ENTROPY AND KINETIC FORMULATIONS OF CONSERVATION LAWS

Transcription

1 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS YUZHOU ZOU Abstract. Entropy an kinetic formulations of conservation laws are introuce, in orer to create a well-pose theory. Existence an uniqueness results are proven for both formulations. Convergence of approximations an compactness of solutions are also proven using the kinetic formulation. Contents 1. Introuction 1. Kruzkov s entropy formulation.1. Distributional solutions.. Entropy solutions an equivalent notions 4.3. Uniqueness 6.4. Existence 1 3. Kinetic Formulation 1 4. Existence an Uniqueness Statement an Outline Solving a linear approximation Properties of approximate solutions an the approximate measure Convergence of approximate solutions Uniqueness 5. Convergence in the Diffusion Approximation 3 6. Compactness an Averaging Lemmas 4 Acknowlegments 5 eferences 5 1. Introuction Conservation laws are prevalent in physics an are some of the most basic examples of nonlinear first-orer PDEs. It is known that many conservation laws lack classical solutions for all time, even if the initial ata is smooth. A less restrictive notion of solution is thus neee for conservation laws to be well-pose. In this paper, we will explore two alternative equivalent notions of solutions: entropy solutions an solutions to the kinetic formulation, or kinetic solutions. We will prove that both notions lea to conservation laws being well-pose, an using the kinetic formulation we will also erive results concerning the convergence of solutions to approximations of conservation laws, as well as results concerning the compactness of a family of entropy solutions. We assume the reaer is familiar with Lebesgue integration an basic aspects of the theory of istributions. Date: August 8, 15. 1

2 YUZHOU ZOU. Kruzkov s entropy formulation.1. Distributional solutions. It is well known (see [1]) that the problem u t +iv x (A(u)) = may not have classical solutions for all positive time, even if A an the initial ata u are both smooth. In orer to provie a well-pose theory of conservation laws, we introuce alternative notions of solutions in orer to obtain the existence an uniqueness of solutions. Definition.1. Let T >, an enote = [, T ]. Let A : be C 1, an let a = A. We say that a boune locally integrable function u : is a istributional solution to the problem (.1) u t + iv x (A(u)) =, u(, x) = u (x) if it satisfies the above equations in the sense of istributions, i.e. if for all f CC ([, T ) ) we have uf t + A(u) x f x + u (x)f(, x) x =. For the rest of this section, we will use CC ( ) to enote C functions with compact support strictly insie (, T ) ; in particular, these functions are zero at times t = an t = T. Note that all classical (i.e. C 1 ) solutions to the PDE are istributional solutions, as can be seen from integrating the equation by parts against any smooth test function. Furthermore, any istributional solution which is C 1 is also a classical solution. In aition, the integral equality in the efinition of the istributional solution yiels the following conition on the iscontinuities of u, known as the ankine-hugoniot conition: Proposition.. Suppose u is a istributional solution to (.1) an is C 1 in everywhere except along a finite number of C 1 surfaces, where no two surfaces intersect on a set of positive measure. Let (t, x) be a point belong to exactly one of the above curves. If ˆn is the normal vector to the surface at (t, x), then (u 1 u, A(u 1 ) A(u )) ˆn = where u 1 an u are the limits of u approaching (t, x) from either sie of the iscontinuity surface. In particular, if = 1, an the iscontinuity curve is parametrize by x = x(t), then ẋ(t)(u 1 u ) = A(u 1 ) A(u ). Conversely, if u is a function which satisfies the PDE in (.1) at every point in except along a finite number of C 1 surfaces, with no two surfaces intersecting on a set of positive measure, an each non-intersection point of the surfaces satisfies the above relation, then u is a istributional solution to (.1). Proof. We prove this in one imension for simplicity. Let u be a istributional solution, an assume first that we only have one curve of iscontinuity C. Let L an be the regions to the left an right, respectively, of C in the (t, x) plane, an let u 1 an u be the respective limits. Since u is smooth in L an, we know that u t + iv x (A(u)) = in the interior of L an. By the ivergence theorem, we have uf t + A(u)f x x = (u 1 f, A(u 1 )f) ˆn l s f(u t + iv x (A(u))) x = (u 1 f, A(u 1 )f) ˆn 1 s L C for all f CC ( ). Similarly, we have uf t + A(u)f x x = (u f, A(u )f) ˆn s. L C C

3 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS 3 We thus have = L uf t + A(u)f x + uf t + A(u)f x = f(u 1, A(u 1 )) ˆn 1 + (u, A(u )) ˆn s). Letting ˆn = (ˆt, ˆx) with ˆx >, we have ˆn 1 = ˆn = ˆn, an ẋ(t) = ˆt ˆx. We have f((u 1 u )ˆt + (A(u 1 ) A(u ))ˆx) s =. Since this hols for all f, it follows that C C (u 1 u )ˆt + (A(u 1 ) A(u ))ˆx = = ẋ(t)(u 1 u ) = ˆt ˆx (u 1 u ) = A(u 1 ) A(u ), thus proving the relation in the case of one curve. For the case of multiple curves, we aapt the above proof by testing with f whose support oes not intersect any of the other curves. The converse also follows easily from the above calculations, by noting that every point along a single curve of iscontinuity satisfying the above relation will also satisfy the relation (u 1 u )ˆt + (A(u 1 ) A(u ))ˆx =. Denoting the curves by {C i }, we thus have f[(u 1 u )ˆt + (A(u 1 ) A(u ))ˆx] s =. i C i If we enote the regions separate by the curves by { j }, we have uf t + A(u)f x x = uf t + A(u)f x x π j T j = (uf, A(u)f) ˆn s f(u t + iv x (A(u))) x j = i j thus implying that u is a istributional solution. C i f[(u 1 u )ˆt + (A(u 1 ) A(u ))ˆx] s =, While the istributional notion of a solution oes not require the solution to be C 1 an hence amits a larger class of possible solutions, it has the isavantage of amitting too many solutions, in that solutions are not generally unique. For example, let A(u) = u on (, T ). (The corresponing PDE u t +uu x = is commonly known as Burgers equation.) The ankine-hugoniot conitions require the curves of iscontinuity to satisfy ẋ(t)(u 1 u ) = A(u 1 ) A(u ) = (u 1 u )(u 1 + u ) j = ẋ(t) = u 1 + u. For the initial ata u, it is clear that u is a istributional solution to the problem. However, consier the function 1 < x < t u 1 (t, x) = 1 t < x <. otherwise The initial ata for u 1 is also u. It has three curves of iscontinuity (namely at x = an x = ± t ), an it is easy to check that the ankine-hugoniot conition is satisfie along each curve. It follows that u 1 is also a istributional solution to Burgers equation with u, showing that the problem oes not amit a unique solution for u.

