynolds number. Inviscid limit problem: behaviour of the solutions u ε relative to v, solution of the incompressible Euler equation t v + v v + q = 0 div v = 0 (2) v t=0 = v 0 when ε goes to 0 high-reynolds approximation. The Euler equation is of order one, so only one BC is required: v 2 y=0 = 0. 5 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

6 As Navier-Stokes solutions can converge to Euler solutions in the whole space, a Prandtl boundary layer expansion 1 is proposed: ( ) u ε y (t, x, y) v(t, x, y) + V t, x, (3) ε 0 ε Smithsonian National Air and Space Museum 1 Prandtl, Verhand. III Intern. Math. Kongresses Heidelberg, / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

7 As Navier-Stokes solutions can converge to Euler solutions in the whole space, a Prandtl boundary layer expansion 1 is proposed: ( ) u ε y (t, x, y) v(t, x, y) + V t, x, (3) ε 0 ε Smithsonian National Air and Space Museum Boundary layer stability in a normed function space (E, ) if sup u ε (t) v(t) ε 0 0. t [0,T ] 1 Prandtl, Verhand. III Intern. Math. Kongresses Heidelberg, / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

8 Issues with the inviscid limit problem with Dirichlet BCs: if u = (u 1, u 2 )(t, x, y) = (ũ, εṽ)(t, x, y = ε 1/2 y), the boundary layer is governed by the ill-posed 2 Prandtl equation: t ũ + ũ x ũ + ṽ y ũ y y ũ = ( tv + v v) 1 y=0 x ũ + y ṽ = 0 (ũ, ṽ) y=0 = 0 lim y + ũ = v 1 y=0 ; 2 Gérard-Varet & Dormy, J. Amer. Math. Soc., Grenier, Comm. Pure Appl. Math., Kato, Seminar on nonlinear PDEs, / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

9 Issues with the inviscid limit problem with Dirichlet BCs: if u = (u 1, u 2 )(t, x, y) = (ũ, εṽ)(t, x, y = ε 1/2 y), the boundary layer is governed by the ill-posed 2 Prandtl equation: t ũ + ũ x ũ + ṽ y ũ y y ũ = ( tv + v v) 1 y=0 x ũ + y ṽ = 0 (ũ, ṽ) y=0 = 0 lim y + ũ = v 1 y=0 ; even if Prandtl is well-posed, ansatz (3) can be invalid due to creation of vorticity inside the boundary layer 3, and control of the vorticity (turbulence) near the boundary is crucial 4. 2 Gérard-Varet & Dormy, J. Amer. Math. Soc., Grenier, Comm. Pure Appl. Math., Kato, Seminar on nonlinear PDEs, / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

10 Our boundary condition: the Navier, or slip, boundary condition 5, a mixed BC that allows the fluid to slip along the boundary at a speed proportional to the normal stress: u 2 y=0 = 0 and y u ε 1 y=0 = 2a ε u ε 1 y=0 (4) The slip phenomenon is observed at microscopic levels, for instance in capillary blood vessels. 5 Navier, Mém. Acad. Roy. Sci. Inst. France, / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

11 Let a ε = a/ε β for a, β R. β > 0: approximation of Dirichlet BCs as ε 0, transition between the stable β = 0 case and the unstable vorticity in the Dirichlet (a ε = ) case. With a > 0 and β = 0: stability in L 2, using energy estimates 6. 6 Iftimie & Planas, Nonlinearity, / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

12 Let a ε = a/ε β for a, β R. β > 0: approximation of Dirichlet BCs as ε 0, transition between the stable β = 0 case and the unstable vorticity in the Dirichlet (a ε = ) case. With a > 0 and β = 0: stability in L 2, using energy estimates 6. We extend this result: Theorem 1.1 (Diff. Integral Eqns. 2014) Let u ε 0 be a converging sequence in L2 (Ω), whose limit v 0 is in H s (Ω) for some s > 2. Then, for every time T > 0, the sequence of global weak solutions to the Navier-Stokes equation u ε with initial condition u ε 0 and Navier boundary condition as above, converges in L ([0, T ], L 2 (Ω)) towards the unique strong solution v of the Euler equation with initial condition v 0 if either a > 0 and β < 1, or a < 0 and β 1/2. 6 Iftimie & Planas, Nonlinearity, / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

