Remarks on the stochastic transport equation with H¨older drift 3 1 0 2 F. Flandoli1, M. Gubinelli2, E. Priola3 n (1) Dipartimento diMatematica, Universit`a diPisa, Italia a (2) CEREMADE (UMR 7534), Universit´eParis Dauphine,France J 7 (3) Dipartimento diMatematica, Universit`a diTorino, Italia 1 January 18, 2013 ] P A Abstract . h We consider a stochastic linear transport equation with a globally t a H¨oldercontinuousandboundedvectorfield. Oppositetowhathappensin m thedeterministiccasewhereshocksmayappear,weshowthattheunique [ solution startingwith aC1-initialcondition remains ofclass C1 inspace. We also improve some results of [8] about well-posedness. Moreover, we 1 proveastabilitypropertyforthesolutionwithrespecttotheinitialdatum. v 2 1 1 Introduction 0 4 . Theaimofthispaperistwofold. Ononeside,wereviewideasandrecentresults 1 abouttheregularizationbynoiseinODEsandPDEs(Section1). Ontheother, 0 we give detailed proof of two new results of regularization by noise, for linear 3 1 trasportequations,relatedto those ofthe paper [8](Theorem7 and the results : of section 4). v i X 1.1 The ODE case r a A well known but still always surprising fact is the regularization produced by noise on ordinary differential equations (ODEs). Consider the ODE in Rd d X(t)=b(t,X(t)), X(0)=x ∈Rd 0 dt with b : [0,T]×Rd → Rd. If b is Lipschitz continuous and has linear growth, uniformlyint,thenthereexistsauniquesolutionX ∈C [0,T];Rd . Butwhen b is less regular there are well-known counterexamples, like the case d = 1, (cid:0) (cid:1) b(x) = 2sign(x) |x|, x = 0 where the Cauchy problem has infinitely many 0 solutions: X(t)=0, X(t)=t2, X(t)=−t2, andothers. The functionb ofthis p example is Ho¨lder continuous. 1 Consider now the stochastic differential equation (SDE) dX(t)=b(t,X(t))dt+σdW (t), X(0)=x ∈Rd (1) 0 with σ ∈R and {W (t)} a d-dimensional Brownian motion on a probability t≥0 space (Ω,F,P). We say that a continuous stochastic process X(t,ω), t ≥ 0, ω ∈Ω,adaptedto the filtration{FW} ofthe Brownianmotion,isa solution t t≥0 if it satisfies the identity t X(t,ω)=x + b(s,X(s,ω))ds+σW (t,ω), t≥0, 0 Z0 for P-a.e. ω ∈Ω. In the Lipschitz case we have again existence and uniqueness ofsolutions. Butnow,wehavemore: ifσ 6=0andb∈L∞ [0,T]×Rd;Rd then thereisexistenceanduniquenessofsolutions,[19]. Theresultistrueevenwhen (cid:0) (cid:1) b ∈ Lq 0,T;Lp Rd;Rd with d + 2 < 1, p,q ≥ 2 [14] (the assumptions can p q be properly localized). Recently, we have proved in [8] the following additional (cid:0) (cid:0) (cid:1)(cid:1) result,whichwillbeusedbelow(the functionspacesaredefinedinSection1.4). Theorem 1 If σ 6= 0 and b ∈ L∞ 0,T;Cα Rd;Rd , α ∈ (0,1), then there b exists a stochastic flow of diffeomorphisms φ =φ(t,ω) associated to the SDE, t with Dφ(t,ω) and Dφ−1(t,ω) of cla(cid:0)ss Cα′ fo(cid:0)r every(cid:1)α(cid:1)′ ∈(0,α). By stochastic flow of diffeomorphisms we mean a family of maps φ(t,ω) : Rd →Rd such that: i) φ(t,ω)(x ) is the unique solution of the SDE for every x ∈Rd; 0 0 ii) φ(t,ω) is a diffeomorphisms of Rd. For several results on stochastic flows under more regular conditions on