Kinetic Approach to Stochastic Conservation Laws

Transcription

1 Kinetic Approach to Stochastic Conservation Laws Martina Hofmanová Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 1623 Berlin web: July 3, 215

2 Contents Chapter 1. What is the right notion of solution? Method of characteristics Linear case Nonlinear case Inviscid Burgers equation in 1D Weak solutions Nonuniqueness Well-posedness for entropy solutions Kinetic formulation Examples of kinetic defect measure Burgers equation Linear case 12 Chapter 2. Stochastic conservation laws Kinetic formulation The concept of Young measures Generalized kinetic solution Existence Energy estimate Estimate of the kinetic measures Estimate of the Young measures Uniqueness 25 Bibliography 29 i

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4 CHAPTE 1 What is the right notion of solution? In this chapter, we will study the following deterministic scalar equation (1.1) t u + div B(u) =, u() = u. Here t + = [, ), x N. The coefficient B = (B 1,..., B N ) : N called the flux function and the flux term div B(u) is defined as is N N div B(u) = xj B j (u) = B j(u) xj u. j=1 For notational simplicity we denote b = (b 1,..., b N ) = (B 1,..., B N ). The solution u : + N is a scalar function and equation (1.1) is a conservation law for u, i.e. it describes conservation of the quantity u. Why u is conserved can be seen easily: let N = 1 and let us calculate how the amount of u inside the interval [a, b] changes in time. It holds d dt b a u(t, x) dx = b a t u(t, x) dx = b a j=1 x B(u) dx = B(u(t, a)) B(u(t, b)) and therefore we see that the amount of u inside [a, b] changes only via the inflow and outflow across the boundary of [a, b]. Conservation laws represent the very basic physical phenomena such as balance of mass, energy etc. Our main interest lies in solving (1.1), that is, we want to prove existence, uniqueness of a solution as well as continuous dependence on the initial data. In Chapter 2, we will add some stochastic perturbation on the right hand side of (1.1) which would further complicate the problem. However, as our stochastic perturbation will be rather general and the deterministic case will still be included as a special case, it is necessary to understand the difficulties in solving (1.1) first. To this end, let us start with some particular examples Method of characteristics One of the basic techniques for solving PDEs is the method of characteristics. It is especially useful for first order equations. The main idea is to reduce the PDE at hand to a system of ODEs that can be solved by elementary techniques. Therefore, one has to search for curves along which the PDEs rewrites as ODEs Linear case. Let us restrict ourselves to N = 1 and let us consider (1.1) with the flux function B being linear, i.e. B(u) = cu for some c. Then 1

5 2 1. WHAT IS THE IGHT NOTION OF SOLUTION? (1.1) rewrites as (1.2) t u + c x u =, u() = u. In that case, the characteristic curve is governed by ˆx(t) dt hence the solution starting from x is given by ˆx(t, x) = x + ct. We have the following result. Lemma Let u(t, x) := u (x ct). If u C 1 (), then u C 1 ( + ) and it is a classical solution to (1.2). The proof is simple and follows immediately from the chain rule formula. It follows that in this case, (1.1) possesses an explicit solution and it is the so-called travelling wave. Indeed, the shape of does not change in time as the solution is given by translation of the initial profile u() = u. This behavior is particular for the linear case and as we will see later, the main difficulties in conservation laws arise in nonlinear cases where new features appear Nonlinear case. Let us keep the assumption N = 1 and let b = B be nonlinear. (1.1) rewrites as (1.3) = c t u + b(u) x u =, u() = u. Following the method of characteristics, we obtain the following formula for the characteristic curve ˆx(t) = b(u(ˆx(t)). dt Unlike the previous example, where the speed was constant, here it changes in time and also according to position in space. Consequently, also the shape that we see at time t differs from the one at time and moreover shocks may appear. What else can we deduce? Lemma Let u C 1 ( + ) be a solution to (1.3). Then u is constant along characteristics. Accordingly, characteristics are straight lines. Proof. To show the first claim, we calculate: d dt u(t, ˆx(t)) = tu + x u dˆx(t) = t u + x u b(u) = dt which implies that u(t, ˆx(t, x)) = u(, ˆx(, x)) = u (x) for all t +. Characteristics are then given by ˆx(t, x) = x + and the proof is complete. t b(u(s, ˆx(s, x))) ds = x + b(u (x))t Let us now show some of the consequences of the above result.

6 1.1. METHOD OF CHAACTEISTICS Inviscid Burgers equation in 1D. Inviscid Burgers equation is an example of (1.1), where the flux function is given by B(u) = 1 2 u2, it reads as (1.4) t u + u x u =, u() = u. Using Lemma 1.1.2, the characteristic lines are given by ˆx(t, x) = x + u (x)t hence they are determined by the initial condition u. For certain initial conditions this gives a method of solving (1.4) but for other not. Let u. Then one can draw the characteristic lines and identify the solution u at each point (t, x), see Figure 1. Figure 2 shows the characteristic lines corresponding to a nondecreaing piecewise constant initial condition. t t u (x) x u (x) x x x ˆx(t) Figure 1 ˆx(t) Figure 2 u (x) x u (x) x x x Figure 3 In fact, it is a generic problem. Figure 4 Proposition If B > and u is not monotone increasing then (1.3) does not have a smooth solution. It follows from the above considerations that we have to extend our notion of solution. Indeed, the notion of classical solution is apparently too strong as already for some simple examples such solutions do not exist. The difficulties are coming from the nonlinearity in the flux but also from absence of second order terms that could provide smoothing. Consequently, we loose regularity even for arbitrarily smooth initial conditions and flux functions and it is necessary to work within the class of discontinuous functions and to consider distributional solutions.

7 4 1. WHAT IS THE IGHT NOTION OF SOLUTION? 1.2. Weak solutions In this part, we weaken the notion of solution and study its properties. Definition A measurable function u : + N is called a weak solution to (1.1) if for every test function ϕ C 1 ( N ) it holds ( N ) u t ϕ + B j (u) xj ϕ dx dt + u (x) ϕ(, x) dx =. N N j=1 Clearly, weak solution does not have to be smooth or not even continuous. Let us now derive some conditions that have to be satisfied at points of jumps. Theorem (ankine-hugoniot condition). Let N = 1. Let N be an open neighborhood in +. Suppose that a curve C : t (α, β) ˆx(t) divides N into two pieces N l and N r. Let u be a weak solution to (1.3) such that (i) u is a classical solution in both N l and N r, (ii) u has a jump discontinuity at C, (iii) the jump is continuous along C. For every p C, let s = ˆx (p) be the slope of C at p. Then the following relation holds true s[u] = [B(u)], provided [u] = u r u l denotes the size of the jump of u and similarly [B(u)] = B(u r ) B(u l ) is the size of the jump of B(u). Proof. Let ϕ Cc 1 ( ) be a test function whose support is included in N. Since u is a weak solution, we have using the Green s identity = u t ϕ + B(u) x ϕ dx dt N l = u t ϕ + B(u) x ϕ dx dt + u t ϕ + B(u) x ϕ dx dt N N = u t ϕ + B(u) x ϕ dx dt u l ϕˆx dγ + B(u l )ϕ dγ N l C C u t ϕ + B(u) x ϕ dx dt + u r ϕˆx dγ B(u r )ϕ dγ. N r C C The first and the fourth term on the right hand side are equal to due to assumption (i). Summing the remaining terms we obtain and the claim follows. = β α r ( [u]ˆx [B(u)] ) ϕ dt Nonuniqueness. In this part we will show that the concept of weak solution is in general not enough to guarantee uniqueness. Let us continue with the example of inviscid Burgers equation in 1D. We consider a piecewise constant initial condition: let u l, u r, u l u r, and set { u l, x <, u (x) = u r, x >.