13 Further extensions. The theorem can be extended to higher dimensions by starting with the Leray energy inequality. 7 Sueur, J. Math. Fluid Mech., / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

14 Further extensions. The theorem can be extended to higher dimensions by starting with the Leray energy inequality. The first point of the theorem (with a 0) has been extended to the isentropic compressible model by Sueur 7. 7 Sueur, J. Math. Fluid Mech., / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

15 1 Stability and instability in the 2D incompressible model 2 11 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

16 Set a > 0 and β = 1/2: (4) is written y u 1 y=0 = 2a ε u 1 y=0. (5) 8 Grenier, Comm. Pure Appl. Math., / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

17 Set a > 0 and β = 1/2: (4) is written y u 1 y=0 = 2a ε u 1 y=0. (5) Shear flow: vector field of the form u sh (y) = (u sh (y), 0); stationary solution to the incompressible Euler equation (with constant pressure). It is linearly unstable if there exist λ C with R(λ) > 0, k R and ψ H 1 0 (R+ ) such that u(t, x, y) = e λt (e ikx ψ(y)) solves the Euler equation linearised around u sh : t u + u u sh + u sh u + p = 0 div u = 0 u 2 y=0 = 0, For the Euler equation, linear instability implies nonlinear instability 8. (6) 8 Grenier, Comm. Pure Appl. Math., / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

18 Theorem 1.2 (Diff. Integral Eqns. 2014) Let Y = y/ ε. We consider a family of boundary-layer scale initial conditions written as u sh (Y ), with u sh a linearly unstable shear profile for the Euler equation. We generate a reference solution to the Navier-Stokes equation as follows: t u sh (t, Y ) YY 2 u sh(t, Y ) = 0 u sh (0, Y ) = u sh (Y ) Y u sh (t, 0) = 2au sh (t, 0) 13 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

19 Theorem 1.2 (Diff. Integral Eqns. 2014) Let Y = y/ ε. We consider a family of boundary-layer scale initial conditions written as u sh (Y ), with u sh a linearly unstable shear profile for the Euler equation. We generate a reference solution to the Navier-Stokes equation as follows: t u sh (t, Y ) YY 2 u sh(t, Y ) = 0 u sh (0, Y ) = u sh (Y ) Y u sh (t, 0) = 2au sh (t, 0) Then, for any n N, there exist δ 0 > 0 and ε 0 > 0 such that, for every 0 < ε < ε 0, there exist initial data u0 ε such that u ε 0(x, y) u sh (Y ) H s (Ω) Cεn for some s > 0, which generates a solution to the Navier-Stokes equation with boundary condition (5) that satisfies, at a time T ε n ε ln(ε), u ε (T ε, x, y) u sh (T ε, Y ) L (Ω) δ / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

20 Main ideas of the proof. After rescaling the equation (t, x, y) = ε 1/2 (t, x, y), construct an approximate solution to the Navier-Stokes equation written as: u ap (t, x, y) = u sh ( εt, y) + N ε jn Uj i (t, x, y) + ε jn+1/4 Uj b (t, x, y/ε 1/4 ). j=1 (7) Note the ε 1/4 amplitude of the boundary layer terms, hinted by the asymptotic expansion for fixed Navier BCs 9. It will allow to bound u ap independently of ε. Each Uj i and Uj b are built using another asymptotic expansion in powers of ε 1/8, e.g. U1 i = 8n 1 m=0 εm/8 um. i 9 Iftimie & Sueur, Arch. Ration. Mech. Anal., / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