b see [15]. Let us give an idea of the proof assuming σ = 1. Introduce the vector valued non homogeneous backwardparabolic equation ∂U 1 +b·∇U + ∆U =−b+λU on [0,T] ∂t 2 U(T,x)=0 with λ ≥ 0. By parabolic regularity theory we have the following result (cf. Theorem 2 in [8]): Theorem 2 If b ∈ L∞ 0,T;Cα Rd;Rd , α ∈ (0,1), then there exists a b unique bounded and locally Lipschitz solution U with the property (cid:0) (cid:0) (cid:1)(cid:1) ∂U ∈L∞ 0,T;Cα(Rd;Rd) , D2U ∈L∞ 0,T;Cα Rd;Rd⊗Rd⊗Rd . ∂t b b (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) Moreover, for large λ one has, for any (t,x)∈[0,T]×Rd, 1 |∇U(t,x)|≤ . 2 2 If X(t) is a solution of the SDE, we apply Itoˆ formula to U(t,X(t)) and get t t U(t,X(t))=U(0,x )+ LU(s,X(s))ds+ ∇U(s,X(s))dW (s) 0 Z0 Z0 where LU = ∂U +b·∇U + 1∆U. Hence, being LU =−b+λU, ∂t 2 t t U(t,X(t))=U(0,x )+ (−b+λU)(s,X(s))ds+ ∇U(s,X(s))dW (s) 0 Z0 Z0 and thus t t b(s,X(s))ds=U(0,x )−U(t,X(t))+ λU(s,X(s))ds 0 Z0 Z0 t + ∇U(s,X(s))dW (s). Z0 In other words, we may rewrite the SDE as t X(t)=x +U(0,x )−U(t,X(t))+ λU(s,X(s))ds 0 0 Z0 t + ∇U(s,X(s))dW (s)+W (t). Z0 The advantage is that U is twice more regular than b and ∇U is once more regular. All terms in this equation are at least Lipschitz continuous. From the new equation satisfied by X(t) it is easy to prove uniqueness, for instance. But, arguing a little bit formally, it is also clear that we have differentiability of X(t) with respect to the initial condition x . Indeed, if 0 D X(t) denotes the derivative in the direction h, we (formally) have h D X(t)=h+D U(0,x )−∇U(t,X(t))D X(t) h h 0 h t + λ∇U(s,X(s))D X(s)ds h Z0 t + D2U(s,X(s))D X(s)dW (s). h Z0 Alltermsaremeaningful(forinstancethetensorvaluedcoefficientD2U(s,X(s)) is bounded continuous), ∇U(t,X(t)) has norm less than 1/2 (hence the term ∇U(t,X(t))D X(t) contracts) and one can prove that this equation has a h solution D X(s). Along these lines one can build a rigorous proof of differen- h tiability. We do not discuss the other properties. Remark 3 A main open problem is the case when b is random:b = b(ω,t,x). In this case, strong uniqueness statements of the previous form are unknown (when b is not regular). 3 1.2 The PDE case WehaveseenthatnoiseimprovesthetheoryofODEs. IsitthesameforPDEs? We have several more possibilities, several dichotomies: linear ր equations: ց non linear uniqueness (weak solutions) ր problems: ց blow-up (regular solutions) additive (like for ODEs) ր noise: ց bilinear multiplicative. Let us deal with two of the simplest but not trivial combinations: linear trans- portequations,boththeproblemofuniqueness ofweakL∞ solutionsandofno blow-up ofC1-solutions,theimprovementsofthedeterministictheoryproduced by a bilinear multiplicative noise. The linear deterministic transport equation is the first order PDE in Rd ∂u +b·∇u=0, u| =u t=0 0 ∂t whereb:[0,T]×Rd →Rd isgivenandwelookforasolutionu:[0,T]×Rd →R. Definition 4 Assume b,divb ∈ L1 = L1 ([0,T]×Rd), u ∈ L∞ Rd . We loc loc 0 say that u is a weak L∞-solution if: (cid:0) (cid:1) i) u∈L∞ [0,T]×Rd ii) for all θ(cid:0)∈C∞ Rd (cid:1)one has 0 (cid:0) (cid:1) t