8 1.2. WEAK SOLUTIONS 5 The ankine-hugoniot condition in this case reads as hence necessarily s(u + u ) = (u +) 2 2 (u ) 2 2 = 1 2 (u + u )(u + + u ) s = u + + u 2 whenever a weak solution u jumps from u to u + over a curve with slope s. Therefore, defining (1.5) u(t, x) = { u l, x < u l+u r 2 t, u r, x > u l+u r 2 t, we obtain a weak solution that fulfils the ankine-hugoniot condition. On the other hand, setting u l, x < s 1 t, a, s 1 t < x <, (1.6) u(t, x) = a, < x < s 2 t, u r, x > s 2 t, with s 1 = u l a, s 2 = u r + a, a > max{u l, u r } 2 2 defines the whole family of weak solutions u a with the same initial condition and satisfying the ankine-hugoniot condition. And therefore uniqueness does not hold true and some non-physical solutions might appear. Therefore, we have to look for a criterion that, on the one hand, would ensure uniqueness and, on the other hand, it would select the correct physical solution. Our first idea is to consider an additional conservation law that is satisfied by any smooth solution and that could play the role of a selector. To be more precise, we want to find condition on functions η : and q = (q 1,..., q N ) : N such that any smooth solution u automatically satisfies (1.7) t η(u) + div q(u) = To this end, let u be a classical solution to (1.1). To derive the new conservation law, let us multiply (1.1) by η (u) and use the chain rule formula. It leads to N t η(u) + η (u)b j(u) xj u = j=1 and therefore setting q j ( ) = η ( )B j ( ) for all j = 1,..., N we obtain (1.7). Let us continue with our investigation. For instance, what are the consequences of the ankine-hugoniot condition for the new conservation law (1.7)? If u is a piecewise smooth solution of (1.7), that is, (1.7) is satisfied only in the corresponding subdomains, the N-dimensional version of ankine-hugoniot condition reads as follows N s[η(u)] = ν j [q j (u)] j=1 where n = ( s, ν 1,..., ν N ) is the ourward normal that is normalized in such a way that (ν 1,..., ν N ) = 1. The problem here is that, in general, ankine-hugoniot

9 6 1. WHAT IS THE IGHT NOTION OF SOLUTION? condition for (1.7) is not compatible with the original ankine-hugoniot condition for (1.1). The set of all the conditions is just too restrictive. Another idea (that in the end will fix the above flaw) is that any physically reasonable solution should be a limit of viscous approximation of the form (1.8) t u ε + div B(u ε ) = ε u ε, u ε () = u. In other words, we added a parabolic perturbation, i.e. a second order term that depends on a small parameter ε, we want to pass to the limit as ε and in the limit we should see the correct physical solution of (1.1). To solve the approximation (1.8) we can use the classical theory for parabolic PDEs which is very well developed. Let us now multiply (1.8) by η (u ε ), the same way as we did with (1.1) above. The chain rule now yields N t η(u ε ) + η (u ε )B j(u ε ) xj u ε = εη (u ε ) u ε j=1 which can further rewritten as t η(u ε ) + div q(u ε ) = ε η(u ε ) εη (u ε ) u ε 2 Assume now that η C 2 () is convex. Consequetly, and we obtain εη (u ε ) u ε 2 t η(u ε ) + div q(u ε ) = ε η(u ε ). The above inequality can be now considered as a parabolic perturbation of the inequality (1.9) t η(u ε ) + div q(u ε ) and this finally gives the right additional conditions that ensure uniqueness and select the correct physical solution among all weak solution. Definition Let η C 1 () be a convex function. If there exists q j C 1 () such that for all v (1.1) η (v)b j(v) = q j(v), j = 1,..., N, then (η, q) is called an entropy-entropy flux pair of the conservation law (1.1). emark The structure condition (1.1) only becomes important when dealing with systems of conservation laws. In the case of scalar conservation, q j can be easily defined by the left hand side of (1.1) and therefore (1.1) does not impose any further restriction on η, B. Finally we have all in hand to define the notion of entropy solution. Definition A weak solution to (1.1) is called an entropy solution if for every entropy-entropy flux pair (η, q) the so-called entropy inequality (1.9) is fulfilled in the following sense: for all ϕ D( N ), ϕ, it holds ( N ) η(u) t ϕ + q j (u) xj ϕ dx dt + η(u )ϕ() dx. N N j=1

10 1.3. WELL-POSEDNESS FO ENTOPY SOLUTIONS 7 emark To establish uniqueness, it is necessary to have all the entropy inequalities corresponding to all the admissible entropy-entropy flux pairs (η, q). Let us now go back to our example of one-dimensional inviscid Burgers equation (1.4) and verify if our selection criterion applies. Since B(u) = u 2, the structure condition (1.1) reads as q (u) = η (u)u. ecall that if a solution jumps over a straight line from U L to U then due to ankine-hugoniot condition for (1.4) the slope has to satisfy s = U L+U 2. The ankine-hugoniot condition for the entropy inequalities (1.9) is s[η(u)] [q(u)] hence we obtain 1 2 (U L + U ) ( η(u ) η(u L ) ) U λη (λ) dλ. U L Let η(u) = (u c ) 2 where c = U L+U 2. Then the above left hand side vanishes and we get 1 6 (U U L ) 3. Consequently we deduce that no jumps of the form U L < U are allowed. egarding the solutions constructed in (1.5) and (1.6) we obtain that the only suitable solution is the one from (1.5) with u l u r. Exercise What does the above statement that no jumps of the form U L < U are allowed suggest about the possible definition of the solution in the region not determined by characteristics in Figure 2? 1.3. Well-posedness for entropy solutions In this section, we present a classical well-posedness result for entropy solutions that is due to Kružkov [6]. Theorem Let b i = B i L loc () and u L 1 ( N ) L ( N ). Then there exists a unique entropy solution u C( + ; L 1 ( N )) L ( + N ). Moreover, if u 1, u 2 are two entropy solutions corresponding to initial conditions u 1,, u 2,, respectively, then the following L 1 -contraction property holds true u 1 (t) u 2 (t) L 1 x u 1, u 2, L 1 x t +. This is a classical result that dates back to 197. Concerning uniqueness, one needs all the entropy inequalities except for the case of one-dimensional conservation law with convex (or concave) fluxes and BV solutions. We skip the proof of this result and continue with our discussion about possible concepts of solution. Later on, we prove in detail a well-posedness result in the stochastic setting that basically includes Theorem (1.3.1) as a special case. Since the L 1 -contraction property gives continuous dependence on the initial condition in L 1, it is natural to ask whether it is possible to establish a pure L 1 - well-posedness theory. Namely, to consider unbounded initial conditions in L 1 ( N ). Here it turns out that the concept of entropy solutions is not sufficient. To be more precise, let us for instance consider a quadratic flux function B, such as in the Burgers equation. If u() L 1 ( N ) then B(u) is not even locally integrable in

