Splash singularity for a free-boundary incompressible viscoelastic fluid model

Transcription

1 Splash singularity for a free-boundary incompressible viscoelastic fluid model Pierangelo Marcati (joint work with E.Di Iorio, S.Spirito at GSSI) Workshop 2016 Modeling Computation of Shocks Interfaces Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie, Paris December 6-8, 2016

2 Presentation Outline

3 Splash Singularity in 2-D

4 Splash Singularity This definition appeared in 2011 in a paper on 2-D Water Waves of Angel Castro, Diego Cordoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano They exhibit initial data for the 2D water wave equation for which the ness of the interface breaks down in finite time. Moreover, by a stability result together with numerics they found that starting from a graph, turn over collapse in a splash singularity (self intersecting curve in one point) in finite time.

5 (Highly Incomplete) references Water Waves: A. Castro, D. Cordoba, C. Fefferman, F. Gancedo, M. Gomez-Serrano ( ) Incompressible Euler: A. Castro, D. Cordoba, C. Fefferman, F. Gancedo, M. Gomez-Serrano (2013 Ann.Math); D. Cout S. Shkoller (2014); D. Cordoba, A. Enciso, N. Grubic (2014) Incompressible Navier Stokes: A. Castro, D. Cordoba, C. Fefferman, F. Gancedo,M. Gomez-Serrano (2015) Impossibility for Vortex sheet: D. Cout S. Shkoller, (2014) arxiv: No splash singularities for two-fluid interfaces C. Fefferman, A.D. Ionescu, V. Lie (2013), arxiv:

6 Linear Viscoleastic Maxwell Material Viscoelastic behavior has elastic viscous components modeled as linear combinations of springs dashpots. Purely Viscous materials respond to a tangential stress with behavior consistent with Newton s law (tangential force equal to the product of the shear rate the viscosity) Purely Elastic materials respond to a normal stress manifesting a coherent behavior with Hooke s law Let τ, ɛ, η E stress, strain, viscosity Young modulus τ elastic = Eɛ, τ viscous = η dɛ dt Maxwell model for Viscoelastic materials dɛ dt = dɛ viscous dt + dɛ elastic dt = τ η + 1 E dτ dt

7 Upper Convective Maxwell Oldroyd-B model For large deformations consider the convective derivative the streching terms Let τ, v, λ, η 0, D stress tensor, fluid velocity, relaxation time, the material viscosity, the deformation rate (the rate of strain) tensor. Upper - Convective Time Derivative t uc τ = t τ + (v )τ ( v) T τ + τ( v) Upper Convective Maxwell model Oldroyd-B model τ + λ uc t τ = 2η 0 D, 2D = ( v) + ( v) T τ + λ t uc τ = 2η 0 (D + λ s D) λ s = η solv λ η 0

8 Modelling of polymeric fluids Viscoelastic fluids, like all fluids, are governed by the momentum equation u(x, t) Eulerian velocity, p(x, t) pressure, τ(x, t) stress tensor. t u + u u + p = div τ, div u = 0 Newtonian viscous fluids: τ = ν( u + u T ), Polymeric fluids: τ = ν( u + u T ) + τ p.

9 Equation for the extra-stress τ p (Oldroyd-B) The polymers extra-stress τ p satisfies t uc τ p = 1 λ τ p + ν p λ ( u + ut ), ν p polymeric viscosity λ relaxation time γ is the shear rate velocity distance t 1 viscous forces Weissenberg number We elastic forces = νλ Eɛ = λ γ,. We 1 τ p ν p ( u + u T ) We > 1 formation of geometrical singularities. when We 1 (we approximate with the limit We ) t τ p + (u )τ p ( u)τ p τ p ( u) T = 0.

10 Lemma Let F (α, t) = X the deformation tensor, then we have (in α Eulerian coordinates) t F + u F = uf. Let the initial condition τ(α, 0) = τ 0 (α) be positive definite, τ(α, t) = F τ 0 F T. is positive definite too τ satisfies t τ p + (u )τ p ( u)τ p τ p ( u) T = 0.

