Stochastic Hyperbolic PDEq

Transcription

1 Wladimir Neves UFRJ Federal University of Rio de Janeiro Rio de Janeiro - Brazil wladimir@im.ufrj.br Joint work with Christian Olivera IV Workshop em Fluídos e EDP, May 26, 2014

2 Introduction Stochastic (forced) PDEs Stochastic (intrinsic) PDEs. Some important (deterministic) problems. Malliavin calculus for PDE s; Fractional operators. Continuum Physics with new constitutive relations, which allows fractional derivatives. The concept of Simple Materials (Walter Noll).

3 Stochastic (forced) PDEs Following Weinan E, we address some important stochastic (forced) PDEq in hydrodynamics: Stochastic (forced) incompressible Navier-Stokes Eq. t v + div(v v) + p = ν v + f, divv = 0, where f is a stochastic forcing term (usually with zero mean). From Weinan E observations, this is a prototypical SPDE in fluid mechanics related to turbulence.

4 Here, we are not going to enter in discussion about wide range of length scales, hierarchy of vortices, swirls, etc., also Kolmogorov s notion of cascade. We just mention that: One could not view turbulence as an isolated problem! Stochastic (forced) Burgers equation t u + div( u2 ) = ν u + f, 2 which is a prototype of nonlinear waves, and also a canonical example in non-equilibrium statistical physics. Again, following Weinan E, stochastic forcing PDEs provide the natural framework to study long time behavior of complex problems, like turbulence. In particular, from the invariant measures theory.

5 Let us now mention another stochastic forced equations: Scalar conservation laws with stochastic forcing, A. Debussche, J. Vovelle, JFA (2010) t u + diva(u) = φ(u) dw t dt, A C 2, W t = B i t e i, B-Brownian motion. The solvability relies in the kinetic theory (Lions, Perthame, Tadmor). Under some conditions on φ, considering the equation is periodic in space, the stochastic problem is well posed.

6 On nonlinear stochastic balance laws, Gui-Q. Chen, Q. Ding, K. Karlsen, ARMA (2012) t u + divf(u) = σ(x, u) dw t dt. Like us, they said in that paper: We are interested in the effect of noise on discontinuous waves, since these are often the relevant solutions. Once σ(0) = 0, σ(u) σ(v) u v, f C 2, working on BV-framework, the proved well-posedness. Now, let us turn to our approach, that is to say

7 Stochastic (intrinsic) PDEs Our ansatz is based on rational Continuum Physics. Let ψ be a physical quantity, which is a tensor field of order m. Also, we consider the supply and the flux of ψ, denoted respectively σ ψ, φ ψ, which are tensor fields of order m and m + 1. Then, the general stochastic balance equation has the form t ψ + div ( ψ dx t dt φ ) ψ = σψ, with X t given for instance by t X t (x) = x + v(s, X s (x)) ds + V t, 0 where v is the velocity field, and V t is a stochastic process, which is not due necessarily from a Brownian motion.

8 In particular, taking ψ = ρ, and ψ = ρ v, we obtain the following Incompressible non-homogeneous stochastic Navier-Stokes Eq ( t ρ + div ρ ( v + dv t ) ) = 0, dt div v = 0, ( t (ρ v) + div ρ v ( v + dv t ) ) T = ρ g, dt where g is an external body force.

9 Muskat problem Besides the turbulence problem, we are also interested in the Muskat Problem. m(x) porosity. v i (t, x) R d, (i = 1, 2) seepage velocity field. s i (t, x) R, (i = 1, 2) saturation of each i th component, 0 s 1, s 2 1, s 1 + s 2 = 1. (1) Conservation of Mass (Continuity equation): m t (ρ i s i ) + div x (ρ i v i ) = 0, (i = 1, 2) (2) where ρ i is mass density of the i th -phase of liquids.

10 From the incompressibility assumption, we have t (s i ) + div x (v i ) = 0, (i = 1, 2) (3) where we took m 1 without loss of generality. Conservation of Linear Momenta (Darcy s law equation): µ i k 0 k ri (s 1 ) v i = x p i + ρ i g h, (i = 1, 2) (4) where for each component i = 1, 2, p i (t, x) is the pressure, µ i is the dynamic viscosity and k ri is the relative permeability. Moreover, k 0 (x) is the absolute permeability of the porous medium and ρ g h is the external gravitational force, which is used dropped.