11 8 1. WHAT IS THE IGHT NOTION OF SOLUTION? the space variable so div B(u) is not a well-defined distribution. In other words, the equation (1.1) does not make sense as distributional equation and therefore Definition is not meaningful. The same problem would of course hold true for the entropy inequalities where η, q does not necessarily have linear growth. To overcome this difficulties, we introduce a new (and our last one) notion of solution, the so-called kinetic solution. It was first introduced by Lions, Perthame, Tadmor [7] and relies on a new equation, the so-called kinetic formulation, that is derived from the conservation law at hand and that (unlike the original problem) possesses a very important feature - linearity. The two notions of solution, i.e. entropy and kinetic, are equivalent whenever both of them exist, nevertheless, kinetic solutions are more general as they are well defined even in situations when neither the original conservation law nor the corresponding entropy inequalities can be understood in the sense of distributions Kinetic formulation Let us first fix some notation. For ξ and u we define the equilibrium function by 1, < ξ < u, χ u (ξ) = 1, u < ξ <,, otherwise. The above can be rewritten as We have the following easy lemma. χ u (ξ) = 1 <ξ<u 1 u<ξ< = 1 u>ξ 1 >ξ. Lemma Let S : be locally Lipschitz continuous and assume that S L loc (). Then and S (ξ)χ u (ξ) dξ = S(u) S() χ u (ξ) χ v (ξ) dξ = u v. Proof. Both claims follow directly from the definition of the equilibrium function. Theorem Let u C( + ; L 1 ( N )) L ( + ). Then u is an entropy solution to (1.1) if and only if there exists a nonnegative bounded measure m(t, x, ξ) such that t χ u + b(ξ) x χ u = ξ m, (1.11) χ u() = χ u, holds true in D ( + t N x ξ ). Equation (1.11) is called the kinetic formulation of (1.1) and it is the keystone for the notion of kinetic solution that will be introduced soon. This equation is posed in the extended space - it includes a new variable ξ that basically corresponds to the values of the solution u. For that reason is ξ usually called velocity. The measure m is the kinetic defect measure and it is not known in advance, it becomes part of the solution. The main feature of the kinetic formulation is its linearity. In

12 1.4. KINETIC FOMULATION 9 fact, it is a linear transport equation for χ u but clearly the dependence on u is still nonlinear. emark Considering just the left hand side of (1.11) we obtain a linear transport equation t f + b(ξ) f =. That can be solved by the method of characteristics introduced in Section 1.1. (emember that the problems came only with nonlinear equations!) Nevertheless, we are only interested in solutions of the form f = χ u. Therefore, posing this constraint, the kinetic measure m can be regarded as the corresponding Lagrange multiplier. In the case of existence of a classical solution to (1.1), we have the following equivalence which is proved directly by the chain rule formula. Proposition Let u C 1 ( + N ). Then u is a classical solution to (1.1) if and only if it satisfies (1.11) with m. This result further clarifies the name kinetic defect measure. Indeed, the measure m takes account of possible singularities and therefore vanishes for smooth solutions. Proof of Theorem In order to prepare for the proof, let us consider u C( + ; L 1 ( N )) L ( + ) and define the function ξ χ u(t,x) (ζ) dζ C( + ; L 1 loc( N+1 )) and the distribution ξ N ξ (1.12) m(t, x, ξ) = t χ u(t,x) (ζ) dζ + xi b i (ζ)χ u(t,x) (ζ) dζ. Therefore, in D ( + N ) we have i=1 t χ u + b(ξ) x χ u = ξ m. Note that so far u does not have to solve (1.1) in any sense. We can always define a distribution m such that the (1.11) holds true, the nontrivial fact is that m is a nonnegative and bounded measure. Let us now multiply the above by S D() and integrate with respect to ξ. We obtain (1.13) t S (ξ)χ u (ξ) dξ + N xi i=1 S (ξ)b i (ξ)χ u (ξ) dξ = S (ξ)m(t, x, ξ) dξ. The function S will eventually play the role of an entropy. Hence with Lemma and the structure condition q (u) = S (u)b(u) we deduce (1.14) t S(u) + N i=1 xi q i (u) = S (ξ)m(t, x, ξ) dξ. Having these preparations in hand, let us proceed with the proof.

13 1 1. WHAT IS THE IGHT NOTION OF SOLUTION? Step 1: Let u satisfy (1.11) with a nonnegative bounded measure m. Then we have (1.14) for those S such that S D(). By truncation, it is possible to extend the validity of (1.14) to S C 2 () subquadratic, i.e. S(ξ) C(1 + ξ 2 ). Hence for such S that are in addition convex, we obtain the desired entropy inequality N t S(u) + xi q i (u). i=1 If S is not subquadratic, we make use of the boundedness of u. In particular, we go back to (1.13) and observe that since u L ( + N ), the equilibrium function is compactly supported with supp χ u ( ) [ u L x, u L x ]. Consequently, the integration in (1.13) only runs over this interval and therefore it does not matter if we change S for large values of ξ to make it subquadratic. Finally, we conclude that u satisfies all the entropy inequalities and therefore it is an entropy solution. Step 2: Conversely, let u be an entropy solution to (1.1). Then for all S C 2 () convex, it holds (1.15) t N S(u(t, x)) dx. Indeed, since the entropy inequalities hold true in the sense of distributions, let us test by ϕ (x) = ϕ( x ) where ϕ is a smooth cut-off function. It leads to t S(u)ϕ (x) dx q(u) ϕ (x) dx. N N Since both S(u) and q(u) are integrable with respect to x we may use the dominated convergence theorem to pass to the limit as and obtain (1.15). As a consequence, it follows for every p [1, ] that u(t) L p x u L p x. Moreover, if ξ > u L x then from (1.12), the definition of the equilibrium function and the fact that u solves (1.1) m(ξ) = t u L x = t u + N u L x χ u (ζ) dζ + xi b i (ζ)χ u (ζ) dζ N xi B i (u) =. i=1 Hence again if S C 2 () is convex, it can be modified for large values of ξ to be compactly supported and (1.14) yields S (ξ)m(t, x, ξ) dξ and accordingly m is nonnegative. i=1