21 The first term u0 i is chosen as a wavepacket of unstable solutions of (6): u0(t, i x, y) = ϕ 0 (k)e λ(k)t+ikx v 0 (k, y) dk (8) R with ϕ 0 C0 (R) localising the eigenmodes around the most unstable wavenumber. 15 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

22 The first term u0 i is chosen as a wavepacket of unstable solutions of (6): u0(t, i x, y) = ϕ 0 (k)e λ(k)t+ikx v 0 (k, y) dk (8) R with ϕ 0 C0 (R) localising the eigenmodes around the most unstable wavenumber. Denote σ 0 = max{rλ(k) k R} the most unstable eigenvalue, and let g ε (t) = ε n (1 + t) 1/2 e σ0t. Time of study: t T0 ε τ such that g ε (T0 ε ) = 1, and τ > 0 independent of ε. 15 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

23 The first term of u ap carries the growth we are after: u i 0 (t) L 2 (Ω A (t)) C 1(1 + t) 1/4 g ε (t), for Ω A (t) Ω of measure 1 + t. 16 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

24 The first term of u ap carries the growth we are after: u i 0 (t) L 2 (Ω A (t)) C 1(1 + t) 1/4 g ε (t), for Ω A (t) Ω of measure 1 + t. By induction, we then show that ε jn (Uj i (t) + ε 1/4 Uj b (t)) C j (1 + t) 1/4 (g ε (t)) j. L 2 (Ω) This is done by noticing that the interior terms Uj i solve (6) with a source term F j satisfying F j (t) Hs (Ω) C j e γt (1 + t) α with γ > σ 0 and α > 0. A resolvent estimate for the linearised equation then yields that U i j also satisfies this inequality. 16 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

25 Now we examine w = u ε u ap. Choosing N so that Nσ 0 > u ap L ([0,T0 ε τ] Ω) + 1/2, a quick energy estimate on the equation satisfied by w yields w(t) L 2 (Ω) C N+1(1 + t) 1/4 (g ε (t)) N / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

26 Now we examine w = u ε u ap. Choosing N so that Nσ 0 > u ap L ([0,T0 ε τ] Ω) + 1/2, a quick energy estimate on the equation satisfied by w yields w(t) L 2 (Ω) C N+1(1 + t) 1/4 (g ε (t)) N+1. As a result, u ε (t) u sh ( εt) N+1 L (1 + t) 1/4 C 2 (Ω A (t)) 1g ε (t) C j (g ε (t)) j j=2 17 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

27 Now we examine w = u ε u ap. Choosing N so that Nσ 0 > u ap L ([0,T0 ε τ] Ω) + 1/2, a quick energy estimate on the equation satisfied by w yields w(t) L 2 (Ω) C N+1(1 + t) 1/4 (g ε (t)) N+1. As a result, u ε (t) u sh ( εt) N+1 L (1 + t) 1/4 C 2 (Ω A (t)) 1g ε (t) C j (g ε (t)) j (1 + t) 1/4 (C 1/2)g ε (t) for t T ε 0 τ, with τ > τ independent of ε. j=2 17 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

28 Now we examine w = u ε u ap. Choosing N so that Nσ 0 > u ap L ([0,T0 ε τ] Ω) + 1/2, a quick energy estimate on the equation satisfied by w yields w(t) L 2 (Ω) C N+1(1 + t) 1/4 (g ε (t)) N+1. As a result, u ε (t) u sh ( εt) N+1 L (1 + t) 1/4 C 2 (Ω A (t)) 1g ε (t) C j (g ε (t)) j (1 + t) 1/4 δ 0 for t = T ε 0 T, with T > τ chosen independently of ε. j=2 18 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

29 Comparison with the Dirichlet boundary condition. WKB expansion based on the Prandtl ansatz (3): O(1) boundary layers. The study time T ε 0 is shorter. One misses instability in L, ε 1/4 δ 0 replaces δ 0 in the final equality, but we do get instability of the vorticity. 19 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