11 Viscoelastic High Weissenberg number system (no free boundary) The viscoelastic fluid system for high Weissenberg number is t F + u F = uf t u + u u µ u + p = div(ff T ) in R 3 (0, T ) div u = 0, div F = 0 Fang-Hua Lin, Chun Liu & Ping Zhang, On Hydrodynamics of Viscoelastic Fluids CPAM Li Xu, Ping Zhang & Zhifei Zhang, Global solvability of a free boundary three-dimensional incompressible Viscoelastic Fluid System with surface tension ARMA 2013.

12 Free boundary problem for the High Weissenberg number system t F + u F = uf t u + u u u + p = div(ff T ) div u = 0, div F = 0 ( pi + ( u + u T ) + (FF T I))n = 0 in Ω(t) on Ω(t) u(t) t=0 = u 0, F (t) t=0 = F 0 in Ω 0. The boundary condition states the equilibrium of the force fields acting on the interface. Equation for the flux { Ẋ (α, t) = u(x (α, t), t) X (α, 0) = α, α Ω 0

13 Evolution of the domain

14 Arc-Cord condition Splash Curves Definition (Arc-Cord condition) z : R/2πZ R 2 simple closed curve Arc-Cord condition if there exists K > 0 z(α) z(α ) Kdist(α, α ) for α, α R/2πZ. Definition (splash curve) 1 z 1 (α), z 2 (α) are 2π-periodic. 2 z(α) satisfies the Arc-Cord condition at every point but α 1 α 2, with α 1 < α 2, z(α 1 ) = z(α 2 ) dz(α i ) dα 0. 3 z(α) separates the complex plane into a connected fluid region a vacuum region (not necessarily connected). The normal vector n = ( αz 2(α), αz 1 (α)) αz(α) points to the vacuum region. The interface is part of the fluid region.

15 map Consider a conformal map P from the complex plane to an half plane (e.g. P(z) = z ) image of a splash curve Let z be a splash curve a branch of the function P on the fluid region, the curve z(α) = ( z 1 (α), z 2 (α)) = P(z(α)) satisfies: 1 z 1 (α) z 2 (α) are 2π-periodic. 2 z is a closed contour. 3 z satisfies the Arc-Cord condition.

16 Idea for proving the splash singularity Let P a conformal map P : Ω Ω, where P(z), z C, is a branch of z The initial domain Ω 0 could be nonregular, for instance a splash domain (b), however mapping by P leads to a regular. Ω 0 Since { Ω 0, ũ 0, F 0 } are regular in the conformal coordinates local solution The strategy is take initial domain P 1 ( Ω 0 ) already in splash perturb it use stability

17 Choose suitable initial domain perturb it to get existence Choose in a right way the initial velocity, s.t. ũ 0 (z 1,s ) n > 0 ũ 0 (z 2,s ) n > 0, hence there exists T > 0, s.t. P 1 ( Ω(T )) self-intersects (c). Take a one-parameter family { Ω ε (0), ũ ε (0), F ε (0)}, such that Ω ε (0) = Ω 0 + εb, with b = 1, s.t. P 1 ( Ω ε (0)) is regular there exists a local in time solution

18 Domains local stability in time is needed (stability) let { Ω ε (t), ũ ε (, t), p ε (, t), F ε (, t)} the perturbed solution, then in a suitable norm, hence Ω ε (T ) Ω(T ) O(ε) P 1 ( Ω ε (T )) P 1 ( Ω(T )) so P 1 ( Ω ε (T )) self-intersects. By continuity we have for t = 0, P 1 ( Ω ε (0)) is regular like (a) for t = T, P 1 ( Ω ε (T )) is self-intersecting like (c) define t [0, T ] in the following way t = inf{t ɛ [0, T ] : P 1 ( Ω ε (t)) is splash}. By stability 0 < t < T hence (Ω ɛ, u ɛ, F ɛ, p ɛ ) exists on [0, t ] form a splash singulatity in t = t