14 1.5. EXAMPLES OF KINETIC DEFECT MEASUE 11 It remains to show that m is a bounded measure. We take S(ξ) = ξ2 2 it in (1.14), integrate with respect to x and t and obtain and plug + N+1 m dt dx dξ 1 2 u 2 L 2 x which completes the proof. Finally, we have all in hand to define the notion of kinetic solution. Definition u C( + ; L 1 ( N )) is called a kinetic solution to (1.1) if there exists a nonnegative bounded measure m such that the kinetic formulation (1.11) is satisfied in the sense of distributions over + N. According to Theorem we know that for u C( + ; L 1 ( N )) L ( + N ) it is equivalent to be entropy and kinetic solution. However, the concept of kinetic solution does not require boundedness and therefore is more suitable to deal with initial conditions u L 1 ( N ). The following well-posedness result can be found in [7]. Theorem Let u L 1 ( N ) and b L loc (). Then there exists a unique kinetic solution to (1.1) and u C( + ; L 1 ( N )). Moreover, if u 1, u 2 are two entropy solutions corresponding to initial conditions u 1,, u 2,, respectively, then the following L 1 -contraction property holds true u 1 (t) u 2 (t) L 1 x u 1, u 2, L 1 x t Examples of kinetic defect measure Burgers equation. Let us continue with our example of one-dimensional Burgers equation, in particular with the solution constructed in (1.5), and let us calculate its kinetic defect measure. Due to (1.12) we shall calculate t χ u + ξ x χ u in the sense of distributions over +. Let ϕ D( + ). Then using Green s identity and the fact that u is piecewise constant, we have 1 t χ u, ϕ = χ u t ϕ dξ dx dt x ξ = χ ul t ϕ dξ dx dt χ ur t ϕ dξ dx dt and similarly = So we obtain that and ξ ξ {x<st} {x=st} ξ χ ul ϕ( s) dξ dx dt = ( s)(χ ur χ ul )δ x=st, ϕ ξ {x>st} {x=st} ξ x χ u, ϕ = ξ(χ ur χ ul )δ x=st, ϕ. ξ m = (ξ s)(χ ur χ ul )δ x=st m(t, x, ξ) = δ x=st ξ (ζ s)(χ ur χ ul ) dζ. 1 By, we denote the duality between distributions and test functions. χ ur ϕ s dξ dx dt

15 12 1. WHAT IS THE IGHT NOTION OF SOLUTION? In particular, we see that m is indeed supported by the shock. Besides, it can be shown that m(t, x, ξ) = m(ξ)δ x=st sgn(u r u l ) where =, ξ min{u l, u r } m(ξ) =, ξ max{u l, u r } <, otherwise. Hence again we observe the contradiction for the case of u l < u r and these jumps are not allowed Linear case. Let us now consider the linear case B(u) = cu, c, and let { u l, x < st, u(t, x) = u r, x > st, be a piecewise constant solution. From ankine-hugoniot condition we deduce that s(u r u l ) = c(u r u l ) so s = c. The computation of the kinetic measure from Subsection applies and we obtain that m. Therefore, in this case u and χ u satisfy the same equation and the kinetic measure vanishes even though the solution is not continuous.

16 CHAPTE 2 Stochastic conservation laws Finally, we get to the original goal and that is the study of scalar conservation laws with stochastic forcing. Namely, we are interested in the following problem (2.1) du + div B(u) dt = Φ(u) dw, u() = u. Here we restrict ourselves to a finite time interval [, T ] with T > and consider periodic boundary conditions: x T N where T N = [, 1] N is the N-dimensional torus. As before, we have a flux function B = (B 1,..., B N ) : N, we denote b = B and, in addition, we assume that B is of class C 2 and b has a polynomial growth. Let us fix a stochastic basis (Ω, F, (F t ) t, P) with a complete right-continuous filtration. Let P denote the predictable σ-algebra on Ω [, T ] associated to (F t ). That is, P is generated by sets of the form A {} where A F and A (s, t] where s < t T and A F s. ecall that P is generated by left-continuous and adapted processes. The initial datum may be random in general so it is F -measurable and we assume that u L p (Ω; L p (T N )) for all p [1, ). In fact, this assumption can be immediately weakened to the L p -integrability for p p and p would be determined by the growth of b. However, comparing this to our discussion in Chapter 1 we would like to establish well-posedness also for initial conditions in L 1 which is the natural set-up for kinetic solutions. This is indeed possible and the proof then consists of two parts. First, we assume the L p -integrability as above and in the second step we take suitable approximation of the initial condition and pass to the limit. We will only focus on the first part, the proof of the second one can be found in [2]. Note that in the stochastic setting, due to the active white noise term, the solution is never bounded in ω and therefore the boundedness assumption is replaced by L p -integrability. The driving process W is a cylindrical Wiener process in a separable Hilbert space U, which is formally given by W (t) = k 1 β k (t)e k where (e k ) k 1 is a complete orthonormal system in U and (β k ) k 1 are mutually independent real-valued Wiener processes relative to (F t ). ecall that this sum does not converge in any good probabilistic sense hence some suitable conditions upon the coefficient Φ are needed so that the stochastic integral in (2.1) makes sense. egarding W itself, in order to get a process with continuous trajectories, 13

17 14 2. STOCHASTIC CONSEVATION LAWS one introduces an auxiliary space U U via { U = v = α k e k ; α 2 } k k 2 <, k 1 k 1 endowed with the norm v 2 U = k 1 α 2 k k 2, v = k 1 α k e k. Note that the embedding U U is Hilbert-Schmidt. Moreover, trajectories of W are P-a.s. in C([, T ]; U ) (see [1]). The assumptions upon the coefficient Φ are the following. If u L 2 (T N ) than Φ(u) is a mapping from U to L 2 (T N ) and it is defined by Φ(u)e k = g k (, u( )) where g k C(T N ) satisfy (2.2) (2.3) G 2 (x, ξ) := g k (x, ξ) 2 C(1 + ξ 2 ), k 1 g k (x, ξ) g k (y, ζ) 2 C( x y 2 + ξ ζ 1+α ), k 1 for some α > and for all x, y T N, ξ, ζ. Exercise Check that under the above assumptions Φ maps L 2 (T N ) to L 2 (U; L 2 (T N )), the space of Hilbert-Schmidt operators from U to L 2 (T N ). Consequently, given a predictable process u L 2 (Ω [, T ], P, dp dt; L 2 (T N )), the stochastic integral in (2.1) is a well-defined process taking values in L 2 (T N ) and it holds Φ(u) dw = t Φ(u) dβ k = t g k (u(s)) dβ k. k 1 k 1 Later on, we will consider our equation in the sense of distributions. In that case we have for a test function ϕ C (T N ) Φ(u) dw, ϕ = t gk (u(s)), ϕ dβ k. k Kinetic formulation The aim of the present section is to introduce the notion of kinetic solution to (2.1). ecall, that in the deterministic case this was based on the so-called kinetic formulation of the corresponding conservation law (2.4) t u + div B(u) = on + N. The kinetic formulation is an equation satisfied by the equilibrium function χ u = 1 u>ξ 1 >ξ and reads as follows t χ u + b(ξ) χ u = on + N ξ. Its includes the additional variable ξ and is solved in the sense of distributions D ( + N ξ ). egarding derivation of the kinetic formulation, if u C 1 ( + N ) is a classical solution to (2.4), then it is enough to multiply (2.4) by δ u=ξ and use the chain rule formula. Indeed, at least formally, t 1 u>ξ = δ u=ξ t u