30 Robustness of the method. This linear instability implies nonlinear instability argument requires: a most unstable eigenmode for the linearised system, a resolvent estimate for the linearised system, and good energy estimates for the equation solved by the difference between the exact and approximate solutions. 10 Rousset & Tzvetkov, multiple references 11 arxiv / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

31 Robustness of the method. This linear instability implies nonlinear instability argument requires: a most unstable eigenmode for the linearised system, a resolvent estimate for the linearised system, and good energy estimates for the equation solved by the difference between the exact and approximate solutions. Thus it has also been used to obtain transverse instability of solitary waves in numerous settings: NLS, KP-I, water-waves 10, Euler-Korteweg 11, Rousset & Tzvetkov, multiple references 11 arxiv / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

32 1 Stability and instability in the 2D incompressible model 2 21 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

33 3D isentropic, compressible Navier-Stokes equation on the half space Ω = {X = (x, y, z) R 2 ]0, + [}: { t ρ + div (ρu) = 0 ρ t u + ρu u = div Σ + ρf, (9) modelling the motion of a viscous, non-heat-conducting fluid with typically supersonic speeds. 22 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

34 3D isentropic, compressible Navier-Stokes equation on the half space Ω = {X = (x, y, z) R 2 ]0, + [}: { t ρ + div (ρu) = 0 ρ t u + ρu u = div Σ + ρf, (9) modelling the motion of a viscous, non-heat-conducting fluid with typically supersonic speeds. Σ is the stress tensor: Σ = εµ(ρ)( + t )u + ελ 0 (ρ)div ui 3 P(ρ)I 3. (10) The viscosity coefficients µ > 0 and µ + λ 0 > 0 can depend smoothly on the density. The pressure P(ρ) is given by a barotropic law: P(ρ) = kρ γ with k > 0 and γ > / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

35 F is a smooth force term. We solve for t R with the fluid at rest for negative times: Navier boundary condition: (ρ, u, F ) = (1, 0, 0) for t < 0. u 3 z=0 = 0 and (ε 1 Σ n + au) τ z=0 = 0, where v τ = (v 1, v 2 ) is the tangential part of a 3D vector v, and a R. 23 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

36 F is a smooth force term. We solve for t R with the fluid at rest for negative times: Navier boundary condition: (ρ, u, F ) = (1, 0, 0) for t < 0. u 3 z=0 = 0 and [µ(ρ) z u τ ] z=0 = 2au τ z=0, (11) where v τ = (v 1, v 2 ) is the tangential part of a 3D vector v, and a R. 24 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

37 Global existence is assured for weak solutions with constant viscosity coefficients, and only under certain conditions for variable coefficients 12. Strong solutions are only local in time 13. In the constant coefficient case, convergence of weak solutions to a strong solution of the Euler equations has been obtained by Sueur. 12 Bresch, Desjardins & Gérard-Varet, J. Math. Pures Appl., Solonnikov, Zap. Naučn. Sem. LOMI Steklov., / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

38 Method: a priori estimates in conormal Sobolev spaces, similar to works on the incompressible Navier-Stokes system with Navier BCs (fixed boundary and free surface 14 ). This avoids constructing approximate solutions. 14 Masmoudi & Rousset, Arch. Ration. Mech. Anal., 2012 & preprint 26 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

39 Method: a priori estimates in conormal Sobolev spaces, similar to works on the incompressible Navier-Stokes system with Navier BCs (fixed boundary and free surface 14 ). This avoids constructing approximate solutions. Conormal regularity is necessary to get uniform estimates for boundary layers. Indeed, we expect the ansatz (ρ, u)(t, y, z) (r, v)(t, y, z) + ε(r, V )(t, y, z/ ε), with (r, v) solving the isentropic Euler equation: Lipschitz control at best. 14 Masmoudi & Rousset, Arch. Ration. Mech. Anal., 2012 & preprint 26 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

40 Consider this set of derivatives that are tangent to Ω: Z 0 = t, Z 1,2 = y1,2, Z 3 = φ(z) z = z 1 + z z, 27 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