19 Choice of the initial data {u 0, F 0 } The initial data {u 0, F 0 } must satisfy the compatibility condition t( u 0 + u T 0 )n = τ(f 0 F T 0 I)n (1) Let U a neighborhood Ω given by the parametrization x(s, λ) = z(s) + λz s (s), where z(s) is the parametrization Ω z s (s) = 1. Construct a stream function on U as follows ψ(x(s, λ)) = ψ(s, λ) = ψ 0 (s) + λψ 1 (s) λ2 ψ 2 (s). Define u 0 = ψ, then div u 0 = 0.

20 Extend t, n : { T (s, λ) = xs (s, λ) = z s (s) + λzss (s) = (1 λk(s))z s (s) N(s, λ) = x λ = zs, By the LHS of (1) in λ = 0 s 2 ψ 0 (s) ψ 2 (s) = T (F 0 F0 T I)N (2) u 0 n = s ψ 0 (s) independent from F 0 (depends only on ψ), then for all F 0 we can choose ψ 0 (s) ψ 2 (s) by (2). Moreover, we can choose u 0 n > 0.

21 Existence of splash singularity for the free boundary problem for the Navier-Stokes equations For the local existence we use a technique introduced by T. Beale in The initial value problem for the Navier-Stokes equations with a free surface, CPAM The analysis of the self intersection via conformal maps is a somehow very classical idea resumed recently (for Navier Stokes) by A.Castro, D. Cordoba, C.Fefferman, F.Gancedo J. Gomez-Serrano, arxiv: Other Methods on Navier Stokes D. Cout, S. Shkoller, arxiv: v1.

22 Transformation We define A conformal map P : Ω Ω, where P(z), z C, as a branch of z, the conformal velocity ũ( X (α, t), t) = u(p 1 ( X (α, t), t) u(x (α, t), t) = ũ(p(x (α, t), t)), the conformal deformation tensor is F ( X (α, t), t) = F (P 1 ( X (α, t), t) F (X (α, t), t) = F (P(X (α, t), t)), for the derivatives we will use where A kj = Xj P k P 1. ( Xj u i ) P 1 = A kj X k ũ i,

23 ly transformed system The transformed system in Ω(t) t Fij + (A rk ũ k r ) F ij = r ũ i A rk Fkj t ũ i + (A rk ũ k r )ũ i Q 2 ũ i + A ri r p = (A rk F kl r ) F il Tr( ũa) = 0 ( pi + ( ũa + ( ũa) T ) + ( F F T I))A 1 ñ = 0 ũ t=0 (t) = ũ 0, F t=0 (t) = F 0. where Q 2 = P z P 1 2. The transformed flux equation d dt X (α, t) = (A X )(ũ X ) in Ω(t) X (α, 0) = α in Ω(0)

24 Transformation To have a fixed boundary problem, we transform in coordinates: ṽ(α, t) = ũ X (α, t) q(α, t) = p X (α, t) G(α, t) = F X (α, t). differentiating X j ũ i = ζ lj l ṽ i, where ζ lj is the lj-th element of ( α X ) 1 l = αl.

25 system in the conformal domain The system is t G ij = A kj X ζ sr s ṽ i G kj t ṽ i Q 2 X ζ sr s ( ζ jr j ṽ i ) + A ri X ζ sr s q = = A rk X G kl ζsr s Gil Tr( α ṽ( α X ) 1 A X ) = 0 ( qi + (( α ṽ( α X ) 1 A X ) + ( α ṽ( α X ) 1 A X ) T + +( G G T I))A 1 X J X ñ0 = 0 ṽ(α, 0) = ṽ 0 (α) = ũ 0 (α), G(α, 0) = G 0 (α) = F 0 (α). ( ) 0 1 where J X = J X J, with J =, this is due to the 1 0 fact that ñ = JA Ω(t) Jn.