18 2.1. KINETIC FOMULATION 15 and the similarly for the space derivatives. In the stochastic setting, all the calculations of this form have to go through the Itô formula. To make everything rigorous, let us assume that u C([, T ]; C 1 (T N )) a.s. is a solution to (2.1). In this case, the conservation law (2.1) is even satisfied pointwise, that is, for every x T N and t [, T ] it holds true a.s. (2.5) u(t, x) = u (x) t div B(u(s, x)) ds + k 1 t g k (x, u(s, x)) dβ k (s). In the sequel, we denote by, ξ the duality between D( ξ ) and Cc ( ξ ). In order to derive the equation satisfied by 1 u>ξ in the sense of distributions, we will consider a test function θ Cc () and derive an equation satisfied by 1 u(t,x)>ξ, θ ξ. Since 1 u>ξ, θ ξ = u(t,x) θ (ξ) dξ = θ(u(t, x)), writing down the equation for 1 u(t,x)>ξ, θ ξ is the same as applying the Itô formula to (2.5) and the function θ. Note that at this point it was important to have the equation (2.5) satisfied pointwise. We obtain [ d 1 u(t,x)>ξ, θ ξ = θ (u(t, x)) div B(u(t, x)) dt + ] g k (x, u(t, x)) dβ k k θ (u(t, x)) k 1 g(x, u(t, x)) 2 dt. Every term on the right hand side should be rewritten as some distribution applied to the test function θ. ( u ) θ (u) div B(u) = θ (u)b(u) u = div b(ξ)θ (ξ) dξ = div b1 u>ξ, θ ξ = b 1 u>ξ, θ ξ, θ (u)g k (u) = g k δ u=ξ, θ ξ, Altogether, θ (u)g 2 (u) = G 2 δ u=ξ, θ ξ = ξ (G 2 δ u=ξ ), θ ξ d 1 u(t,x)>ξ, θ ξ = b 1 u>ξ, θ ξ dt + k 1 g k δ u=ξ, θ ξ dβ k 1 2 ξ(g 2 δ u=ξ ), θ ξ dt and denoting f = 1 u>ξ we may finally read off the kinetic formulation of (2.1) (2.6) df + b f dt = δ u=ξ Φ dw 1 2 ξ(g 2 δ u=ξ ) + ξ m. As in the deterministic case, we added ξ-derivative of a kinetic measure m on the right hand side, although it did not pop up from our calculation as we assumed rather high regularity of u. emark In the stochastic setting it appears to be more convenient to write down the kinetic formulation for 1 u>ξ rather then χ u = 1 u>ξ 1 >ξ. However, since δ u=ξ = ξ 1 u>ξ, (2.6) can be rewritten as df + b f dt = ξ fφ dw ξ(g 2 ξ f) + ξ m,

19 16 2. STOCHASTIC CONSEVATION LAWS which is different from the same equation with f replaced by χ u unless Φ() =. Kinetic solution to (2.1) is then, roughly speaking, a distributional solution to (2.6) with some kinetic measure m. Let us give the precise definitions. Definition (Kinetic measure). A mapping m from Ω to M + b ([, T ] T N ), the set of nonnegative bounded measures over [, T ] T N, is said to be a kinetic measure provided (i) m is measurable in the following sense: for each ψ C ([, T ] T N ) the mapping m(ψ) : Ω is measurable, (ii) m vanishes for large ξ: if B c = {ξ ; ξ } then lim E m( [, T ] T N B c ) =, (iii) for any ψ C (T N ), the process t ψ(x, ξ) dm(s, x, ξ) is predictable. T N [,t] Definition (Kinetic solution). Assume that, for all p [1, ), u L p (Ω [, T ], P, dp dt; L p (T N )) L p (Ω; L (, T ; L p (T N ))). Then u is said to be a kinetic solution to (2.1) with initial datum u provided there exists a kinetic measure m such that the pair (f = 1 u>ξ, m) satisfies, for all ϕ Cc ([, T ) T N ), P-a.s., T f(t), t ϕ(t) dt + f, ϕ() T + f(t), b ϕ(t) dt = k 1 T 1 2 ( ) ( ) g k x, u(t, x) ϕ t, x, u(t, x) dx dβk (t) T N T T N G 2( x, u(t, x) ) ξ ϕ ( t, x, u(t, x) ) dx dt + m( ξ ϕ). Note that a kinetic solution is in fact not a stochastic process in the usual sense: it is a class of equivalence in L p (Ω [, T ]; L p (T N )). In particular, it is a priori not defined for every time t. This is connected to the fact that the kinetic formulation is solved in the sense of distributions D ([, T ] T N ), i.e. the solution that we obtain is weak also in time and consequently it is defined only almost everywhere in time. Nevertheless, it can be proved that in this class of equivalence there exists a representative with good continuity properties, namely, u C([, T ]; L p (T N )) a.s. for every p [1, ). In proof of existence of a kinetic solution to (2.1), another notion of solution will be useful and that is called generalized kinetic solution. Before we give its definition, let us present a short intermezzo about Young measures The concept of Young measures. Let us consider a sequence (v n ) which is bounded in L (T N ). It follows from the Banach-Alaoglu theorem that there exists a subsequence, still denoted by (v n ) which converges to some limit, say v in L (T N ) weak*. To be more precise, for every ϕ L 1 (T N ) v n, ϕ v, ϕ.