11 From the above considerations and, denoting λ i = µ i /(k 0 k ri (s 1 )) (i = 1, 2), we have λ i v i = x p i. (5) Then, from equations (3) and (5), we are in condition to formulate the Muskat problem. One remarks that, in each part of the medium, where just exist one component, the saturation s i is obviously constant and equals one.

12 Muskat problem (original formulation): λ o v o = x p o, div x (v o ) = 0, in Q o, λ i v w = x p i, div x (v w ) = 0, in Q w, p o = p w, v w n t = v o n t, on Γ t, where the under-scrip o, w stand respectively for oil and water. Further, we have assumed that, p c = 0, i.e. the capillarity pressure is zero on Γ t. Q o, Q w regions of oil and water respectively. Γ t free boundary between Q w and Q o. n t unitary normal field to Γ t. Therefore, Muskat problem (original formulation) is a time-dependent elliptic-diffraction problem with a free-boundary. (6)

13 Muskat problem (weak formulation): First, we define on Q w Γ t Q o : u(t, x) := s w (t, x), hence s o (t, x) = 1 u(t, x). v(t, x) := v w (t, x) + v o (t, x) total velocity. Moreover, we define the pressure p on Q w Γ t Q o as p(t, x) := { pw in Q w Γ t, p o in Q o. (7) After some algebraic computation, we have the following system, (also) called Buckley-Leverett system: t u + div x (v g(u)) = 0, div x (v) = 0, h(u) v = x p. (8)

14 From equation (8), it follows that, we have to deal with a scalar non-homogeneous conservation law with t u + div x ϕ(t, x, u) = 0, ϕ(t, x, p) = v(t, x) g(p), (9) and v is expected to be just in L 2! Do not mind w.r.t. g regularity, used to be good enough... NB. The difficulty to show solvability for the above equation is similar to show uniqueness for linear transport equations with drift vector field in L 2!!!

15 Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition Given an initial-data u 0, find u(t, x; ω) R, satisfying ( t u(t, x; ω) + div u(t, x; ω) ( b(t, x) + db t dt (ω))) = 0, u t=0 = u 0, (10) ( (t, x) UT, ω Ω ), where U T = [0, T ] R d, for T > 0 be any fixed real number, (d N), b : R + R d R d is a given vector field (called drift), with div b(t, x) = 0, B t = (B 1 t,..., B d t ) is a standard Brownian motion in R d.

16 Integration is taken (unless otherwise mentioned) in the Stratonovich sense. In fact, we fix a stochastic basis with a d-dimensional Brownian motion ( Ω, F, {F t : t [0, T ]}, P, (B t ) ). The main issue is to prove uniqueness of weak L solution of the Cauchy problem (10) for vector fields b L q ([0, T ], (L p (R d )) d ), p, q <, p 2, q > 2, and d p + 2 (11) q < 1, which is known in the fluid dynamic s literature as the Ladyzhenskaya-Prodi-Serrin condition, with in place of <.

17 We do not assume any differentiability, nor boundedness of the vector field b. The uniqueness result is established using the transportation property of the continuity equation for divergence free vector fields. Therefore, we have sharpened the answer of the following question: Why noise improves the deterministic theory for transport/continuity equations? Our result is the first one in this direction! In fact, that noise could improve the theory of transport equations was first discovered by F. Flandoli, M. Gubinelli, E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., (2010). More precisely, the condition assumed is Hölder continuity and boundedness of b, and an integrability condition on the divergence.

18 We recall that the Ladyzhenskaya-Prodi-Serrin condition (11) was first considered by Krylov, Röckner (Prob. Theory Relat. Fields, (2005)). In that paper, they proved the existence and uniqueness of strong solutions for SDE t X s,t (x) = x + b(r, X s,r (x)) dr + B t B s, (12) s where given t [0, T ] and x R d, it was shown that ( T ) P b(t, X t ) dt = = 0. 0 More recently, Fedrizzi, Flandoli proved the α-hölder continuity of the stochastic flow x X s,t for any α (0, 1). Moreover, they prove that it is a stochastic flow of homeomorphism.