u(t,x)θ(x)dx= u (x)θ(x)dx+ u(s,x)div(b(s,x)θ(x))dxds 0 ZRd ZRd Z0 ZRd Existence of weak L∞-solutions is a general fact, obtained by weak-star com- pactness methods. When b ∈ L∞ 0,T;Lip Rd;Rd , uniqueness can be b proved,andalsoexistenceofsmoothersolutionswhenu issmoother. Moreover, 0 (cid:0) (cid:0) (cid:1)(cid:1) one has the transport relation u(t,φ(t,x))=u (x) 0 4 where φ(t,x) is the deterministic flow associated to the equation of character- istics d φ(t,x)=b(φ(t,x)), φ(0,x)=x. dt When b is less than Lipschitz continuous, there are counterexamples. For in- stance, for d=1, b(x)=2sign(x) |x| the PDE has infinitely many solutions from anypinitial condition u . These 0 solutions coincide for |x|>t2, where the flow is uniquely defined, but they can be prolonged almost arbitrarily for |x|<t2, for instance setting u(t,x)=C for |x|<t2 with arbitrary C. Remarkable is the result of [5] which states that the solution is unique when (we do not stress the generality of the behavior at infinity) ∇b∈L1 [0,T]×Rd;Rd , (2) loc divb∈L1 0(cid:0),T;L∞ Rd,Rd(cid:1) . (3) There are generalizations of this r(cid:0)esult (for(cid:0)instanc(cid:1)e(cid:1)[1]), but not so far from it. In these cases the flow exists and is unique but only in a proper generalized sense. The assumption (3) is the quantitative one used to prove the estimate (for simplicity we omit the cut-of needed to localize) t u2(t,x)dx= u2(x)dx+ ds u2(s,x)divb(s,x)dx 0 ZRd ZRd Z0 ZRd t ≤ u2(x)dx+ kdivb(s,·)k ds u2(s,x)dx 0 ∞ ZRd Z0 ZRd which implies, by Gronwall lemma, u2(t,x)dx = 0 when u = 0 (this im- Rd 0 plies uniqueness, since the equation is linear). The assumption (2) apparently R hasnorolebutitisessentialtoperformthesecomputationsrigorously. Onehas to prove that a weak L∞-solution u satisfies the previous identity. In order to apply differentialcalculus to u, one can mollify u but then a remainder,a com- mutator, appears in the equation. The convergence to zero of this commutator (establishedbythesocalledcommutator lemma of[5])requiresassumption(2). We have recalled these facts since they are a main motiv below. The problemofno blow-upofC1 orW1,p solutionis openfor the determin- istic equation, under essentially weaker conditions than Lipschitz continuity of b. The equation satisfied by first derivatives v = ∂u involves derivatives of b k ∂xk as a potential term ∂v ∂b ∂u k 0 +b·∇v + v =0, v | = k i k t=0 ∂t ∂x ∂x i k i X 5 and L∞ bounds on ∂b seem necessary to control v . Again there are simple ∂xi k counterexamples: in the case d=1, b(x)=−2sign(x) |x|, p the equation of characteristics has coalescing trajectories (the solutions from ±x meet at x = 0 at time |x |) and thus, if we start with a smooth initial 0 0 conditionu suchthatatsomepointx satisfiesu (x )6=u (−x ),thenattime 0 p 0 0 0 0 0 t = |x | the solution is discontinuous (unless u is special, the discontinuity 0 0 0 appears immediately, for t>0). p Consider the following stochastic version of the linear transport equation: ∂u dW +b·∇u+σ∇u◦ =0, u| =u . t=0 0 ∂t dt ThenoiseW isad-dimensionalBrownianmotion,σ ∈R,theoperation∇u◦dW dt has