4 4 YUZHOU ZOU.. Entropy solutions an equivalent notions. To avoi the issue of non-uniqueness, we introuce a ifferent notion of solution more restrictive than simply satisfying the equation in the sense of istributions. Definition.3. We say that a boune locally integrable function u : is an entropy solution of the problem u t + iv x (A(u)) =, u(, x) = u (x) on if, for all k, we have the inequality (.) t ( u k ) + sgn(u k)iv x(a(u)) in the sense of istributions, i.e. for any nonnegative f CC ( ) we have (.3) u(t, x) k f t + sgn(u(t, x) k)(a(u(t, x)) A(k)) x f x, an there exists E [, T ] such that [, T ]\E has measure zero, the function u(t, ) is efine almost everywhere in for all t E, an for any ball B we have (.4) lim u(t, x) u (x) x =. t,t E B emark.4. To see that (.) an (.3) are equivalent, integrate (.) by parts against any nonnegative f C C ( ), noting that iv x (sgn(u k)(a(u) A(k))) = sgn(u k)iv x (A(u) A(k)) + x (sgn(u k)) (A(u) A(k)) = sgn(u k)iv x (A(u)) + δ(u k) x u (A(u) A(k)) = sgn(u k)iv x (A(u)), where δ is the Dirac elta. The last equality following from the fact that δ(u k) = whenever A(u) A(k). We first show that all entropy solutions are istributional solutions, showing that the notion of entropy solutions is more strict than that of istributional solutions. Proposition.5. Let u L ( ) satisfy (.3). Then u is a istributional solution to the problem (.1). Proof. Let f C C ( ) an k. Choosing k > u L, we have u k = k u an sgn(u k) = 1, an hence (.3) becomes (k u)f t (A(u) A(k)) x f x. Since f C C ( ), we have an hence f t = an x f =. Thus we have kf t + A(k) x f x = uf t A(u) x f x = uf t + A(u) x f x.

5 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS 5 Similarly, choosing k < u L yiels Thus, we have (u k)f t + (A(u) A(k)) x f x = uf t + A(u) x f x = uf t + A(u) x f x. for all nonnegative f CC ( ), which is enough to conclue that u satisfies (.1) in the sense of istributions. emark.6. If u W 1,1 ( ) an u is a istributional solution, then u is an entropy solution as well, since we may apply the chain rule to obtain t ( u k ) + sgn(u k)iv x(a(u)) = sgn(u k)(u t + iv x (A(u))) =. In general, istributional solutions nee not be entropy solutions as well. Consier the function u 1 liste above in the Burgers equation example. It is not an entropy solution, since it is easy to verify that for f C C ( ) we have u T 1 u 1 f t + sgn(u 1 ) f x x = f(t, ). Hence, if f in, an f is not ientically zero, then the right-han sie is negative, showing x= that u 1 is not an entropy solution. Of course, u 1 is not in W 1,1 ( ). One way to see this is to note that T u 1 f t x = f( x, x) x which implies that the right-han sie is if f C C ( \{(t, x) : t = x }). This shows that the istributional time erivative of u 1 is zero almost everywhere outsie of {(t, x) : t = x } an hence zero almost everywhere in, implying that it cannot be in L 1 ( ) as u 1 is not constant in time. Finally, we prove a property of entropy solutions which allows an alternative characterization of entropy solutions that, while appearing more restrictive at first, is logically equivalent. Proposition.7. Let u be an entropy solution to (.1). Then the inequality t (S(u)) + iv x(η(u)) hols for all convex S, where η satisfies η = S A. Note that the efinition is equivalent to the above inequality holing for all S of the form S(u) = u k, k. Hence, entropy solutions can be efine by the solutions satisfying the above inequality for all convex S, an not just those of the form S(u) = u k. Proof. Notice that F (k) = k / is the funamental solution to the Laplace equation in 1 imension. This implies that S (u) = (S /) (u), an hence S (k) S(u) = u k k + au + b T

6 6 YUZHOU ZOU for some constants a an b in the sense of istributions. From Proposition.5, we know that u is a istributional solution, an so if S is linear (say S(u) = au + b), then Hence, assume that S(u) = an hence η(u) = CC ( ) we have S(u)f t + η(u) x f x = since S convex implies S. t (S(u)) + iv x(η(u)) = a(u t + iv x (A(u))) =. S (k) u k k. Then η (u) = S (u)a (u) = S (k) sgn(u k)a (u) k, S (k) sgn(u k)(a(u) A(k)) k, up to an aitive constant. For any f = S (k) S (k) u k f t + S (k) sgn(u k)(a(u) A(k)) x f x k u k f t + sgn(u k)(a(u) A(k)) x f x k,.3. Uniqueness. In efining the notion of entropy solutions, we aime to create a notion of solution where existence hel in orer to fix the main rawback to the notion of istributional solutions. We will now show that entropy solutions are inee unique. Theorem.8. Let u, v be entropy solutions with corresponing initial ata u an v. satisfy u L, v L M, an let N = max a(u). Then, for any >, we have S τ u M u(τ, x) v(τ, x) x for almost every < τ < T = min(t, /N), where is the cross-section of the plane t = τ of the cone B S τ = {x : x Nτ} u (x) v (x) x C = {(t, x) : x Nt, t T }. Note that if we take, we obtain the L 1 contraction property u(t, ) v(t, ) L 1 ( ) u v L 1 ( ). Let M The proof will follows Kruzkov s oubling variables proof in [3]. We procee by proving two lemmas: Lemma.9. Let g CC ( ) be nonnegative. Then (.5) u(t, x) v(τ, y) (g t + g τ )+ sgn(u(t, x) v(τ, y))(a(u(t, x)) A(v(τ, y))) ( x g + y g) x τ y. Proof. We wish to apply the inequality (.3) by by integrating over the variables t, x, τ, y, integrating over x an using v(τ, y) as a constant, an then integrating over τ y, treating u(t, x) as a constant. We thus have u(t, x) v(τ, y) g t + sgn(u(t, x) v(τ, y))(a(u(t, x)) A(v(τ, y))) x g x τ y

7 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS 7 an v(τ, y) u(t, x) g τ + sgn(v(τ, y) u(t, x))(a(v(τ, y)) A(u(t, x))) y g τ y x Aing the two inequalities yiels the esire lemma. Lemma.1. Let f C C ( ) be nonnegative. Then u v f t + sgn(u v)(a(u) A(v)) f x. Proof. Let supp f K for some compact K. We apply Lemma.9 with ( t + τ g(t, x, τ, y) = f, x + y ) ( t τ ρ ɛ, x y ) where ρ ɛ is the stanar mollifier on +1, supp ρ ɛ B ɛ, an ɛ is chosen small enough for g to be well-efine. For notational purposes, let t = t+τ x+y t τ x y, x =, t =, an x =. We have an I now claim that (.6) (g t + g τ )(t, x, τ, y) = f t ( t, x) ρ ɛ ( t, x) ( x g + y g)(t, x, τ, y) = ( f ( t, x)) ρ ɛ ( t, x) u(t, x) v(τ, y) f t ( t, x) ρ ɛ ( t, x) x τ y ɛ +1 u(t, x) v(t, x) f t (t, x) x an sgn(u(t, x) v(τ, y))(a(u(t, x)) A(u(τ, y))) ( f ( t, x) ρ ɛ ( t, x)) x τ y (.7) ɛ +1 sgn(u(t, x) v(t, x))(a(u(t, x)) v(t, x)) f(t, x) x. For simplicity, we prove (.6). We re-write the integral on the left han sie as [ u(t, x) v(τ, y) f t ( t, x) u(t, x) v(t, x) f t (t, x)] ρ ɛ ( t, x) x τ y (.8) + u(t, x) v(t, x) f t (t, x)ρ ɛ ( t, x) x τ y. After a change-of-variables, the secon integral becomes +1 u(t, x) v(t, x) f t (t, x)ρ ɛ (τ t, y x) x τ y = +1 (( u v f t ) ρ ɛ ) (τ, y) τ y ɛ +1 u(τ, y) v(τ, y) f t (τ, y) τ y.