19 We assume Definition b L 1 loc (U T ), div b L 1 loc (U T), u 0 L (R d ). (13) A stochastic process u L (U T Ω) is called a weak L solution of the Cauchy problem (10), when for any ϕ Cc (R d ), the real value process u(t, x)ϕ(x)dx has a continuous modification which is a F t -semimartingale, and for all t [0, T ], we have P-almost sure u(t)ϕdx = u 0 ϕ dx + R d R d + t 0 t 0 R d u(s) i ϕ dx db i s. R d u(s) b i (s) i ϕ dxds (14)

20 Lemma Under condition (24), there exits a week L solution u of the Cauchy problem (10). The proof follows according to a minor modification of the arguments in DiPerna, Lions seminal paper (Invent. Math., 1989) see Proposition II.1 (taking only test functions defined on R d ). Now, we consider the uniqueness result under the divergence-free condition, that is div b = 0 (15) (understood in the sense of distributions), and also the Ladyzhenskaya-Prodi-Serrin condition (11).

21 Theorem Assume conditions (11), and (15). If u, v L (U T Ω) are two weak L solutions for the Cauchy problem (10), with the same initial data u 0 L (R d ), then for each t [0, T ], u(t) = v(t) almost everywhere in R d Ω. Proof s idea.under condition (11) we have suitable regularity of the stochastic characteristics. Indeed, under the divergence-free condition the continuity equation turns to transport equation. Therefore, the main feature of the transport equation, which is the transportation property, it is used by the authors to show uniqueness, completely different from the renormalization (due commutators).

22 Now we are concerned with the initial-boundary value problems for stochastic transport equation in bounded domains. For a given stochastic perturbation of the drift vector field, and the initial-boundary data in L, we prove existence and uniqueness of weak L solutions with non-regular coefficients. The existence result, which is by no means a trivial adaptation, relies on a strong stochastic trace theorem. Moreover, the uniqueness of weak solutions is obtained under suitable conditions, which allow vacuum.

23 Given an initial-boundary data u 0, u b, find u(t, x; ω) R, satisfying ( t u(t, x; ω) + b(t, x) + A(x) db ) t dt (ω) u(t, x; ω) = 0 in U T, u = u 0 in {t = 0} U, u = u b on Γ T, (16) where U be an open and bounded domain of R d (d N). The real d d matrix value function A is symmetric, and nonsingular, so that det A(x) 0. We denote by Γ the boundary of U, with the outside normal field to U at r Γ denoted by n(r), and Γ T = (0, T ) Γ. We define the influx boundary zone Γ T := { (t, r) Γ T : (b n)(t, r) < 0 }. (17)

24 One of the premieres studies of linear transport equations (deterministic case) in bounded domains was done by Bardos (Ann. Sci. École Norm. Sup., 1970). On that extended paper Bardos consider the regular case, where the vector field b has Lipschitz regularity, and it was introduced the correct notion of the Dirichlet boundary condition. Then, we mention the work of Mischler (Comm. Part. Diff. Eq., 2003), who consider weak solutions for the Vlasov equation (instead of the transport equation) posed in bounded domains. If u is not sufficiently regular, in particular we are seeking for measurable bounded functions, the restriction to negligible Lebesgue sets is not, a priori, defined. Therefore, one has to deal with the traces theory to ensure the correct notion of Dirichlet boundary condition.

25 In the same direction as Mischler, Boyer (Diff. Integral Eq., 2005) establish the trace theorems with respect to the measure dµ = (b n)drdt, and show the existence and uniqueness of solutions for the transport equation using the Sobolev framework of DiPerna, Lions cited before. More recently, Crippa, Donadello, Spinolo (arxiv: v1) studied the initial-boundary value problems for continuity equations with total bounded variation coefficients, hence analogue framework to Ambrosio (Invent. Math., 2004).

26 Let us now focus on the stochastic case. First, Funaki (J. Math. Soc. Japan, 1979) studied the random transport equation with very regular coefficients for bounded domains. To the knowledge of the authors, nothing has already been done for stochastic transport equations with low regularity coefficients in bounded domains. Here, we deal with the problem (16) and show the existence and uniqueness of weak L -solutions for Dirichlet data.