41 Consider this set of derivatives that are tangent to Ω: Z 0 = t, Z 1,2 = y1,2, Z 3 = φ(z) z = z 1 + z z, For m N and p {2, }, we introduce the notations f (t) 2 m,p = (Z α f )(t) 2 L p (Ω), for T 0. α N 4, α m and f m,p,t = sup f (t) m,p t [0,T ] 27 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

42 The solutions U = (ρ 1, u) we consider will satisfy T E m (T, U) = U 2 m,2,t + zu τ 2 m 1,2,T + z (ρ, u 3 )(s) 2 m 1,2 0 ds T + z u τ 2 1,,T + ( z ρ, tz ρ)(s) 2 1, 0 ds < / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

43 The solutions U = (ρ 1, u) we consider will satisfy T E m (T, U) = U 2 m,2,t + zu τ 2 m 1,2,T + z (ρ, u 3 )(s) 2 m 1,2 0 ds T + z u τ 2 1,,T + ( z ρ, tz ρ)(s) 2 1, The force term will satisfy, for every T 0, N m (T, F ) = and F (t, x) = 0 for t < 0. 0 ds < +. sup ( F (t) 2 m,2 + zf (t) 2 m 1,2 + F (t) 2 2, ) < + t [ T,T ] 28 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

44 Theorem 2.1 (submitted, ) Under the conditions stated previously, for ε 0 > 0 and m 7, there exists T > 0 such that, for every 0 < ε < ε 0, there exists a unique solution U to the isentropic Navier-Stokes system (9) with Navier boundary condition (11) in the class of functions satisfying E m (T, U) < +. Moreover, there is uniformly no vacuum on this time interval. The family of solutions (U ε ) 0<ε<ε0 constructed above converges in L 2 ([0, T ] Ω) and in L ([0, T ] Ω) towards the unique solution of the isentropic Euler equation ((9) with ε = 0 and the non-penetration boundary condition) in the same class. 15 arxiv / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

45 Remarks. The results extend to less regular forces or different initial conditions, providing the compatibility conditions yield uniform bounds on N m (0, U). 16 Wang & Williams, Ann. Inst. Fourier, / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

46 Remarks. The results extend to less regular forces or different initial conditions, providing the compatibility conditions yield uniform bounds on N m (0, U). Wang and Williams 16 proved similar results in the constant coefficient case by constructing approximate solutions, and needs prior existence of the solution to the Euler system. 16 Wang & Williams, Ann. Inst. Fourier, / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

47 Idea of the proof: a priori estimates + compactness argument. 31 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

48 Idea of the proof: a priori estimates + compactness argument. Theorem 2.2 Under the conditions of Theorem 2.1, for ε 0 > 0, there exist T > 0 and a positive, increasing function Q : R + R + such that, for every 0 < ε < ε 0 and t T, t E m (t, U) + z u 3 2 1,,t + ε u(s) 2 m,2 + 2 u τ (s) 2 m,2 0 ds (t + ε)q(e m (t, U) + N m (t, F )). 31 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

49 Idea of the proof: a priori estimates + compactness argument. Theorem 2.2 Under the conditions of Theorem 2.1, for ε 0 > 0, there exist T > 0 and a positive, increasing function Q : R + R + such that, for every 0 < ε < ε 0 and t T, t E m (t, U) + z u 3 2 1,,t + ε u(s) 2 m,2 + 2 u τ (s) 2 m,2 0 ds (t + ε)q(e m (t, U) + N m (t, F )). Throughout, ρ is assumed to be uniformly bounded from below - this is checked by a bootstrap argument using the fact that, along the flow of u, t ρ = ρdiv u, and div u is bounded. 31 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

50 The energy estimates on U use the symmetrisable hyperbolic-parabolic structure of the equation. The system on U α = Z α U is (DA 0 )(ρ) t U α + 3 (DA j )(U) xj U α ε(0, div (µ u) α +(µ+λ 0 ) div u α )) j=1 = (0, (ρf + εσ( U)) α ) C α, with the DA j (U) symmetric matrices, σ( U) the remainder of the order-two terms and C α containing commutators between Z α and the operators involved in the equation (e.g. [Z α, DA j xj ]U). Basic inequalities are used in the energy estimate. 32 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