26 Iteration scheme- fluid part We observed that it is possible to separate the equation for ṽ from the equation for G. t ṽ (n+1) Q 2 ṽ (n+1) + A T q (n+1) = f (n) Tr( ṽ (n+1) A) = g (n) ( q (n+1) I + (( ṽ (n+1) A) + ( ṽ (n+1) A) T ))A 1 ñ 0 ) = h (n) ṽ(α, 0) = ṽ 0. where f (n), g (n), h (n) contain all the missing terms in the previous time step, for instance in f (n) in h (n) there are G (n) terms.

27 Iteration for the deformation tensor This implies t G (n+1) ij G(α, 0) = G 0. G (n+1) (α, t) = G 0 + For the flux we get = A kj X (n) ζsr s ṽ (n) i t 0 G (n) kj (A X (n) ζ(n) ṽ (n) G (n) )(α, τ) dτ. X (n+1) (α, t) = α + t 0 (A X (n) ṽ (n) )(α, τ) dτ. Assuming the convergence as n, in the limit we find the solution of the system.

28 Function spaces These are the spaces where we will use for the : H ht,s ([0, T ]; Ω) = L 2 ([0, T ]; H s (Ω)) H s 2 ([0, T ]; L 2 (Ω)), Hpr ht,s ([0, T ]; Ω) = {q L ([0, T ]; H 1 (Ω)) : q H ht,s 1 ([0, T ]; Ω), q H ht,s 1 2 ([0, T ]; Ω)}, H ht,s ([0, T ]; Ω) = L 2 ([0, T ]; H s (Ω)) H s+1 2 ([0, T ]; H 1 (Ω)), F s+1 ([0, T ]; Ω) = L 1 ([0, T ]; H s+1 (Ω)) H 2 ([0, T ]; H γ (Ω)), 4 for s 1 ε < γ < s 1, F s ([0, T ]; Ω) = L 1 ([0, T ]; H s (Ω)) H 2 ([0, T ]; H γ 1 (Ω)), 4 for s 2 ε < γ 1 < s 2. f L 14 = sup t 1 4 f (t) t [0,T ]

29 Idea for solving the linear system for ṽ This theory was developed by T. Beale. The idea is to start with the homogeneous system t v Q 2 v + A T q = f Tr( va) = 0 ( qi + ( va) + ( va) T ) AT Q 2 n = 0 v(α, 0) = v 0. Weak formulation of the problem Projection R on H 0 σ = {v H 1 : Tr( va) = 0}

30 Theorem The main result Let f H ht,s 1, 2 < s < 3 such that Rf (0) = 0. Then v H ht,s+1 + q H ht,s 1 + q H ht,s 1 2 with C indipendent on T. Write the linear problem as t u + S A u = Rf Prove the thm by using the following results: 1 S 1 A f H s+1 C f H s 1, 2 (λ + S A ) 1 Rf H s+1 C where 1 s 3, λ C R(λ) 0. C f H ht,s 1, ( ) Rf H s 1 + λ s 1 2 Rf L 2

31 Inhomogeneous linear problem For the inhomogeneous problem t v Q 2 v + A T q = f Tr( va) = g ( qi + ( va) + ( va) T )A 1 n = h v(α, 0) = v 0 Let s introduce the compatibility conditions for the initial data: { Tr( v0 A) = g(0) (A 1 n) ( v 0 A + ( v 0 A) T )A 1 n = h(0)(a 1 n) (3)

32 For this system we define spaces X space of Y space of data. { } X := (v, q) : v H ht,s+1, q Hpr ht,s Y := {(f, g, h, v 0 ) : f H ht,s 1, g H ht,s, h H ht,s 1 2 ( Ω [0, T ]), v0 H s (Ω) (3) are satisfied} Theorem Let 2 < s < 5 2. Then L : X Y has a bounded inverse: (v, q) X C (f, g, h, v 0 ) Y

33 In order to have the constant C independent of the time we define X 0 := {(v, q) X : v(0) = 0, t v(0) = 0, q(0) = 0} Y 0 := {(f, g, h, 0) Y : f (0) = 0, g(0) = 0, t g(0) = 0, h(0) = 0} Theorem L : X 0 Y 0 is invertible for 2 < s < 5 2. Moreover, L 1 is bounded uniformly if T is bounded above.