20 2.1. KINETIC FOMULATION 17 Since we are interested in nonlinear problems, we want to know if also B(v n ) converges in some sense and what is the limit. If B was continuous, then also (B(v n )) is bounded in L (T N ). Hence we may apply the Banach-Alaoglu theorem again and get a subsequence (B(v n )) and B L (T N ) such that B(v n ) w B in L (T N ). Is it possible to represent B in terms of v? In other words, under what conditions B = B(u)? To get some feeling about what the difficulties are, set v n (x) = sin(nx) and B(v) = v 2. We make use of the following lemma which is based on Fourier series expansion. Lemma Let v L 2 (, 2π) be 2π-periodic. Let v n (x) = v(nx). Then w 1 v n 2 a in L 2 (, 2π), where a = 1 π 2π v(x) dx. Applying this result to our example above, in particular, to both v n and B(v n ), we obtain that w 1 2π v n sin(x) dx = in L 2 (, 2π), 2π and (v n ) 2 w 1 2π sin 2 (x) dx = 1 in L 2 (, 2π). 2π 2 Hence B = 1 2 and B(u) =. In general, the problems originate in oscillatory behavior of (v n ) and one can only say that B(w lim v n ) w lim B(v n ). Therefore, weak convergence is not enough to pass to the limit in nonlinear terms. However, limits of composite functions can be described by a family of probability measures that is called a Young measure. Definition (Young measure). Let (X, λ) be a measure space. Let P 1 () denote the set of probability measures on. A mapping ν from X to P 1 () is said to be a Young measure if, for all ψ C b (), the map z ν z (ψ) from X into is measurable. We say that a Young measure ν vanishes at infinity if, for all p 1, ξ p dν z (ξ) dλ(z) <. X Let us go now go back to our problem of convergence of B(v n ). Theorem There exists a Young measure ν : T N P 1 () which represents the sequence (v n ) in the following sense: For any B C() it holds that where B(v n ) w B in L (T N ), B(x) = B(ξ) dν x (ξ) for a.e. x T N.

21 18 2. STOCHASTIC CONSEVATION LAWS Important consequence of this result is that the Young measure ν corresponding to the weak*-converging sequence (v n ) describes the limit of B(v n ) for every continuous B without taking any further subsequence (unlike our first naive approach using the Banach-Alaoglu theorem). Intuitively, ν x gives the limit probability distribution of the values of v n in the neighborhood of x, as n. Theorem It holds true that v n converges to v strongly in L 2 (T N ) if and only if ν x = δ u(x). In that case, B(x) = B(u(x)) for a.e. x T N Generalized kinetic solution. Let us now go back to the kinetic formulation (2.6). The observation that the Dirac masses appearing on the right hand side can be understood as Young measures directly leads to a a more general notion of kinetic solution: roughly speaking, we replace δ u=ξ by a general Young measure and define f by ξ f = δ u=ξ. The following definition makes this statement rigorous. Definition Let (X, λ) be a measure space. A measurable function f : X [, 1] is said to be a kinetic function if there exists a Young measure ν on X that vanishes at infinity, such that for a.e. z X and for all ξ f(z, ξ) = ν z (ξ, ). Clearly, 1 u>ξ considered before is the kinetic function corresponding to the Young measure δ u=ξ on X = Ω [, T ] T N. Definition (Generalized kinetic solution). Let f : Ω T N [, 1] be a kinetic function. A measurable function f : Ω [, T ] T N [, 1] is said to be a generalized kinetic solution to (2.1) with initial datum f if (f(t)) is predictable and is a kinetic function such that the corresponding Young measure ν satisfies for every p [1, ) E ess sup ξ p dν t,x (ξ) dx C p t [,T ] T N and there exists a kinetic measure m such that for every ϕ Cc ([, T ) T N ) it holds true T f(t), t ϕ(t) dt + f, ϕ() T + f(t), b ϕ(t) dt = k 1 T 1 2 T N T g k (x, ξ)ϕ(t, x, ξ) dν t,x (ξ) dx dβ k (t) T N G 2 (x, ξ) ξ ϕ(t, x, ξ) dν t,x (ξ) dx dt + m( ξ ϕ). We will see in the proof of existence that the latter notion of solution allows for a much simple proof. In particular, weak convergence of some approximate solutions is sufficient in order to pass to the limit. ecall, that according to considerations made in Subsection this is not enough to pass to the limit in nonlinear terms and therefore one would expect that such a notion of solution is too weak to preserve uniqueness. Nevertheless, the key result is the so-called eduction theorem. It states that not only it is possible to establish uniqueness for generalized kinetic solutions but, moreover, any generalized kinetic solution is actually a kinetic

22 2.2. EXISTENCE 19 solution, that is, the corresponding Young measure is Dirac. In view of Theorem we thus recover the strong convergence of the approximate solutions. Theorem (Uniqueness, eduction). There is at most one kinetic solution to (2.1). Besides, any generalized kinetic solution is actually a kinetic solution, i.e. if f is a generalized kinetic solution with initial datum 1 u>ξ then there exists a kinetic solution u such that u = 1 u>ξ. Moreover, if u 1, u 2 are kinetic solutions with initial data u 1,, u 2, then for all t [, T ] E u 1 (t) u 2 (t) L 1 x E u 1, u 2, L 1 x. The existence result reads as follows. Theorem (Existence). There exists a kinetic solution to (2.1) which is the strong limit of a parabolic approximation (u ε ): for all p [1, ) lim ε E uε u p =. L p t,x 2.2. Existence This section is devoted to the proof of Theorem which is based on the vanishing viscosity method and proceeds in several steps. First, the existence of a unique solution to the parabolic approximation is established, then uniform a priori estimates are derived and in the last step, the passage to the limit is performed. To begin with, let us consider the following viscous approximation of (2.1) (2.7) du ε + div B ε (u ε ) dt = ε u dt + Φ ε (u ε ) dw, u ε () = u ε. Here u ε is a suitable smooth approximation of u, the approximate flux functions B ε are smooth and compactly supported and the approximation Φ ε is defined as follows. ecall that Φ(u)e k = g k (u) where g k are continuous functions. Let gk ε be a smooth compactly supported approximation of g k if k < 1 ε and let gε k = if k 1 ε. Set Φ ε (u)e k = gk ε (u). In particular, we approximate the infinite dimensional noise in (2.1) by finite dimensional noises. On this level, we can consider the approximate coefficients as nice as we want so that we can not only solve (2.7) but also derive sufficient regularity properties of the solution that are needed later on in order to derive the kinetic formulation. In particular, we will show that solutions to (2.7) satisfy the corresponding kinetic formulation of (2.7) and we pass to the limit there. (emember that the original conservation law (2.1) may not be well defined in the sense of distributions.) Let ε (, 1) be fixed. Existence and uniqueness for (2.7) is classical and can be obtained by semigroup approach. Namely, let S be the semigroup generated by ε, then we rewrite (2.7) in its mild form as u ε (t) = S(t)u ε t S(t s) div B ε (u ε ) ds + t S(t s)φ ε (u ε ) dw and use the Banach fixed point theorem to obtain solutions (for instance) in L 2 (Ω [, T ]; L 2 (T N )). For more details we refer the reader to [1]. To establish higher regularity, we use the result of [5] which implies that the unique mild solution to (2.4) in fact belongs to L p (Ω; C([, T ]; W l,q (T N ))) for all p (2, ), q [2, ), l N. Therefore, the Sobolev embedding applies and yields that u ε C([, T ]; C 1 (T N ))