51 We now need estimates on z U: these are obtained component by component. 33 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

52 We now need estimates on z U: these are obtained component by component. The norms of z u 3 are easily controlled by using the equation of conservation of mass: ρ z u 3 = t ρ u ρ ρ( x u 1 + y u 2 ) 33 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

53 To deal with z u τ, we use a modified vorticity W = (curl u) τ 2a µ(ρ) u τ, so that we have ρ t W + ρu W εµ W = H and W = 0 on the boundary. 34 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

54 To deal with z u τ, we use a modified vorticity W = (curl u) τ 2a µ(ρ) u τ, so that we have ρ t W + ρu W εµ W = H and W = 0 on the boundary. The commutator estimates are standard, except for [Z α, ρu 3 z ]W, in which a Hardy-type inequality is used to avoid the appearance of 2 u. 34 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

55 To deal with z u τ, we use a modified vorticity W = (curl u) τ 2a µ(ρ) u τ, so that we have ρ t W + ρu W εµ W = H and W = 0 on the boundary. The commutator estimates are standard, except for [Z α, ρu 3 z ]W, in which a Hardy-type inequality is used to avoid the appearance of 2 u. L bounds are obtained using a variant of the maximum principle, on W in the constant coefficient case, on (curl u) τ otherwise. 34 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

56 The mass equation is used to control R = z ρ: lε( t R + u R) + γp(ρ)r = h, with l = 2µ + λ 0. The conormal energy estimates are L 2 in time. 35 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

57 The mass equation is used to control R = z ρ: with l = 2µ + λ 0. lε( t R + u R) + γp(ρ)r = h, The conormal energy estimates are L 2 in time. By following the flow of u, we reduce the equation to the ODE lε t R + γpr = h, and the Duhamel formula yields the L estimates. 35 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

58 1 Stability and instability in the 2D incompressible model 2 36 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

59 A recent preprint obtains a similar result 17 using the natural energy structure of the Navier-Stokes system to get estimates on (u, P), using H 1 co bounds on P to control ρ 1,. 17 Wang, Xin & Yong, arxiv / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

60 A recent preprint obtains a similar result 17 using the natural energy structure of the Navier-Stokes system to get estimates on (u, P), using H 1 co bounds on P to control ρ 1,. They thus exhibit weaker boundary layers on ρ compared to u, and, on the domain T (0, 1), they obtain the rates of convergence: U ε (t) U E (t) L 2 Cε 2 and U ε (t) U E (t) L Cε 2/5. 17 Wang, Xin & Yong, arxiv / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

61 Try the same approach as ours on the full Navier-Stokes equation: t ρ + div (ρu) = 0 ρ t u + ρu u = div Σ + ρf ρ t θ + ρu θ + Pdiv u = εκ θ + Σ : u + ρu F. (12) Internal energy proportional to the temperature θ Neumann boundary condition for θ Ideal gas pressure law P(ρ, θ) = kρθ 38 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

62 Disaster strikes! The component-by-component examination of the normal derivatives has worked for the incompressible and isentropic models, but fails in this case! The problem: the terms involving div u in the equations on ρ and θ. The field z V = ( z ρ, z u 3, z θ) cannot be decoupled. 39 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

63 Disaster strikes! The component-by-component examination of the normal derivatives has worked for the incompressible and isentropic models, but fails in this case! The problem: the terms involving div u in the equations on ρ and θ. The field z V = ( z ρ, z u 3, z θ) cannot be decoupled. Work in progress: we can get Hco m 1 bounds: Green function study? W 1, co estimates on z V, but we still need 39 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions

64 Thank you for your attention. 40 / 40 Matthew Paddick Boundary layers for Navier-Stokes equations with slip boundary conditions