27 The initial-boundary value problem is much harder to solve than the Cauchy one, hence we exploit new improvements due the perturbation of the drift vector field by a Brownian motion. The solvability in the weak sense for the Cauchy problem is easily established under the mild assumption of local integrability for b and divb. On the other hand, the existence result establish here on bounded domains, considering BV regularity to the drift, relies strongly on the passage from the Stratonovich formulation into Itô s one, which is a new deeply result. It is also core for the existence s proof, the strong stochastic trace result.

28 The uniqueness result obtained does not assumed L control on the divergence of the vector field b, as it is used to be in the deterministic case. We have assumed just a boundedness from above, which means that vacuum is allowed to occur. Moreover, we just assume a boundedness of b w.r.t. the spatial variable. Despite we have used some special technics to show uniqueness for the stochastic case, in particular the features of the Hamilton-Jacobi-Belmann equation, the uniqueness result under the same assumptions follows as well in the deterministic case.

29 Classical solutions Given s [0, T ] and x R d, we consider X s,t (x) = x + t s b(r, X s,r (x)) dr + B t B s, (18) where X s,t (x) = X (s, t, x), also X t (x) = X (0, t, x). In particular, for m N and 0 < α < 1, let us consider b L 1 ((0, T ); C m,α (R d )). (19) It is well know that, under condition (19), the stochastic flow X s,t is a C m -diffeomorphism. Moreover, the inverse Y s,t := Xs,t 1 satisfies the following backward stochastic differential equations, Y s,t = y t s b(r, Y r,t ) dr (B t B s ), (20) for 0 s t. Usually, Y is called the time reversed process of X.

30 Given (t, x) U T and the time reversed process Y s,t given by (20), we consider and define S = {s (0, t)/ Y (s, t, x) / U T } τ(t, x) := sup S. (21) Clearly S could be an empty set, and in this case we set τ = 0. Now, we define Ȳs,t on [0, t] Ū as Ȳ s,t (x) := Y s,t (x) for s [max{0, τ}, t]. (22) Then, from the above considerations, we may apply a standard computation to prove the following

31 Lemma For m 3, 0 < α < 1, let u 0 C m,α (U), u b C m,α (Γ T ) be respectively initial-boundary data satisfying compatibility conditions, and assume (19). Then, the IBVP problem (16) has a unique solution u(t,.) for 0 t T, such that, it is a C m -semi-martingale given by u(t, x) := { u0 (Ȳτ,t(x)), if τ(t, x) = 0, u b (τ, Ȳ τ,t (x)), if τ(t, x) > 0, (23) where Ȳ is the stochastic process defined in (22), with τ given by (21).

32 Existence of Solution SPDE We shall always assume that b L 1 ((0, T ); BV loc (R d )) and divb L 1 loc ((0, T) Rd ). (24) First, we consider a distributional solution to (16), and reeling on that, we establish a stochastic trace result, which should be more refined than deterministic one. The requirement is due the stochastic boundary terms, which are integrated with respect to drdt instead of the measure dµ.

33 Definition Let u 0 L (U), u b L (Γ T ; µ ) be given. A stochastic process u L (U T Ω) is called a distributional L solution of the IBVP (16), when for each test function ϕ Cc (U), the real value process U u(t, x)ϕ(x)dx has a continuous modification which is a F t -semimartingale, and for all t [0, T ], we have P-almost sure u(t)ϕdx = U + + U t u 0 ϕ dx + 0 U t 0 U t 0 U u(s) div b(s) ϕ dxds u(s) xi ϕ dx db i s. u(s) b i (s) xi ϕ dxds Hereafter the usual summation convention is used. (25)

34 Following Flandoli, Gubinelli, Priola, see Lemma 13, we can reformulate equation (25) in Itô s form as follows: A stochastic process u L (U T Ω) is a distributional L solution of the SPDE (16) if, and only if, for every test function ϕ Cc (U), the process u(t, x)ϕ(x)dx has a continuous modification, which is a F t -semimartingale, and satisfies the following Itô s formulation for all t [0, T ] U u(t)ϕdx = + + t 0 U t 0 U U u 0 ϕ dx + t 0 U u(s) div b(s) ϕ dxds u(s) xi ϕ dx db i s u(s) b i (s) xi ϕ dxds t 0 U u(s) ϕ dxds. (26)