8 8 YUZHOU ZOU It suffices to show that the first integral in (.8) vanishes as ɛ. To o so, we make the following estimate: u(t, x) v(τ, y) f t ( t, x) u(t, x) v(t, x) f t (t, x) u(t, x) v(τ, y) u(t, x) v(t, x) f t ( t, x) + u(t, x) v(t, x) f t ( t, x) f t (t, x) v(t, x) v(τ, y) f t L + u(t, x) v(t, x) f t L ( t, x) f t L + ɛ f t L u(t, x) v(t, x). Note that we may assume ( t, x) < ɛ since supp ρ ɛ B ɛ. Furthermore, uner this assumption we have supp f t ( t, x) K +B ɛ = {x+y x K, y ɛ}. Hence, we can take the limits of integration in the integral to be (K + B ɛ ) (K + B ɛ ). We thus have [ u(t, x) v(τ, y) f t ( t, x) u(t, x) v(t, x) f t (t, x)] ρ ɛ ( t, x) x τ y (.9) (K+B ɛ) (K+B ɛ) Since (K+B ɛ) (K+B ɛ) = +1 it follows that (K+B ɛ) (K+B ɛ) ( f t L v(t, x) v(τ, y) + ɛ f t L u(t, x) v(t, x) )ρ ɛ ( t, x) x τ y. u(t, x) v(t, x) ρ ɛ ( t, x) x τ y ɛ f t L u(t, x) v(t, x) ρ ɛ (τ t, y x) x τ y ɛ +1 (K+B ɛ) (K+B ɛ) u(t, x) v(t, x) ρ ɛ ( t, x) x τ y ɛ. K u(t, x) v(t, x) x Furthermore, we have (.1) (K+B ɛ) (K+B ɛ) f t L ρ L f t L v(t, x) v(τ, y) ρ ɛ ( t, x) x τ y K+B ɛ 1 ɛ +1 (t,x)+b ɛ v(t, x) v(τ, y) τ y x. By the Lebesgue Differentiation Theorem, almost every (t, x) is a Lebesgue point, i.e. 1 ɛ ɛ +1 v(t, x) v(τ, y) τ y (t,x)+b ɛ (t,x)+b ɛ for almost every (t, x). Since 1 1 ɛ +1 v(t, x) v(τ, y) τ y ɛ +1 (t,x)+b ɛ M τ y = M B 1 an the outer integral in (.1) is taken over a boune set, we can apply the Dominate Convergence Theorem to conclue that the integral in (.1) vanishes as ɛ. Hence, the first integral in (.8) vanishes, so we arrive at (.6). The statement in (.7) can be proven similarly.

9 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS 9 Hence, applying Lemma.9 with our choice of g, an letting ɛ, we have +1 u(t, x) v(t, x) f t (t, x) + sgn(u(t, x) v(t, x))(a(u(t, x)) A(v(t, x))) f(t, x) x, which proves the esire lemma. With these two lemmas proven, we may now procee to prove the main theorem. Proof of Theorem.8. Formally, Lemma.1 implies that t ( u v ) + iv x(sgn(u v)(a(u) A(v))) in the sense of istributions. Furthermore, we have A(u) A(v) N u v by the efinition of N, an hence N u v sgn(u v)(a(u) A(v)) ν for any unit vector ν. It follows that u v x = S t S t S t S t ( u v ) x N t ( u v ) x + t S t S t u v S sgn(u v)(a(u) A(v)) ˆn S t ( u v ) + iv x(sgn(u v)(a(u) A(v))) x. We can formalize the argument as follows: let µ(t) = S t u(t, x) v(t, x) x, let E µ be the set of Lebesgue points of µ, an let E u an E v be the subsets of [, T ] involve in the efinition of entropy solutions. Let E = E µ E u E v. Then [, T ]\E has measure zero. It suffices to show that µ is ecreasing on E, an that µ(t) approaches µ() = u (x) v (x) x for t E approaching. B To show that µ(t) µ(), note that u(t, x) v(t, x) x u(t, x) u (x) x + v(t, x) v (x) x + u (x) v (x) x. B B S t For t E approaching, the first an secon terms on the right-han sie vanish, so we obtain the esire result. We now show that µ is ecreasing on E. Let ρ ɛ be the stanar mollifier in, an let χ ɛ (x) = x ρ ɛ (y) y. Note that χ ɛ (x) = for x < ɛ an χ ɛ (x) = 1 for x > ɛ. For t 1 < t E, let f(t, x) = (χ ɛ (t t 1 ) χ ɛ (t t ))(1 χ ɛ ( x + Nt + ɛ )) with ɛ chosen so that ɛ < Nt. Notice that supp f [t 1 ɛ, t + ɛ] C, an that f. Furthermore, f is clearly infinitely ifferentiable wherever x, an at x =, either t t, in which case Nt + ɛ < ɛ, an hence χ ɛ ( x + Nt + ɛ ) = in a neighborhoo of (t, x), or t > t, in which case f = in a neighborhoo of (t, x). Hence, f C C ( ), so we may apply Lemma.1. We have f t (t, x) = (ρ ɛ (t t 1 ) ρ ɛ (t t ))(1 χ ɛ ( x +Nt +ɛ ))+(χ ɛ (t t 1 ) χ ɛ (t t ))( Nρ ɛ ( x +Nt +ɛ )) an ( f(t, x) = (χ ɛ (t t 1 ) χ ɛ (t t )) ρ ɛ ( x + Nt + ɛ ) x ). x B

10 1 YUZHOU ZOU Hence, we have sgn(u v)(a(u) A(v)) f(t, x) N u v (χ ɛ (t t 1 ) χ ɛ (t t ))ρ ɛ ( x + Nt + ɛ ) (.11) = u v ((ρ ɛ (t t 1 ) ρ ɛ (t t ))(1 χ ɛ ( x + Nt + ɛ )) f t ). Applying Lemma.1 to f an combining with inequality (.11), we obtain (.1) u(t, x) v(t, x) (ρ ɛ (t t 1 ) ρ ɛ (t t ))(1 χ ɛ ( x + Nt + ɛ )) x. As ɛ, we have 1 χ ɛ ( x + Nt + ɛ ) an hence we have (.13) T µ(t)(ρ ɛ (t t 1 ) ρ ɛ (t t )) = Since t 1 is a Lebesgue point of µ, we have T µ(t)ρ ɛ (t t 1 ) µ(t 1 ) = T { 1 if x < Nt if x > Nt S t u(t, x) v(t, x) (ρ ɛ (t t 1 ) ρ ɛ (t t )) x. T t 1+ɛ t 1 ɛ = ρ L ɛ (µ(t) µ(t 1 ))ρ ɛ (t t 1 ) µ(t) µ(t 1 ) ρ L ɛ t t 1 <ɛ µ(t) µ(t 1 ) ɛ. A similar result hols for t. Hence, the left-han sie of (.13) converges to µ(t 1 ) µ(t ) as ɛ, so (.13) implies that µ(t 1 ) µ(t ) for t 1 < t, thus proving the theorem..4. Existence. We can also establish a result regaring the existence of entropy solutions. The iea is to consier, for ɛ >, the solution to u ɛ of the problem u t + iv x (A(u)) = ɛ u, u t= = u It is well known[] that this equation amits a unique classical solution u ɛ if u is boune an has sufficient boune erivatives. Hence, for regular enough u, it suffices to show that the family {u ɛ } is compact, in orer to extract a subsequence ɛ n an a limit u such that u ɛn u, with u satisfying the esire entropy inequalities, while for u L we can approximate by smooth initial ata u h, thus getting a family {u ɛ,h } which converges to some u as h an ɛ. The proof involves fining equicontinuity estimates on u ɛ an is similar to the corresponing proof for kinetic solutions escribe later in this paper, so the proof, which can be foun in [3], will be omitte. In fact, we will investigate the rate of convergence of the parabolic approximation u ɛ to the entropy solution u later in this paper. 3. Kinetic Formulation We now turn our attention to a reformulation of conservation laws which generalizes the notion of entropy solutions. In the kinetic formulation, a function χ(ξ, u) is introuce to turn the nonlinear conservation law into a linear equation on the nonlinear quantity χ(ξ, u). This structure provies a metho to construct solutions by approximating with solutions of a linear equation, as well as