23 2 2. STOCHASTIC CONSEVATION LAWS a.s. which is exactly the regularity we required in Section 2.1 for the derivation of the kinetic formulation. Hence we proceed similarly and obtain the following result. Proposition (Kinetic formulation of (2.7)). Let u ε be the unique solution to (2.7) and set f ε = 1 uε >ξ. Then for all ϕ C 2 c ([, T ) T N ) it holds T f ε (t), t ϕ(t) dt + f ε, ϕ() T + f ε (t), b ϕ(t) T dt + ε f ε (t), ϕ(t) dt = k 1 T 1 2 T N T g k (x, ξ)ϕ(t, x, ξ) dν ε t,x(ξ) dx dβ k (t) T N G 2 (x, ξ) ξ ϕ(t, x, ξ) dν ε t,x(ξ) dx dt + m ε ( ξ ϕ), where f ε = 1 u ε >ξ, νt,x ε = δ u ε (t,x) and for φ C b ([, T ] T N ) m ε (φ) = ε φ(t, x, u ε (t, x)) u ε 2 dxdt. [,T ] T N Proof. Motivated by the calculations in Section 2.1, we take θ Cc (), fix x T N and apply the Itô formula to 1 uε (t,x)>ξ, θ = θ(u ε (t, x)). Let us show what happens with the second order term, the rest being the same as before. We obtain θ (u ε )ε u ε = ε θ(u ε ) ε u ε 2 θ (u ε ) = ε 1 uε >ξ, θ ξ ε u ε 2 δ uε =ξ, θ ξ and altogether = ε 1 u ε >ξ, θ ξ + ε ξ ( u ε 2 δ u ε =ξ), θ ξ d 1 u ε >ξ, θ ξ = b ε 1 u ε >ξ, θ ξ dt + k 1 g ε kδ uε =ξ, θ ξ dβ k 1 2 ξ(g 2 εδ u ε =ξ), θ ξ dt + ε 1 u ε >ξ, θ ξ + ξ m ε, θ ξ. Finally we set θ(ξ) = ξ ϕ1 (ζ)dζ and test the above by ϕ 2 C (T N x ) to complete the proof. emark The previous result show that the concept of kinetic solution extends to parabolic equations. In particular, it also applies to degenerate parabolic equations and allows to prove well-posedness. As the next step, we will pass to the limit in the kinetic formulation of (2.7) to obtain the kinetic formulation of (2.1). To this end, we need to derive some estimates uniform in ε Energy estimate. Lemma For all p [2, ) we have the following estimate uniform in ε E sup u ε (t) p L C( 1 + E u p x p Lx). p t T

24 2.2. EXISTENCE 21 Proof. Let us apply the infinite dimensional Itô formula to the function F (v) = v p. A straightforward calculation yields L p x F (v) = p v p 2 v L p (T N ), F (v) = p(p 1) v p 2 Id L(L p (T N ), L p (T N )), hence t u ε (t) p L = u p p x L p u ε p 2 u ε, div B ε (u ε ) ds p x + εp t u ε p 2 u ε, u ε ds + p k t 2 p(p 1) t u ε p 2 u ε, g ε k(u ε ) dβ k (s) T N u ε p 2 G 2 ε(u ε )dxds = I I 4. Let us estimate each of the terms separately. First, we set and observe that I 1 = p = p t t H ε (ξ) = ξ ζ p 2 B ε (ζ)dζ [ u ε p 2 u ε] B ε (u ε )dxds = p T N div H ε (u ε )dxds =, T N t due to periodic boundary conditions. Next, we have and due to (2.2) I 4 C t I 2 = εp t T N u ε p 2 u ε 2 dxds T N u ε p 2 (1 + u ε 2 )dxds C T N u ε p 2 u ε B ε (u ε )dxds ( 1 + t u ε (s) p L p x ds ). Consequently, after taking expectation (since the stochastic integral vanishes), we apply the Gronwall lemma and obtain (2.8) E u ε (t) p L p x C( 1 + E u p L p x). In order to complete the proof it is needed to first take supremum over t [, T ] and then expectation. The only change appears in the estimation of the stochastic integral, where we make use of the Burkholder-Davis-Gundy inequality, the Schwartz

25 22 2. STOCHASTIC CONSEVATION LAWS inequality, the assumption (2.2) and the weighted Young inequality: t E sup u ε p 2 u ε gk(u ε ε )dxdβ k (s) t T k 1 T N ( T ( ) 2 1/2 CE u ε p 1 gk(u ε ε ) dx ds) k 1 T N ( T CE u ε p 2 2 u ε p 2 L 2 2 g ε k (u ε ) 2 x L xds 2 k 1 ( T ) CE u ε p Lx( 1 + u ε p ) 1/2 p L p x ds ( ) 1/2 ( T CE sup u ε p L p x t T 1 2 E sup u ε (1 p L + C + E p x t T ) 1/2 ) ( 1 + u ε p ) 1/2 L p x ds T ) ( 1 + u ε p ) L p x ds. Finally, we put the first term on the right hand side to the left hand side, estimate the second one by (2.8) and apply the Gronwall lemma Estimate of the kinetic measures. ecall that the approximate kinetic measures m ε are defined by (2.9) dm ε = ε u ε 2 δ u ε(dξ)dxdt. Our aim is to show that (m ε ) is bounded in L 2 w(ω; M b ([, T ] T N )), the space of weak-star measurable mappings from Ω to the space of bounded Borel measures with the norm given by the total variation. Since it follows that M b ([, T ] T N ) = ( C ([, T ] T N ) ) L 2 w(ω; M b ([, T ] T N )) = ( L 2 (Ω; C ([, T ] T N )) ) and therefore we then apply the Banach-Alaoglu theorem and obtain existence of a weak-star convergent subsequence. The proof of the necessary bound follows from the calculation in Lemma Indeed, E m ε 2 M b = E m ε ([, T ] T N ) 2 T 2 = E ε u ε 2 dxdt T N where the term inside the square corresponds to I 2 from Lemma Hence we proceed similarly, take the square and expectation and estimate the remaining terms on the right hand side, that is, I 3 and I 4 to deduce E m ε 2 M b C ( 1 + E u 2 L 2 x). The Banach-Alaoglu now gives a subsequence (m n ) and m L 2 w(ω; M b ) such that (2.1) m n m in L 2 w (Ω; M b ).