35 Lemma Under condition (24), there exits a distributional L solution u of the stochastic IBVP (16).

36 Stochastic Trace Definition Let u be a distributional L -solution of the IBVP problem (16). A stochastic process γu L ([0, T ] Γ Ω) is called the stochastic trace of the distributional solution u, if for each test function ϕ Cc (R d ), Γ γu(t, r)ϕ(r)dr is an adapted real value process, which satisfies for any β C 2 (R) and all t [0, T ]

37 β(u(t)) ϕ dx = U U t β(u 0 ) ϕ dx 0 U t 0 U t 0 Γ t 0 t 0 Γ U β(u(s)) b(s) ϕ dxds β(u(s)) div b(s) ϕ dxds β(γu) ϕ b(s) n drds β(u(s)) xi ϕ dx db i s β(γu) ϕ n i dr db i s. (27)

38 Proposition Assume condition (24), and let u be a distributional L -solution of the IBVP problem (16). Then, there exits the stochastic trace γu. The proof relies in the Cauchy sequences on the boundary, starting with a suitable technique of global approximation. For any ε > 0 fixed, 0 τ ε, and y U, we define y ε := y + λ ε h(y), for λ > 0 sufficiently large. Then, we take a standard mollifier ρ ε, and for any u L 1 loc (U T ), we define the following (space) global approximation u ε (t, y) (u n ρ ε )(t, y) := u(t, z)ρ ε (y ε z) dz. U

39 Stochastic transport equation IBVP Definition Let u 0 L (U), u b L (Γ T ; µ ) be given. A stochastic process u L (U T Ω) is called a weak L solution of the IBVP (16), when for each test function ϕ Cc (R d ), the process U u(t, x)ϕ(x)dx has a continuous modification which is a F t -semimartingale, and satisfies for all t [0, T ]

40 t u(t)ϕdx = u 0 ϕ(x) dx + U U 0 t + u(s) div b(s) ϕ dxds U t 0 Γ t 0 U u b (s) ϕ(r)(b j n j ) drds u(s) j ϕ dx db j s. U t u(s) b j (s) j ϕ dxds γu(s) ϕ (b j n j ) + dr ds 0 Γ t 0 Γ γu(s, r) ϕ(r) n j (r) dr db j s (28)

41 Theorem Under condition (24), there exits a weak L solution u of the IBVP (16). We follow our strategy used to show existence of distributional L -solutions, that is to say applying the stochastic characteristics. Now the proof become showier, since we have to deal with the boundary terms. The Co-area and Area Formulas are used to handle them.

42 Uniqueness We prove uniqueness following the concept of renormalized solutions introduced by DiPerna, Lions. The BV framework is the one adopted in the sequel, where we make extensive use of the ideas from Ambrosio. First, we have the following Lemma Assume condition (24). Let u be a distributional L -solution of the stochastic IBVP (16), and define v := E(β(u)) for any β C 2 (R). Then, for each u 0 L (U) the function v satisfies t v(t, x) + b(t, x) v(t, x) = 1 v(t, x) 2 in D ([0, T ) U). (29)

43 Next, we pass to the uniqueness theorem. It should be assumed more conditions on the vector function b, which are all explicit in the following Theorem Let b be a vector field satisfying condition (24), and for a.e. t [0, T ] b(t, x) α(t), divb(t, x) γ(t), (30) for some nonnegative functions α, γ L 1 loc (R). If u, v L (U T Ω) are two weak L solutions of the IBVP (16), with the same initial-boundary data u 0 L (U), u b L (Γ T ; µ ), then u v almost sure in U T Ω.

44 Stochastic scalar multidimensional conservation laws. Work in progress, relying on the L 1 -semigroup, and kinetic theory. Stochastic Keyfitz-Kranzer multidimensional system. Work in progress, prototype of multidimensional system of conservation laws, which satisfies Brenner s commutator condition.

45 THANK YOU.