11 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS 11 nice estimates on such approximations an compactness results on solutions without requiring compactness of initial ata. We will follow the approach of Perthame[5] in introucing the kinetic formulation, existence results, convergence estimates, an compactness results. We first introuce a simple, yet important function whose properties are critical in forming the kinetic formulation: Definition 3.1. The function χ : is efine by 1 if < ξ < u χ(ξ, u) = 1 if u < ξ < otherwise We prove a few basic properties: Proposition 3.. We have (1) S (ξ)χ(ξ, u) ξ = S(u) S() for S locally Lipschitz, i.e. S L loc, an in particular χ(ξ, u) ξ = u, () χ(ξ, u) χ(ξ, v) ξ = u v, (3) ξ (χ(ξ, u)) = δ(ξ) δ(ξ u), an (4) u (χ(ξ, u)) = δ(ξ u) for ξ. The last two statements are mae in the sense of istributions. The proof of these properties is an easy exercise to verify. We now consier an entropy solution u to u t +iv(a(u)) =, an efine the istribution m(t, x, ξ) by m(t, x, ξ) = ξ ξ χ(ζ, u(t, x)) ζ + iv x a(ζ)χ(ζ, u(t, x)) ζ. t It turns out that m has some interesting properties. For example, by ifferentiating both sies in ξ, we have the istributional equation m ξ = t (χ(ξ, u)) + a(ξ) x(χ(ξ, u)). Furthermore, we can show that m is nonnegative. Multiplying m by ϕ(ξ), where ϕ CC (), an integrating by parts yiels ϕ (ξ)m(t, x, ξ) ξ = ϕ(ξ)χ(ξ, u(t, x)) ξ + iv x ϕ(ξ)a(ξ)χ(ξ, u(t, x)) ξ t In particular, if we choose ϕ approaching S for S convex, we obtain, using the properties above, that S (ξ)m(t, x, ξ) ξ = S (ξ)χ(ξ, u(t, x)) ξ + iv x η (ξ)χ(ξ, u(t, x)) ξ t = t (S(u(t, x)) S()) + iv x(η(u(t, x)) η()) = t (S(u(t, x))) + iv x(η(u(t, x))). Since this hols for all S convex (i.e. for all S ), it follows that m is nonnegative.

12 1 YUZHOU ZOU Inspire by these results, we efine the kinetic formulation of the conservation law as follows: Definition 3.3. Let u C( + ; L 1 ( )). We say that u is a kinetic formulation to the equation u t + iv x (A(u)) =, u(, x) = u (x) if there exists a nonnegative boune measure m C (, w M 1 ( + )) such that t (χ(ξ, u(t, x))) + a(ξ) x (χ(ξ, u(t, x))) = m (t, x, ξ), ξ χ(ξ, in the sense of istributions. u(, x)) = χ(ξ, u (x)) Note that this efinition oes not require a L boun on u, an as such can be applie to initial ata which is not L. emark 3.4. If u is regular enough (say W 1,1 ), then applying the chain rule just as we i in Proposition.5 yiels t (χ(ξ, u)) + a(ξ) x(χ(ξ, u)) = δ(ξ u)(u t + a(ξ) x u) = δ(ξ u)(u t + iv x (A(u))) =. Hence, for u W 1,1, the corresponing measure is ientically zero. emark 3.5. If u is in L, then the corresponing measure m is compactly supporte in ξ, with support containe in ξ u L. In particular, the measure associate with entropy solutions are compactly supporte in ξ. To see this, note that for ξ > u L we have m(t, x, ξ) = ξ ξ χ(ζ, u(t, x)) ζ + iv x a(ζ)χ(ζ, u(t, x)) ζ t = χ(ζ, u(t, x)) ζ + iv x a(ζ)χ(ζ, u(t, x)) ζ t = t (u(t, x)) + iv x(a(u(t, x)) A()) =. The secon equality follows from the fact that χ(ζ, u) = for ζ not between an u, while the last equality follows from the fact that u satisfies the conservation law in the sense of istributions. 4. Existence an Uniqueness 4.1. Statement an Outline. We have the following existence/uniqueness theorem: Theorem 4.1. Let u L 1 ( ), an let A be locally Lipschitz, i.e. a (L loc ()). Then there exists a unique istributional istribution u C( + ; L 1 ( )) to the kinetic formulation. The proof will follow the following steps: We first solve the equation f λ t + a(ξ) xf λ = λ(χ(ξ, u λ ) f λ ), f λ t= = χ(ξ, u ) for every λ > through a fixe point argument, where u λ = f λ ξ. For every λ >, we fin a (nonnegative boune) measure m λ such that an hence m λ ξ = λ(χ(ξ, u λ ) f λ ) f λ t + a(ξ) xf λ = m λ ξ.

13 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS 13 This is similar to the esire equation in the kinetic formulation, except that we o not know if f λ has the structure of χ(ξ, u) for some u. Hence, we will show that {m λ } is uniformly boune for all λ >, an that the initial conition f λ t= = χ(ξ, u ) imposes extra conitions on the sign an bouns of f λ, in the hope that f λ χ(ξ, u λ ) = 1 m λ λ ξ λ in some way. We then argue that as λ, the sequences {u λ }, {f λ }, an {m λ } converge to u, f, an m, so that f t + a(ξ) xf = m ξ. an that f λ χ(ξ, u), i.e. f = χ(ξ, u), thus proving the existence of the esire solution. 4.. Solving a linear approximation. We first investigate properties of the solutions of the problem (4.1) f t + a(ξ) x f + λf = g, f = f t= in + ξ, where λ is a fixe positive parameter. Theorem 4.. Let f L 1 ( ξ ), g L 1 ((, T ) x ξ ), an a (L loc ()). Then there exists a istributional solution f C( + ; L 1 ( ξ )) to (4.1) given by the formula t f(t, x, ξ) = f (x a(ξ)t, ξ)e λt + Furthermore, this solution satisfies the properties f(t, x, ξ) x ξ + λ f(t, x, ξ) x ξ = ξ an ξ ξ f(t, x, ξ) x ξ + λ ξ e λs g(t s, x a(ξ)s, ξ) s. f(t, x, ξ) x ξ ξ ξ g(t, x, ξ) x ξ g(t, x, ξ) x ξ. Proof. For f an g smooth, we have ( e λt f(t, x + a(ξ)t, ξ) ) = e λt (f t + a(ξ) x f + λf)(t, x + a(ξ)t, ξ) = e λt g(t, x + a(ξ)t, ξ). Integrating from s = to s = t gives the esire formula. For f an g in L 1, let {fn} an {g n } be sequences of smooth functions such that fn f in L 1 ( +1 ) an g n g in L 1 ((, T ) +1 ). Letting {f n } enote the corresponing solutions, we note that the preceing formula gives t f n (t, x, ξ) f m (t, x, ξ) = (fn fm)(x a(ξ)t, ξ)e λt + e λs (g n g m )(t s, x a(ξ)s, ξ) s. Hence, integrating along x an ξ, an making an appropriate change of variables, we obtain t f n (t, ) f m (t, ) L1 ( +1 ) e λt fn fm L 1 + e λs g n (s, ) g m (s, ) L 1 s.