26 2.2. EXISTENCE Estimate of the Young measures. As we intend to pass to the limit in the kinetic formulation from Proposition 2.2.1, we need also to get compactness of the approximate Young measures ν ε. It is based on the following result. Proposition Let (X, λ) be a finite measure space such that L 1 (X) is separable. 1 Let (ν n ) be a sequence of Young measures such that for some p 1 (2.11) sup ξ p dνz n (ξ)dλ(z) <. n X Then there exists a subsequence, still denoted by (ν n ), and a Young measure ν such that for all h L 1 (X) and for all φ C b () h(z) φ(ξ)dνz n (ξ)dλ(z) h(z) φ(ξ)dν z (ξ)dλ(z). X X Corollary Let (X, λ) be a finite measure space such that L 1 (X) is separable. Let (f n ) be a sequence of kinetic functions such that the corresponding Young measures (ν n ) satisfy (2.11). Then there exists a subsequence, still denoted by (f n ), and a kinetic function f such that f n f in L (X ). In our case, we apply the above results to X = Ω [, T ] T N, f ε = 1 u ε >ξ and ν ε = δ u ε =ξ, hence the estimate (2.11) reads as T sup E u ε p dxdt < n T N which holds true due to the energy estimate Lemma Therefore, we have all in hand to pass to the limit and to obtain a generalized kinetic solution to (2.1). It only remains to verify that m is a kinetic measure. The first and the third requirement of Definition are technical and not difficult and hence they are left to the reader. Let us verify the point (ii). We observe that it follows, once we show, for some p > 2, that (2.12) E ξ p 2 dm(t, x, ξ) C. [,T ] T N Indeed, in that case Em([, T ] T N B) c = E 1 p 2 [,T ] T N [,T ] T N 1 ξ dm(t, x, ξ) ξ p 2 dm(t, x, ξ) C. p 2 In order to show (2.12), we consider the approximate kinetic measures m ε given by (2.9) and write T E ξ p 2 dm ε (t, x, ξ) = E ε u ε 2 u ε p 2 dxdt. [,T ] T N T N This has already appeared in the term I 2 in Lemma 2.2.3, so we know that E ξ p 2 dm ε (t, x, ξ) C [,T ] T N 1 It is sufficient to assume that the σ-algebra is countably generated.

27 24 2. STOCHASTIC CONSEVATION LAWS uniformly in ε. Since φ(ξ) = ξ p 2 is not a good test function for the convergence (2.1), we have to truncate: let h δ be a smooth cut-off function on. We have E ξ p 2 dm(t, x, ξ) lim inf E ξ p 2 h δ (ξ)dm(t, x, ξ) δ [,T ] T N lim inf δ lim lim sup E ε [,T ] T N ε E [,T ] T N [,T ] T N ξ p 2 h δ (ξ)dm(t, x, ξ) ξ p 2 h δ (ξ)dm(t, x, ξ) C and the claim follows. Finally, we may apply the eduction theorem and obtain the existence of a kinetic solution with the corresponding kinetic measure m. To complete the proof of Theorem , it remains to verify the strong convergence of u ε to u in L p (Ω [, T ] T N ). Let p = 2. ecall from the theory of Young measures that, roughly speaking, the Young measure corresponding to a weakly converging sequence reduces to a Dirac mass if and only if the sequence converges weakly. Since we have already shown that the limit Young measure ν is a Dirac, the strong convergence from Theorem is expected. Since u ε is bounded in L 2 (Ω [, T ] T N ), due to Banach-Alaoglu, there exists a weakly converging subsequence. The limit is necessarily u. In order to establish the strong convergence, we need to prove convergence of the norms, i.e. (2.13) u ε L 2 ω,t,x u L 2 ω,t,x. To this end, we observe: since f ε converges to f weak-star in L (Ω [, T ] T N ), taking ψ(ξ) = ξ ϕ(ξ) with φ Cc () as a test function and integrating by parts gives T T f ε, ψ = E 1 uε >ξ ξ ϕdξdxdt = E ϕdδ u ε(ξ)dxdt = E T T N T N ϕ(u ε )dxdt E T T N T N ϕ(u)dxdt. This would prove (2.13) provided we could take ϕ(ξ) = ξ 2. However, this is not possible directly, so we truncate first. let (h δ ) be a smooth cut-off function as above and set ϕ δ = ξ 2 h δ (ξ). Then T E T u ε 2 u 2 dxdt E u ε 2( 1 h δ (ξ) ) dxdt T N T N T + E u ε 2 h δ (ξ) u 2 h δ (ξ)dxdt T N T + E u 2( 1 h δ (ξ) ) dxdt = I 1 + I 2 + I 3, T N where I 2 for each δ as ε vanishes. For I 1 and I 3 we make use of Lemma for p > 2 which implies that u ε 2 is uniformly integrable and hence I 1 as δ uniformly in ε. Similarly, since the limit u satisfies the same a priori estimates is u ε, I 3 as δ uniformly in ε. As a consequence, we obtain the convergence from Theorem for p = 2.

28 2.3. UNIQUENESS 25 If p > 2, we apply the Hölder inequality and the uniform a priori estimate to deduce T T E u ε u p dxdt = E u ε u p 1 u ε u dxdt T N T ( N T ) 1/2 ( T ) 1/2 E u ε u 2(p 1) dxdt E u ε u 2 dxdt T N T ( N T ) 1/2 C E u ε u 2 dxdt T N and the proof of Theorem is complete Uniqueness The proof of uniqueness as well as the reduction theorem are very technical and lengthy. Therefore I will only give some basic ideas and explain the difficulties. ecall that the kinetic formulation (2.6) is satisfied in the following sense: for all ϕ Cc ([, T ) T N ), P-a.s., T f(t), t ϕ(t) dt + f, ϕ() T + f(t), b ϕ(t) dt (2.14) = k 1 T T N T 1 2 T N g k (x, ξ)ϕ(t, x, ξ) dν t,x (ξ) dx dβ k (t) G 2 (x, ξ) ξ ϕ(t, x, ξ) dν t,x (ξ) dx dt + m( ξ ϕ). oughly speaking, (2.6) holds true in the sense of distributions over [, T ] T N. However, it is not clear how to work with this formulation. In particular, one cannot apply the Itö formula. Is it possible to consider a stronger version of (2.6)? Namely, for a test function ϕ Cc (T N ), we want to write f(t), ϕ f, ϕ t f(s), b ϕ ds = k 1 t T N t g k (x, ξ)ϕ(t, x, ξ) dν t,x (ξ) dx dβ k (t) T N G 2 (x, ξ) ξ ϕ(t, x, ξ) dν t,x (ξ) dx dt + m, ξ ([, t)). This was for instance true for the parabolic approximation as can be seen in Proposition Nevertheless, in general, this is not true as the measure can have a shock at time t. Let us consider the first term in (2.14). We observe that testing by t ϕ(t, x, ξ) = δ s=t ψ(x, ξ) we get T f(t), t ϕ(t) dt = f(s), ψ,