34 Navier Stokes iteration In order to apply the previous Theorem, we take an approximation φ = ṽ 0 + t exp( t 2 )(Q 2 ṽ 0 A T q φ ), a new function w (n) = ṽ (n) φ the new system is w (n+1) Q 2 w (n+1) + A T q w (n+1) = f (n) t φ +Q 2 φ A T q φ Tr( w (n+1) A) = g (n) Tr( φa) [ q (n+1) w I + (( w (n+1) A) + ( w (n+1) A) T )]A 1 ñ 0 = w (n+1) t=0 = 0 = h (n) + q φ A 1 ñ 0 (( φa) + ( φa) T )A 1 ñ 0 where f (n), g (n), h (n) contain all the missing terms.

35 Flux deformation gradient iteration The flux the deformation gradient G (n+1) (α, t) = G t t X (n+1) (α, t) = α (A X (n) ζ (n) w (n) G (n) )(α, τ) dτ (A X (n) ζ(n) φ G (n) )(α, τ) dτ, t (A X (n) w (n) )(α, τ) dτ 0 t + 0 (A X (n) φ)(α, τ) dτ.

36 Results for the Navier-Stokes part Theorem (Estimate for the flux (Part 1)) Let X (n) α F s+1, w (n) H ht,s+1 such that X (n) α { X α F s+1 : X α t Aφ dτ F s+1} B Aφ, 0 w (n) H ht,s+1 N. t 0 Aφ dτ F s+1 Then, for small enough T > 0, depending only on N, ṽ 0, X (n+1) α B Aφ.

37 Theorem (Estimate for the flux (Part 2)) Let X (n) α, X (n 1) F s+1, with w (n), w (n 1) H ht,s+1 such that w (n) H M, ht,s+1 w (n 1) M, H ht,s+1 X (n) α M, X (n 1) α M F s+1 F s+1 for some M > 0. Then X (n+1) X (n) F s+1 CT δ ( X (n) X (n 1) F s+1 For a small enough δ. + w (n) w (n 1) H ht,s+1)

38 Estimates for G Theorem (Estimates for G (Part1)) Let G (n) G 0 F s, X (n) α F s+1, w (n) H ht,s+1 such that G (n) G 0 { G G 0 F s : G G 0 t A φ G 0 dτ F s } B, 0 w (n) H ht,s+1 N. t 0 A φ G 0 dτ F s Then, for T > 0 small enough, depending only N, ṽ 0, G 0. G (n+1) G 0 B.

39 Estimates for G (Part 2) Theorem Let G (n) G 0, G (n 1) G 0 F s, with X (n) α, X (n 1) α F s+1 w (n), w (n 1) H ht,s+1 such that w (n) H ht,s+1 M, w (n 1) H ht,s+1 M, X (n) α F s+1 M, X (n 1) α F s+1 M G (n) G 0 F s M, G (n 1) G 0 F s M, for some M > 0. Then G (n+1) G (n) F s CT δ ( G (n) G (n 1) F s + + w (n) w (n 1) H ht,s+1 + X (n) X (n 1) F s+1) For a small enough δ.

40 Theorem (Estimates for ṽ, q (Part 1)) Let X (n) α F s+1, q w (n) such that H ht,s pr w (n) H ht,s+1, X (n) α F s+1 N, G (n) G 0 F s N, ( w (n), q w (n) ) {( w, q) H ht,s+1 Hpr ht,s : w t=0 = 0, t w t=0 = 0, ( w, q) L 1 ( f φ, g φ, h φ ) H ht,s+1 Hpr ht,s L 1 ( f φ, g φ, h φ ) H ht,s+1 Hpr ht,s } B L 1 ( f φ, g φ, h φ ). Then ( w (n+1), q (n+1) w ) B L 1 ( f φ, g φ, h φ ).