14 14 YUZHOU ZOU Thus, {f n } is a Cauchy sequence in C((, T ); L 1 ( +1 )), an thus converges to some f satisfying the same istributional formula. To erive the integral equation, we multiply (4.1) by ϕ (x) = ϕ(x/), where ϕ is a compactly supporte cutoff function, an integrate with respect to x an ξ. As, we have f t ϕ ( ) f, λfϕ λ f, gϕ g, an ( a(ξ) ϕ x f = a(ξ) ) f x ϕ x ξ = 1 a(ξ) f(x) x ϕ( x ) x ξ. This proves the integral equation. Finally, to erive the L 1 inequality, we multiply (4.1) by f η ɛ (f)ϕ (x) an integrate, where η ɛ (f) = ρ ɛ (y) y 1 an ρ ɛ is the stanar mollifier on. By similar arguments as above, as we have a(ξ) η ɛ (f)ϕ x f, while the ϕ term rops out in the other terms, leaing to the equation f t η ɛ (f) + λ fη ɛ (f) = gη ɛ (f). Letting ɛ yiels as esire. ( ) f + λ f = gsgn(f) g (4.) We use the results of this theorem to fin solutions to the problem f λ t + a(ξ) xf λ = λ(χ(ξ, u λ ) f λ ), f λ t= = χ(ξ, u ) To o this, we first fix v C((, T ); L 1 ( )), an consier the solution f to the equation f (4.3) t + a(ξ) xf + λf = λχ(ξ, v), f = χ(ξ, u ) t= Let Φ : C((, T ), L 1 ( )) C((, T ), L 1 ( )) be the operator sening v to the integral of the corresponing f, i.e. Φ(v) : (t, x) u(t, x) = f(t, x, ξ) ξ. We aim to show that Φ is a strict contraction on the Banach space C((, T ), L 1 ( )), with the norm u C((,T );L1 ( )) = sup u(t, ) L1 ( ) t (,T ) in orer to apply the Banach fixe point theorem an obtain a solution to (4.). To o this, let v 1, v C((, T ); L 1 ( )). Letting f = f 1 f, where f 1 an f are the solutions to (4.3) corresponing to v 1 an v, we see that f is a solution to (4.3) with g = χ(ξ, v 1 ) χ(ξ, v ) an f. We thus have (4.4) ξ f(t, x, ξ) x ξ + λ ξ f(t, x, ξ) x ξ λ ξ χ(ξ, v 1 (t, x)) χ(ξ, v (t, x)) x ξ = λ v 1 (t, x) v (t, x) x λ v 1 v C((,T );L1 ( )).

15 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS 15 It follows that e λt ξ f(t, x, ξ) x ξ ( e λt v 1 v C((,T );L 1 ( ))) an, using the fact that f(, x, ξ) x ξ = f x ξ =, we have Hence, we have ξ f(t, x, ξ) x ξ (1 e λt ) v 1 v C((,T );L1 ( )). Φ(v 1 ) Φ(v ) C((,T );L 1 ( )) = sup f 1 (t, x, ξ) f (t, x, ξ) ξ x sup f(t, x, ξ) x ξ t (,T ) t (,T ) ξ (1 e λt ) v 1 v C((,T );L1 ( )) showing that Φ is a strict contraction on C((, T ); L 1 ( )). Hence, for all T >, there exists a fixe point of Φ, i.e. there exists u λ C((, T ); L 1 ( )) an f λ C((, T ); L 1 ( ξ )) such that an f λ t + a(ξ) xf λ = λ(χ(ξ, u λ ) f λ ), f λ t= = χ(ξ, u ) u λ (t, x) = f λ (t, x, ξ) ξ. Moreover, u λ an f λ are efine on (, T ) for all T > an hence can be extene to be efine for all positive time Properties of approximate solutions an the approximate measure. We now prove some properties of the solutions f λ an u λ obtaine above. Since these properties o not epen on the specific value of λ, we rop it in the subscript for ease of notation. Theorem 4.3. Let u an f be as above. Then we have the representational formula t f(t, x, ξ) = χ(ξ, u (x a(ξ)t))e λt + λ Furthermore, we have the following properties: (1) The total mass is conserve, i.e. u(t, x) x =. e λs χ(ξ, u(t s, x a(ξ)s)) s. () For any initial ata u 1 an u, with corresponing u 1, f 1 an u, f, we have an L 1 contraction property u 1 (t, ) u (t, ) L1 ( ) f 1 (t, ) f (t, ) L1 ( +1 ) u 1 u L1 ( ).

16 16 YUZHOU ZOU Hence, for any initial ata u, setting u 1 = u ( +h) an u = u yiels the space-oscillation contraction property u( + h, t) u(, t) L 1 ( ) u ( + h) u ( ) L 1 ( ). (3) f(t, x, ξ) = sgn(ξ)f(t, x, ξ) 1. In other wors, the sign of f matches the sign of ξ, an f 1. (4) If u L ( ), then u(t, ) L u L, an f(t, x, ξ) = for ξ > u L. Proof. Since u an f satisfy the equation f t + a(ξ) xf + λf = g, f = χ(ξ, u ), t= with g = λχ(ξ, u), the representation formula hols, as well as the integral equality f x ξ + λ f x ξ = ξ ξ Since f x ξ = u x, it follows that ξ u(t, x) x + λ ξ u(t, x) x = λ λχ(ξ, u) x ξ = λ u(t, x) x, u x. thus proving mass conservation. The L 1 contraction property follows similarly, since f 1 f satisfies (4.3) with g = χ(ξ, u 1 ) χ(ξ, u ), an hence ( ) f 1 f x ξ + λ f 1 f x ξ λ χ(ξ, u 1 ) χ(ξ, u ) x ξ = λ u 1 u x λ f 1 f x ξ ( Hence, f1 f x ξ ), an since (f 1 f ) = u 1 u t=, the secon inequality in () follows. The first inequality follows easily from the fact that u = f ξ. To prove the sign property on f, we note χ 1, an the sign of χ(ξ, u) matches the sign of ξ. Using the representational formula, we see that the sign of f λ also matches the sign of ξ, while t f λ (t, x, ξ) e λt + λ e λs s = 1 as esire. To show the L bouns on u, we aim to show that the set {v C((, T ); L 1 ( )) : v L ((,T ) ) u L ( )}, a close subset of C((, T ); L 1 ( )), is invariant uner Φ, an hence the fixe point (i.e. u) must lie in that subset. For v with v L u L, consier the associate solution f v. We have t f v (t, x, ξ) = χ(ξ, u (x a(ξ)t))e λt + λ e λs χ(ξ, v(t s, x a(ξ)s, ξ)) s.

17 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS 17 From this formula, we see immeiately that f v = whenever ξ u L, an, using similar arguments as those use to prove (3), we obtain the sign property f v = sgn(ξ)f v 1. We can thus write Φ(v)(t, x) = u v (t, x) = f v (t, x, ξ) ξ = u L f v (t, x, ξ) ξ u L f v (t, x, ξ) ξ. Notice that both integrals on the right-han sie are positive, an since f v 1, both integrals are boune by u L. Thus u v, as a ifference of two positive terms, must have absolute value boune by the maximum of the two terms, an hence Φ(v)(t, x) u L for all t an x. This shows that u L ((,T ) ) u L ( ). Finally, from the representational formula for f, we clearly see that f(t, x, ξ) = for ξ > u L. We now show that we can fin a function m λ such that m λ ξ = λ(χ(ξ, u λ ) f λ ) = f λ t + a(ξ) xf λ. This brings the equation to a form similar to that in the kinetic formulation. We obtain m λ through the formula m λ (t, x, ξ) = λ ξ From this, we can prove several properties: χ(ζ, u λ (t, x)) f λ (t, x, ζ) ζ. Proposition 4.4. The function m λ satisfies the following properties: (1) m λ is nonnegative. () For any convex S with S () =, we have +1 S (ξ)m λ (t, x, ξ) x ξ S(u (x)) S() x. In particular, for S(ξ) = (ξ ξ ) ± (ξ ) we have m λ (t, x, ±ξ ) x (u (x) ξ ) ± x µ(±ξ ) where µ(ξ) u L 1 an lim µ(ξ) =, an if ξ u L, then for S(ξ) = ξ we have m λ L 1 1 u L. (3) If u L, then m λ (t, x, ξ) = for ξ > u L. Proof. Note that if u λ, then by the sign property of f λ we have χ(ζ, u λ ) f λ (t, x, ζ) = f λ (t, x, ζ) for ζ, Similarly, for < ζ < u λ, the integran is 1 f λ (t, x, ζ), while for ζ > u λ the integran is f λ (t, x, ζ). Hence, χ(ζ, u λ ) f λ (t, x, ζ) is nonnegative for ζ < u λ an nonpositive