simultaneously two features: it is a scalar product between the vectors ∇u and dW, and has to be interpreted in the Stratonovich sense. The noise has a dt transport structure as the deterministic part of the equation. It is like to add the fast oscillating term σdW to the drift b: dt dW b(x)−→b(x)+σ (t). dt ConcerningStratonovichcalculus andits relationwithItoˆ calculus, see [15]. We recall the so called Wong-Zakai principle (proved as a rigorous theorem in several cases): when one takes a differential equations with a smooth approx- imation of Brownian motion, and then takes the limit towards true Brownian motion, the correct limit equation involves Stratonovich integrals. Thus equa- tions with Stratonovich integrals are more physically based. Definition 5 Assume b,divb ∈ L1 , u ∈ L∞ Rd . We say that a stochastic loc 0 process u is a weak L∞-solution of the SPDE if: (cid:0) (cid:1) i) u∈L∞ Ω×[0,T]×Rd ii) for all θ(cid:0)∈ C∞ Rd , (cid:1) u(t,x)θ(x)dx is a continuous adapted semi- 0 Rd martingale (cid:0) (cid:1) R iii) for all θ ∈C∞ Rd , one has 0 (cid:0) (cid:1) t u(t,x)θ(x)dx= u (x)θ(x)dx+ u(s,x)div(b(s,x)θ(x))dxds 0 ZRd ZRd Z0 ZRd t +σ u(s,x)∇θ(x)dx ◦dW (s). Z0 (cid:18)ZRd (cid:19) The following theorem is due to [8]. 6 Theorem 6 If σ 6=0 and b∈L∞ 0,T;Cα Rd;Rd , divb∈Lp([0,T]×Rd), (4) b for someα∈(0,1)an(cid:0)dp>d∧(cid:0)2, then(cid:1)t(cid:1)hereexistsauniqueweakL∞-solutionof the SPDE. If α∈(1/2,1) then we have uniqueness only assuming divb∈L1 . loc Moreover, it holds u(t,φ(t,x))=u (x) 0 whereφ(t,x)is thestochasticflowofdiffeomorphisms associated totheequation dφ(t,x)=b(t,φ(t,x))dt+σdW (t), φ(0,x)=x given by Theorem 2. Thus we see that a suitable noise improves the theory of linear transport equation from the view-point of uniqueness of weak solutions. One of the aims of this paper is to prove a variant of this theorem, under different assumptions onb. Itrequiresanewformofcommutatorlemmawithrespecttothoseproved in [5] or [8]. Let us come to the blow-up problem. The following result can be deduced from[8,AppendixA]inwhichwehaveconsideredBV -solutionsforthetrans- loc portequation. InSection2wewillgiveadirectproofoftheexistencepartwhich is of independent interest. Theorem 7 If σ 6=0, b∈L∞(0,T;Cα(Rd;Rd)), b for some α ∈ (0,1) and u ∈ C1 Rd , then there exists a unique classical C1- 0 b solution for the transport equation with probability one. It is given by (cid:0) (cid:1) u(t,x)=u φ−1(x) (5) 0 t where φ−1 is the inverse of the stochastic(cid:0)flow φ (cid:1)=φ(t,·). t t The main claim of this theorem is the regularity of the solution for positive times,whichisnewwithrespecttothedeterministiccase. Theuniquenessclaim is known, as a particular case of a result in BV , see Appendix 1 of [8]. loc Notice that, for solutions with such degree of regularity (BV or C1), no loc assumption on divb is required; divb does not even appear in the definition of solution (see below). On the contrary, to reach uniqueness in the much wider class of weak L∞-solutions, in [8] we had to impose the additional condition (4) on divb, for some p > d∧2 (divb also appears in the definition of weak L∞-solution); this happens also in the deterministic theory. 