41 Theorem (Estimates for ṽ, q (Part 2)) Let X (n) α, X (n 1) F s+1, G (n) G 0, G (n 1) G 0 F s, w (n), w (n 1) H ht,s+1, with w (n) (n 1) t=0 = w t=0 = 0, t w (n) t=0 = t w (n 1) t=0 = 0, q w (n), q w (n 1) Hpr ht,s such that all these function are bounded, in their norm, by a constant M > 0. Then w (n+1) w (n) H ht,s+1 + q (n+1) w q w (n) H ht,s pr CT δ ( X (n) X (n 1) F s+1 + w (n) w (n 1) H ht,s+1+ + q (n) w q w (n 1) H ht,s + G (n) G (n 1) pr F s ) For a small enough δ.

42 Existence Theorem Putting together all the results above applying the Contraction Mapping Principle we get Theorem For 2 < s < 2.5, 1 < γ < s 1 for T sufficiently small we have that { w (n) } n=0, { q(n) w } n=0, { G (n) } n=0 { X (n) } n=0 are Cauchy sequences respectively in H ht,s+1 ([0, T ], Ω 0 ), Hpr ht,s ([0, T ], Ω 0 ), F s ([0, T ], Ω 0 ) F s+1 ([0, T ], Ω 0 ). The limit of the sequence is the desired solution.

43 We pick Ω ε (0) = Ω 0 + εb, with b = 1, for instance b = e 2, such that P 1 ( Ω ε (0)) is a regular domain. We take the difference between ( w, q, X, G) ( w ε, q ε, X ε, G ε ) we get t ( w w ε ) Q 2 ( w w ε ) + A T ( q w q w,ε ) = F ε Tr( ( w w ε )A) = K ε [ ( q w q w,ε )I + ( w w ε )A + ( ( w w ε )A) T ]A 1 ñ 0 = H ε w 0 = 0, where F ε, K ε, H ε contain all the other terms.

44 t X X ε = bε + G G ε = bε + 0 t 0 ) (A X ṽ A X ε ṽ ε dτ, (A X ζ ṽ G A X ε ζ ε ṽ ε G ε ) dτ. For δ small enough 2 < s < 2.5, similar as those for the local existence lead to the following results: G G ε L H s + G G ε H 2 Hγ 1 Cε+ + CT δ ( w w ε H ht,s+1 + G G ε L H s + G G ε H 2 H γ 1 + X X ε L H s+1 + X X ε H 2 H γ )

45 w w ε H ht,s+1 + q w q w,ε H ht,s pr F ε H ht,s 1 + K ε Hht,s + H ε H ht,s 1 2 C( F ε H ht,s 1 + K ε H ht,s + H ε H ht,s 1 2 ), Cε + CT δ ( w w ε H ht,s+1 + q w q w,ε H ht,s + pr + G G ε L H s + G G ε H 2 H γ 1+ + X X ε L H s+1 + X X ε H 2 H γ ), X X ε L H s+1 + X X ε H 2 H γ Cε + CT δ ( w w ε H ht,s+1 + X X ε L H s+1 + X X ε H 2 H γ ).

46 Main result Putting together these results we get that for 0 < T < 1 (3C) 1/δ w w ε H ht,s+1 + q w q w,ε H ht,s pr + X X ε L H s+1 + X X ε H 2 H γ + G G ε L H s + G G ε H 2 Hγ 1 3Cε X X ε L Hs+1 3Cε, (4) this means that the two fluxes are close enough so that the domains are close: X ( Ω 0, t) X ε ( Ω 0,ε, t). The domains are close enough so we can apply the idea of splash singularity described in the.

47 Thank You!!!