18 18 YUZHOU ZOU for ζ u λ, assuming u λ. A similar statement also hols for u λ <. It follows that m λ (t, x, ξ) is increasing in ξ for ξ < u λ an ξ > u λ, an furthermore lim m λ(t, x, ξ) = ξ χ(ξ, u λ (t, x)) f λ (t, x, ξ) ξ = u λ (t, x) f λ (t, x, ξ) ξ =. Hence, m λ is nonnegative. To prove (), we multiply the equation m λ ξ = f λ t + a(ξ) x f λ by S (ξ) an integrate in x an ξ. By applying the arguments in Theorem 4., we may assume that the term S (ξ)a(ξ) x f vanishes. We thus have fλ t S (ξ) x ξ = S (ξ) m λ x ξ = S (ξ)m λ x ξ ξ an hence S (ξ)m λ x ξ = ( ) S (ξ)f λ x ξ. Integrating both sies with respect to time gives T S (ξ)m λ x ξ = +1 S (ξ)χ(ξ, u (x)) x ξ S (ξ)χ(ξ, u (x)) x ξ = S(u (x)) x. S (ξ)f λ (T, x, ξ) x ξ Note that S convex an S () = implies that the sign of S (ξ) matches that of ξ, an hence S (ξ)f λ (t, x, ξ) for all ξ. Letting T yiels the general estimate. For S(ξ) = (ξ ξ ) ±, we have S (ξ) = δ(ξ ξ ), so +1 S (ξ)m(t, x, ξ) x ξ = m(t, x, ±ξ ) x, yieling the first inequality. We note that (u (x) ξ ) ± x is clearly boune by u L 1, while the integral goes to as ξ from the Dominate Converge Theorem. Finally, the last inequality follows from noting that S(ξ) = ξ = S (ξ) = 1. To prove (3), we simply note that χ(ξ, u λ ) = f λ = for ξ > u L from Theorem 4.3, so by efinition m λ = for ξ > u L Convergence of approximate solutions. To obtain a kinetic solution from a family of approximate solutions, we seek to show that the family of approximate solutions is compact, in orer to extract subsequences {f λ }, {u λ }, an {m λ } converging to f, u, an m as λ. We also aim to show that f λ χ(ξ, u λ ) as λ, which will imply that f = χ(ξ, u), as esire. To show compactness, we nee uniform bouneness, equicontinuity, an uniform integrability. Uniform bouneness follows from the results of Theorem 4.3, while equicontinuity can be shown using the space-oscillation contraction in Theorem 4.3, combine with the following time continuity estimate: Proposition 4.5. There exists a moulus of continuity ω, inepenent of λ, such that u λ (k, ) u ( ) L1 ( ) f λ (k, ) χ(, u ) L1 ( ξ ) ω(k).

19 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS 19 Consequently, setting u = u(t, x) for some t > yiels the time continuity estimate u λ (t + k, ) u λ (t, ) L 1 f λ (t + k, ) f λ (t, ) L 1 ω(k). Proof. Set u ɛ = u ρ ɛ, where ρ ɛ is the stanar mollifier in. Then u ɛ L Set χ ɛ = χ(ξ, u ɛ), an let N ɛ = sup a(ξ). We have ξ u ɛ L ρ L ɛ u L 1. t (f χ ɛ) + a(ξ) x (f χ ɛ) + λ(f χ ɛ) = (f t + a(ξ) x f + λf) λχ ɛ a(ξ) x χ ɛ = λ(χ(ξ, u) χ ɛ) a(ξ) x χ ɛ. Hence, f χ ɛ satisfies equation (4.1), with g = λ(χ(ξ, u) χ ɛ) a(ξ) x χ ɛ an initial ata χ(ξ, u ) χ ɛ. We note that an From Theorem 4., we have Integrating in time thus gives g L 1 λ χ(ξ, u) χ(ξ, u ɛ) L 1 + x χ ɛ M 1 sup ξ u ɛ L a(ξ) χ(ξ, u) χ(ξ, u ɛ) L 1 = u u ɛ L 1 f χ ɛ L 1. ( f(t) χ ɛ L 1) + λ f χ ɛ L 1 λ f χ ɛ L 1 + N ɛ x u ɛ M 1. f(k) χ(ξ, u ) L 1 χ(ξ, u ) χ(ξ, u ɛ) L 1 + kn ɛ x u ɛ L 1 = u u ɛ L 1 + kn ɛ x u ɛ M 1. Letting ω 1 (k) = sup u ( + h) u L 1, we have h k u u ɛ L 1 u (x y) u (x) ρ ɛ (y) x y ω 1 (ɛ). By a similar argument, we obtain Hence, we have x χ ɛ M 1 = x u ɛ L 1 ρ ɛ L ω 1 (ɛ) ρ L ω 1 (ɛ). ɛ f(k) χ(ξ, u ) L 1 ( 1 + kn ) ɛ ω 1 (ɛ). ɛ Setting ω(k) equal to the infimum of the right-han sie over all ɛ > gives the esire moulus of continuity. We now show the uniform integrability of u λ, which shows the compactness of the family. Proposition 4.6. Let u L, an let N = sup ξ u L a(ξ). Then u λ (t, x) x u (x) x + CNt u L 1. x x Note that we only have to consier u L since, given u L 1, we can always regularize by convolution with some mollifiers such that the resulting convolution approximates u.

20 YUZHOU ZOU Proof. Let ϕ : be a nonnegative smooth function satisfying ϕ(x) = 1 for x 1, ϕ(x) = for x 1, an ϕ L = 1. Set ϕ (x) = ϕ(x/). We have t (f λϕ ) + a(ξ) x (f λ ϕ ) + λf λ ϕ = ϕ (f t + a(ξ) x f + λf λ ) + f λ a(ξ) x ϕ = λχ(ξ, u λ )ϕ + f λ a(ξ) x ϕ. From Theorem 4., we have ( ) f λ ϕ x ξ + λ f λ ϕ x ξ λ χ(ξ, u λ ) ϕ x ξ + f λ a(ξ) x ϕ x ξ. Since χ(ξ, u λ ) x ξ = u λ x f λ x ξ an f λ a(ξ) x ϕ x ξ N ϕ L f λ x ξ N ϕ L u λ (t, ) L 1 CN u L 1 we have ( ) f λ ϕ x ξ CN u L 1 an hence u λ (t, x) x x f λ ϕ x ξ χ(ξ, u ) ϕ x ξ+ CNt u L 1 x u x+ CNt u L 1, as esire. We thus obtain a sequence λ n an u L 1 such that u λn is to show the convergence of {f λn }. u in L 1 ([, T ] ). The next step Proposition 4.7. f λn χ(ξ, u) in L 1 ([, T ] ξ ) for the sequence {λ n } obtaine above. Proof. Assume first that u L. We use the representational formula to have (4.4) t f λ (t, x, ξ) χ(ξ, u(t, x)) = χ(ξ, u (x a(ξ)t))e λt + λ χ(ξ, u(t, x))(e λt + (1 e λt )) = e λt (χ(ξ, u (x a(ξ)t)) χ(ξ, u(t, x))) e λs χ(ξ, u λ (t s, x a(ξ)s)) s t + λ e λs (χ(ξ, u λ (t s, x a(ξ)s)) χ(ξ, u(t, x))) s.