7 1.3 Some other works on regularization by noise The following list does not aim to be exhaustive, see for instance [7] for other results and references: • the uniqueness for linear transport equations can be extended to other weakassumptionsonthe drift, [2],[16]; alsono blow-upholdsfor Lp drift see [6] and [18]; • similar results hold for linear continuity equations, [9], [17]: ∂ρ +div(bρ)=0, ρ| =ρ : t=0 0 ∂t a noise of the form ∇ρ◦ dW prevents mass concentration; dt • analog results hold for the vector valued linear equations ∂M +curl(b×M)=0 ∂t similar to the vorticity formulation of 3D Euler equations or magneto- hydrodynamics, where the singularities in the deterministic case are not shocks but infinite values of M; a noise of the form curl(e×M)◦ dW dt prevents blow-up [12]; • improved Strichartz estimates for a special Schr¨odinger model with noise dW i∂ u+∆u◦ =0 t dt have been proved, which are stronger than the corresponding ones for i∂ u+∆u = 0 and allow to prevent blow-up in a non-linear case when t blow-up is possible without noise, see [3]; • nonlinear transport type equations of two forms have been investigated: 2D Euler equations and 1D Vlasov-Poissonequations; in these cases non- collapse of measure valued solutions concentrated in a finite number of points has been proved, [11], [4]. We conclude the introduction with some notations. 1.4 Notations Usually we denote by D f the derivative in the i-th coordinate direction and i with (e ) the canonical basis of Rd so that D f = e ·Df. For partial i i=1,...,d i i derivatives of any order n ≥ 1 we use the notation Dn . If η : Rd → Rd i1,...,in is a C1-diffeomorphism we will denote by Jη(x) = det[Dη(x)] its Jacobian determinant. Foragivenfunctionf dependingont∈[0,T]andx∈Rd,wewill also adopt the notation f (x)=f(t,x). t 8 LetT >0be fixed. Forα∈(0,1)define the spaceL∞ 0,T;Cα(Rd) asthe b set of all bounded Borel functions f :[0,T]×Rd →R for which (cid:0) (cid:1) |f(t,x)−f(t,y)| [f] = sup sup <∞ α,T |x−y|α t∈[0,T]x6=y∈Rd (|·| denotes the Euclidean norm in Rd for every d, if no confusion may arise). This is a Banach space with respect to the usual norm kfk = kfk + α,T 0 [f] where kfk = sup |f(t,x)|. Similarly, when α = 1 we de- α,T 0 (t,x)∈[0,T]×Rd fine L∞ 0,T;Lip (Rd) . b We write L∞ 0,T;Cα(Rd;Rd) for the space of all vector fields f :[0,T]× (cid:0) (cid:1)b Rd →Rd having all components in L∞ 0,T;Cα(Rd) . (cid:0) (cid:1) b Moreover, for n ≥ 1, f ∈ L∞ 0,T;Cn+α(Rd) if all spatial partial deriva- (cid:0) b (cid:1) tives Dk f ∈ L∞ 0,T;Cα(Rd) , for all orders k = 0,1,...,n. Define the i1,...,ik b (cid:0) (cid:1) corresponding norm as (cid:0) (cid:1) n kfk =kfk + kDkfk +[Dnf] , n+α,T 0 0 α,T k=1 X whereweextendthepreviousnotationsk·k and[·] totensors. Thedefinition 0 α,T of the space L∞ 0,T;Cn+α(Rd;Rd) is similar. The spaces Cn+α(Rd) and b b Cn+α(Rd;Rd) are defined as before but only involve functions f : Rd → Rd b (cid:0) (cid:1) which do not depend on time. Moreover, we say that f : Rd → Rd belongs to Cn,α, n ∈ N, α ∈ (0,1), if