21 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS 1 If we integrate the LHS over x an ξ, the first term in the HS is boune by f L 1e λt, while the secon term is boune by + + t t t λe λs λe λs λe λs χ(ξ, u λ (t s, x a(ξ)s)) χ(ξ, u(t s, x a(ξ)s)) x ξ s χ(ξ, u(t s, x a(ξ)s)) χ(ξ, u(t, x a(ξ)s)) x ξ s χ(ξ, u(t, x a(ξ)s)) χ(ξ, u(t, x)) x ξ s. Substituting λ = λ n an letting n, we note that the first term is, after a change of variables, boune by t λ n e λns χ(ξ, u λn (t s, y)) χ(ξ, u(t s, y)) y ξ s = t λ n e λns u λn (t s, ) u(t s, ) L 1 which vanishes as n since u λn u. The secon term also vanishes as n by a similar argument. To control the thir term, we note that if u is locally Lipschitz an compactly supporte, with N = a(ξ) an supp u [, T ] K, then an hence t sup ξ u L χ(ξ, u(t, x a(ξ)s)) χ(ξ, u(t, x)) x ξ 1 ξ (u(t,x a(ξ)s),u(t,x)) x ξ λe λs Ns sup u(t, x y) u(t, x) x y Ns χ(ξ, u(t, x a(ξ)s)) χ(ξ, u(t, x)) x ξ s N K x u L K x u L x = Ns K x u L t λ n se λns s n. Otherwise, we can regularize u by space convolution an truncation to obtain a locally Lipschitz an compactly supporte u δ such that u(t, ) u δ (t, ) L 1 δ an supp u δ B 1/δ for each t, an the integral with u can thus be controlle by controlling the integral with u δ. Hence, we see that f λn converges to χ(ξ, u) if u is L. For u not in L, we can simply regularize by convolution to obtain L initial ata {u δ } such that u δ u δ L 1, an similarly with u an {f λ }. The conclusion follows from the contraction f λn,δ f λ L 1 u δ u L 1. s, We conclue that f λn χ(ξ, u). To conclue the proof of existence, we note that the functions m λ satisfy the uniform local boun (,) m λ (t, x, ξ) x ξ max ξ µ(ξ) u L 1.

22 YUZHOU ZOU Hence, we can extract a subsequence from {m λn } which converges weakly to some measure m, thus proving the existence of a solution to the kinetic formulation Uniqueness. To finish showing that the kinetic formulation of conservation laws is inee well-pose, we must show that kinetic solutions are unique an epen continuously (in L 1 ) on the initial ata. Both can be shown by showing the following contraction principle: Theorem 4.8. Let u 1 an u be two kinetic solutions with corresponing initial ata u 1 an u. Then u(t, x) v(t, x) x u 1(x) u (x) x. The proof will be sketche below without regar to regularity or rigor. A completely rigorous proof can be foun in [5] an involves regularizing the χ functions by convolution in time an space. Proof. Let m 1 an m be the corresponing measures. Note that, for fixe (t, x), we have u 1 u = χ(ξ, u 1 ) χ(ξ, u ) ξ = χ(ξ, u 1 ) χ(ξ, u ) ξ = χ(ξ, u 1 ) + χ(ξ, u ) χ(ξ, u 1 )χ(ξ, u ) ξ = χ(ξ, u 1 ) + χ(ξ, u ) χ(ξ, u 1 )χ(ξ, u ) ξ since χ = χ an χ(ξ, u 1 ) χ(ξ, u ) can only take the values or 1. It thus suffices to show that ( ) χ(ξ, u 1 ) + χ(ξ, u ) χ(ξ, u 1 )χ(ξ, u ) x ξ. If we multiply the equation t (χ(ξ, u)) + a(ξ) xu = m ξ by sgn(ξ) an integrate, we obtain ( ) χ(ξ, u) x ξ = sgn(ξ) m x ξ = m x. ξ ξ= (As before, the term containing a(ξ) rops out after integration). It follows that ( ) (4.5) χ(ξ, u 1 ) + χ(ξ, u ) x ξ = (m 1 + m ) x. ξ= Similarly, if we multiply the equation t (χ(ξ, u 1)) + a(ξ) x u 1 = m1 ξ of u 1 an u, an a, we obtain by χ(ξ, u ), switch the roles t (χ(ξ, u 1)χ(ξ, u )) + a(ξ) x (χ(ξ, u 1 )χ(ξ, u )) = χ(xi, u ) m 1 ξ + χ(ξ, u 1) m ξ an hence integration yiels ( ) χ(ξ, u 1 )χ(ξ, u ) x ξ = m 1 ξ (χ(ξ, u )) + m ξ (χ(ξ, u 1)) x ξ = m 1 (δ(ξ) δ(ξ u )) + m (δ(ξ) δ(ξ u 1 )) x ξ = (m 1 + m )δ(ξ) x ξ + m 1 δ(ξ u ) + m δ(ξ u 1 ) x ξ (m 1 + m )δ(ξ) x ξ = (m 1 + m ) x. ξ= Subtracting the above inequality from equation (4.5) gives the esire result.

23 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS 3 5. Convergence in the Diffusion Approximation We now prove convergence estimates for solutions of the conservation laws approximation u t + iv x (A(u)) = ɛ u, using the kinetic formulation. In this section, let u be a kinetic solution to u t + iv x (A(u)) = with initial ata u, an let ω an ω 1 be the time moulus of continuity of u an initial moulus of continuity of u, respectively, i.e. an We have the following estimate: ω (k) = sup u(s, ) u L 1 s k ω 1 (k) = sup u ( + h) u L 1. h k Theorem 5.1. Suppose v L ((, T ); L 1 ( )) satisfies the equation t (χ(ξ, v)) + a(ξ) x(χ(ξ, v)) = m ξ + α + β t + iv x(β) + i,j x i x j γ ij for some nonnegative measure m, with initial ata v, where α,β,β, an γ satisfy ᾱ, β, γ M 1 ((, T ) ), β L ((, T ); M 1 ( )) where ᾱ(t, x) = α(t, x, ) L 1 (), an similarly for the other terms. ɛ 1, ɛ >, we have v(t ) u(t ) L 1 v u L 1 + ω 1 (ɛ ) + ω (ɛ 1 ) + ᾱ M 1 + C ɛ β M 1 + C ɛ γ M 1 + ( + CT ) ɛ 1 Then, for T > an any sup t [,T ] The estimate is proven in a similar fashion to that of the uniqueness theorem. We now wish to consier a solution u ɛ to the iffusion problem (u ɛ ) t + iv x (A(u ɛ )) = ɛ u ɛ, u ɛ (, x) = u (x) an estimate the rate of convergence of u ɛ to u as ɛ. We have t (χ(ξ, u ɛ)) + a(ξ) x (χ(ξ, u ɛ )) = δ(ξ u ɛ )((u ɛ ) t + a(ξ) x u ɛ ) β (t, x) x. = δ(ξ u ɛ )((u ɛ ) t + iv x (A(u ɛ ))) = δ(ξ u ɛ )ɛ u ɛ. We can apply Theorem 5.1 either with γ ij = ɛδ ij δ(ξ u ɛ )u ɛ (δ ij being the Kronecker elta) an all other terms equal to zero, or with β = ɛδ(ξ u ɛ ) x u ɛ an all other terms zero. These lea to the following results: Theorem 5.. For any ɛ >, we have If, in aition, u BV, we have u ɛ (T ) u(t ) L 1 ω 1 (ɛ ) + C (ɛ ) (T ɛ u L 1). u ɛ (T ) u(t ) L 1 C u T V T ɛ. Proof. If we let γ ij = ɛδ ij δ(ξ u ɛ )u ɛ, then in Theorem 5.1 we can take ɛ 1 = (since all terms associate with ɛ 1 vanish) an ɛ = ɛ. We thus have u ɛ (T ) u(t ) L 1 ω 1 (ɛ ) + C (ɛ ) γ M 1.

24 4 YUZHOU ZOU T Since γ ij M 1 = ɛδ ij u ɛ M 1 = ɛδ ij u ɛ (T ) L 1 ( ) T ɛδ ij u L 1, it follows that γ M 1 = γ ij M 1 T ɛ u L 1, an the first result follows. If we assume u BV an take i,j=1 β = ɛδ(ξ u ɛ ) x u ɛ, we can again take ɛ 1 = an ɛ = ɛ, this time yieling u ɛ (T ) u(t ) L 1 ω 1 (ɛ ) + C ɛ β M 1 ɛ u T V + C ɛ β M 1. Since β M 1 = ɛ T δ(ξ u ɛ ) x u ɛ ξ x = ɛ x u ɛ (t, x) x x T ɛ u T V, we obtain ( u ɛ (T ) u(t ) L 1 u T V ɛ + CT ɛ ) ɛ. Minimizing the HS with respect to ɛ yiels the esire result. 6. Compactness an Averaging Lemmas We conclue by proving a result on the compactness of entropy solutions. To o so, we will use the following averaging compactness theorem: Theorem 6.1. Let >, an let a satisfy the non-egeneracy conition Consier the transport equation {ξ : ξ <, a(ξ) ζ + α = } = α, ζ S 1. f t + a(ξ) x f = m k= k ξ k (iv t,x,ξg k ), an let {f n } n an {g k,n } n be sequences of functions satisfying this equation. Let ψ C C (), an let ρ n (t, x) = ψ(ξ)f n (t, x, ξ) ξ. If, for some 1 < q <, the sequence {f n } is boune in L q ( + ), an {g k,n } is relatively compact in L q ( + ξ, + ), then the averages {ρ n } are relatively compact in L q ( + ). The proof can be foun in [4] or [5]. The iea is to take the Fourier transform with respect to time an space of the transport equation to obtain (τ + a(ξ) η) ˆf m k (( = ξ k τ, η, ) ) ĝ k ξ an hence ˆf = 1 τ + a(ξ) η k= m k= k (( ξ k τ, η, ) ĝ k ). ξ where τ an η are the time an space Fourier variables, respectively. It follows that ˆf ecays quickly (an hence ensures regularity) whenever the quantity τ + a(ξ) η is far from zero, an the non-egeneracy conitions ensure that this is inee the case almost everywhere. We can now prove the following compactness result on entropy solutions: Theorem 6.. Assume a satisfies the non-egeneracy conition in Theorem 6.1. Let {u n } be a sequence of entropy solutions to u t + iv x (A(u)) = with uniform L 1 an L bouns. Then {u n } is locally relatively compact in L p ( + ) for all 1 p <. emark 6.3. Notice that no compactness requirement is neee for the initial ata, only that the solutions be uniformly boune. Furthermore, the non-egeneracy conition is neee for the result, to rule out the cases of transport equations of the form u t + iv x (au) =, a.

25 ENTOPY AND KINETIC FOMULATIONS OF CONSEVATION LAWS 5 Proof. Let χ n = χ(ξ, u n ). From the kinetic formulation, we have χn t + a(ξ) x χ n = mn ξ. We first localize the functions χ n so that they are uniformly supporte in t an x for every ξ in orer to attain convergence more easily, since we are only seek local compactness results on u n. More specifically, let < t 1 < t, let K be compact, let φ 1 C C (+ ) satisfy φ 1 (t) = 1 for t 1 t t, let φ C C ( ) satisfy φ (x) = 1 for x K, an set f n (t, x, ξ) = φ 1 (t)φ (x)χ n. Then {f n } is uniformly supporte in t an in x, an since {u n } has a uniform L boun, it follows that {χ n } an hence {f n } are uniformly supporte in ξ as well. We have f n t + a(ξ) xf n = φ χ n 1φ χ n + φ 1 φ + φ 1 (a(ξ) x φ )χ n + φ 1 φ a(ξ) χ n t = φ 1 φ m n ξ := m1 n ξ + m n + (φ 1φ + φ 1 a(ξ) x φ )χ n where m 1 n = φ 1 φ m n an m n = (φ 1φ + φ 1 a(ξ) x φ )χ n are uniformly boune measures with uniform compact support. We now claim that there exist Mn 1 an Mn in W 1,1 ( + ξ, + ) such that m i n = iv t,x,ξ Mn. i Inee, if we solve the equations vn i = m i n, an set Mn i = vn, i then m i n = iv t,x,ξ Mn, i an the uniform boun on {m i n} implies that {vn} i uniformly boune in W,1, an hence {Mn} I is uniformly boune in W 1,1. Since {Mn} i are also compactly supporte, we can apply the ellich- Konrachov theorem to conclue that {Mn} i n is compact in L q for 1 q < Hence, if we fix some 1 < q < + +1, we can apply Theorem 6.1 to conclue that {ρ n} is compact in L q ( +1 ), where ρ n = ψ(ξ)f n (t, x, ξ) ξ, for any ψ CC (). Since f n(t, x, ξ) = for ξ > sup u n L, we can choose ψ so that ψ(ξ) = 1 for ξ sup u n L, in which case ρ n (t, x) = φ 1 (t)φ (x)χ(ξ, u n ) ξ = φ 1 (t)φ (x)u n (t, x). It follows that {φ 1 φ u n } is compact in L q ( +1 ), an in particular {u n } is compact in L q (K ), where K = [t 1, t ] K. Hence, there exists u L q (K ) such that u n u, up to subsequence. Since {u n } is uniformly boune in L (K ), it follows that u L (K ). Note that we have the continuous injection L p (K ) L q (K ) L (K ) for all q < p <, an since K has finite measure, we also have the continuous injection L p (K ) L q (K ) for all 1 p < q. It follows that u L p (K ) an u n u in L p (K ) for all 1 p <. This shows that {u n } is relatively compact in L p (K ) for all 1 p <. Since every compact K + is containe in [t 1, t ] K for some < t 1 < an K compact, it follows that {u n } is locally relatively compact in L p for all 1 p <. Acknowlegments. I woul like to thank Professor Takis Souganiis for his guiance as my primary mentor in the stuy of conservation laws. I woul also like to Casey origuez for answering many of my questions regaring integration an istribution theory, an to Professor Peter May for organizing the University of Chicago Mathematics EU. eferences [1] L. C. Evans. Partial Differential Equations. AMS Press. n e. 1. [] A. Frieman. Partial ifferential equations of the parabolic type. Prentice-Hall [3] S. N. Kruzkov. First Orer Quasilinear Equations in Several Inepenent Variables. Mathematics of the USS. Sbornik Vol. 1. No. [4] P.-L. Lions, B. Perthame, an E. Tamor. A kinetic formulation of multiimensional scalar conservation laws an relate equations. J. Amer. Math. Soc Vol.7 [5] B. Perthame. Kinetic Formulation of Conservation Laws. Oxfor University Press..