f is continuous on Rd, n-times differentiable with all continuous derivatives and the derivatives of order n are locally α-H¨older continuous. Finally, C0(Rd) denotes the space of all real continuous functions 0 defined on Rd, having compactsupport and by C∞(Rd) its subspace consisting 0 of infinitely differentiable functions. For any r >0 we denote by B(r) the Euclidean ball centered in 0 of radius r andbyC∞(Rd)the spaceofsmoothfunctions with compactsupportinB(r); r moreover,k·kLpr andk·kWr1,p standfor,respectively,theLp-normandtheW1,p- normonB(r),p∈[1,∞]. Weletalso[f] =sup |f(x)−f(y)|/|x−y|θ. Crθ x6=y∈B(r) We will often use the standard mollifiers. Let ϑ : Rd → R be a smooth test function such that 0 ≤ ϑ(x) ≤ 1, x ∈ Rd, ϑ(x) = ϑ(−x), ϑ(x)dx = 1, Rd supp(ϑ)⊂B(2),ϑ(x)=1whenx∈B(1). Foranyε>0,letϑ (x)=ε−dϑ(x/ε) εR and for any distribution g : Rd →Rn we define the mollified approximation gε as gε(x)=ϑ ∗g(x)=g(ϑ (x−·)), x∈Rd. (6) ε ε If g depends also on time t, we consider gε(t,x) = (ϑ ∗g(t,·))(x), t ∈ [0,T], ε x∈Rd. Recallthat,foranysmoothboundeddomainDofRd,wehave: f ∈Wθ,p(D), θ ∈(0,1), p≥1, if and only if f ∈Lp(D) and |f(x)−f(y)|p [f]p = dxdy <∞. Wθ,p |x−y|θp+d ZZD×D 9 We have W1,p(D)⊂Wθ,p(D), θ ∈(0,1). Inthesequelwewillassumeastochasticbasiswithad-dimensionalBrownian motion (Ω,(F ),F,P,(W )) to be given. We denote by F the completed σ- t t s,t algebra generated by W −W , s≤r ≤u≤t, for each 0≤s<t. u r Let us finally recall our basic assumption on the drift vector field. Hypothesis 1 There exists α∈(0,1) such that b∈L∞ 0,T;Cα(Rd;Rd) . b (cid:0) (cid:1) 2 No blow-up in C1 This section is devoted to prove Theorem 7. Since the solution claimed by this theorem is regular, we do not need to integrate over test functions in the term b·∇u and thus we do not need to require divb ∈ L1 . For this reason, we loc modify the definition of solution. Definition 8 Assumeb∈L1 , u ∈C1 Rd . We say that a stochastic process loc 0 b u ∈ L∞(Ω×[0,T]×Rd) is a classical C1-solution of the stochastic transport (cid:0) (cid:1) equation if: i) u(ω,t,·)∈C1(Rd) for a.e. (ω,t)∈Ω×[0,T]; ii) for all θ ∈ C∞ Rd , u(t,x)θ(x)dx is a continuous adapted semi- 0 Rd martingale; (cid:0) (cid:1) R iii) for all θ ∈C∞ Rd , one has 0 (cid:0) (cid:1) t u(t,x)θ(x)dx= u (x)θ(x)dx− b(s,x)·∇u(s,x)θ(x)dxds 0 ZRd ZRd Z0 ZRd t +σ u(s,x)∇θ(x)dx ◦dW (s). Z0 (cid:18)ZRd (cid:19) IfuisaclassicalC1-solutionanddivb∈L1 [0,T]×Rd ,thenuisalsoaweak loc L∞-solution. Conversely, if u is a weak L∞-solution, u ∈ C1 Rd and (i) is (cid:0) 0(cid:1) b satisfied then u is a classical C1-solution. (cid:0) (cid:1) Before giving the proof we mention the following useful result proved in [8, Theorem 5]: Theorem 9 Assume that Hypothesis 1 holds true for some α ∈ (0,1). Then we have the following facts: (i) (pathwise uniqueness)For every s∈[0,T], x∈Rd, thestochastic equation (1) has a unique continuous adapted solution Xs,x = Xs,x ω ,t∈[s,T], t ω ∈Ω . (cid:0) (cid:0) (cid:1) (ii) (differ(cid:1)entiable flow) There exists a stochastic flow φ of diffeomorphisms s,t for equation (1). The flow is also of class C1+α′ for any α′ <α. 10