Differentiability with respect to initial data for a scalar conservation law François BOUCHUT François JAMES Abstract We linearize a scalar conservation law around an entropy initial datum. The resulting equation is a linear conservation law with discontinuous coefficient, solved in the context of duality solutions, for which existence and uniqueness hold. We interpret these solutions as weak derivatives with respect to the initial data for the nonlinear equation. Proceedings of the seventh Conference on Hyperbolic Problems (Zürich 1998), M. Fey & R. Jeltsch, Eds., International Series of Numerical Mathematics, 129 Birkhäuser, Basel, 1999, 113-118. 1 Introduction Consider the one-dimensional scalar conservation law t u + x f(u) = 0, 0 < t < T, x R, (1) where f is a C 1 convex function, provided with entropy admissible initial data u L (R). Kružkov s results [4] assert that the entropy solution u to (1) lies in L (]0, T [ R) C(0, T ; L 1 loc (R)), and that the following contraction property holds: if u (resp. v) corresponds to the initial data u (resp. v ), then for all R > 0 and any t > 0 u(t, x) v(t, x) dx u (x) v (x) dx, (2) x R x R+Mt where M = max{ f (s), s max( u L, v L )}. This can be interpreted as a continuity result with respect to initial data in L 1 loc. Another kind of stability results was proved by e.g. Majda [5] in the very general setting of a multi-dimensional system of equations: when u is an admissible shock, then the solution corresponding to a smooth perturbation of the shock is also an admissible shock. Mathématiques, Applications et Physique Mathématique d Orléans, UMR CNRS 6628, Université d Orléans, BP 6759, 45067 Orléans Cedex 2, FRANCE, francois.bouchut@ens.fr & francois.james@univ-orleans.fr 1
The aim of this paper is to give a few hints about the analysis of a different point of view. We are going to prove some kind of differentiability with respect to initial data, thus generalizing (2). The derivative of the operator u u is solution to the equation obtained by linearizing (1) in the neighborhood of some given u : t µ + x (a(t, x)µ) = 0, a(t, x) = f (u(t, x)). (3) We shall prove existence and uniqueness for the corresponding Cauchy problem, thus obtaining the first order term of a Taylor expansion of the operator. When applied to the perturbation of a shock, we can recover the perturbed shock as well as an approximation of its position. We would like to mention here that a numerical application of this framework is used by Olazabal [6] in fluid dynamics, where a 2-dimensional perturbation of a 1-dimensional shock is computed by simultaneously discretizing in a convenient way both (1) and (3) (see also [3]). It turns out that the conservation law with discontinuous coefficients (3) has to be solved in the space of measures on R. This leads to the classical problem of defining the product aµ of a discontinuous function by a measure. Moreover, in order to use (3) for the stability analysis of (1), we need to define this product in such a way that we have weak stability when the coefficient a is perturbed. This was achieved in a preceding paper by the authors [1, 2], where the so-called duality solutions are defined. For these solutions, existence and uniqueness hold for the Cauchy problem associated to (3), as well as stability. These results are recalled without proof in Section 2. In Section 3 we interpret the above results as weak differentiability results with respect to an entropy initial datum, when the flux f is strictly convex. Actually, the solution to (3) behaves like a directional or Gâteaux derivative of the operator which to u associates the entropy solution to (1). However, this is not rigorously a derivative, since u lies in L 1 loc (R), and the derivative is continuous only in the weak sense of measures. We also consider the example of an entropy initial shock, and show how to recover Majda s result. 2 Duality solutions We recall here briefly the definition and main properties of the so-called duality solutions, which were introduced by Bouchut and James [1, 2]. For detailed results and proofs, we refer to [2]. Duality solutions are measure-valued solutions µ belonging to the space S M = C([0, T ], M loc (R) σ(m loc (R), C c (R))), where σ denotes the usual weak topology. They are defined as weak solutions, the test functions being Lipschitz solutions to the backward linear transport equation t p + a(t, x) x p = 0, p(t,.) = p T Lip(R). (4) 2
A formal computation shows that t (pµ) + x [a(t, x)pµ] = 0, and thus d p, µ = 0, (5) dt which defines the duality solutions for suitable p-s. It is well known (see e.g. Oleinik [7]) that the existence of solutions to (4) is ensured by the so-called one-sided Lipschitz condition x a α(t) in ]0, T [ R, α L 1 (]0, T [), (6) and that this existence result gives uniqueness for (1) the same will occur for (3). The point here is that there is no uniqueness for (4) even when (6) holds. The corner stone in the construction of duality solutions is therefore the introduction of the notion of reversible solutions to (4). We shall not need here the precise properties of reversible solutions. We only have to know that they can be characterized in various ways: support properties of x p, monotonicity properties, total variation properties, entropy inequalities. We shall admit that there is existence and uniqueness to the backward Cauchy problem (4) in the class of reversible solutions. We now restrict ourselves to those p-s in (5). More precisely, we state the following definition. Definition 2.1 We say that µ S M is a duality solution to (3) if for any 0 < τ T, and any reversible solution p to (4) with compact support in x, the function t p(x, t)µ(t, dx) is constant on [0, τ]. R The main results concerning the Cauchy problem associated to (3) are summarized as follows (see Bouchut & James [2]). Theorem 2.2 (Cauchy problem) 1) Given µ 0 M loc (R), there exists a unique µ S M duality solution to (3), such that µ(., 0) = µ 0. 2) This solution satisfies for any x 1 < x 2 and t [0, T ] µ(t, dx) µ 0 (dx). (7) [x 1,x 2] [x 1 a t,x 2+ a t] 3) There exists a bounded Borel function â, such that â = a almost everywhere, and t µ + x (âµ) = 0 in the distributional sense. The set of duality solutions is clearly a vector space, but it has to be noted that a duality solution is not defined as a solution in the sense of distributions. The product âµ is defined a posteriori, by the equation itself. Remark 2.3 It is useful to notice that L distributional solutions to (3) are duality solutions. 3
We turn now to the most important property of duality solutions, namely weak stability. Theorem 2.4 (Weak stability) Let (a n ) L (]0, T [ R) be a bounded sequence, with a n a in L (]0, T [ R) w. Assume x a n α n (t), where (α n ) is bounded in L 1 (]0, T [), x a α L 1 (]0, T [). Consider a sequence (µ n ) S M of duality solutions to t µ n + x (a n µ n ) = 0 in ]0, T [ R, such that µ n (0,.) is bounded in M loc (R), and µ n (0,.) µ 0 M loc (R). Then µ n µ in S M, where µ S M is the duality solution to t µ + x (aµ) = 0 in ]0, T [ R, µ(0,.) = µ 0. Moreover, â n µ n âµ weakly in M loc (]0, T [ R). Remark 2.5 Duality solutions can be defined if α L 1 (]0, T [) is replaced by α L 1 loc (]0, T [). The same existence result holds, but uniqueness is lost. A weakened form of Theorem 2.4 is valid, up to a subsequence of (µ n ). 3 Differentiability Let us now interpret these results as differentiability results. We first compute some kind of directional derivative of the solution with respect to the initial data. Next, we apply this to the stability analysis of a shock. Theorem 3.1 Assume that f L and f γ > 0. Consider an entropy admissible initial datum u, i.e. x u C for some constant C. Let u λ be the solution of (1) with initial datum u + λδu, where λ > 0 and δu L (R), and set w λ = (u λ u)/λ. Then w λ µ in S M, where µ is the unique duality solution to t µ + x (aµ) = 0, µ(0,.) = δu. (8) Proof. We first have to check that (8) is well-posed. But this follows immediately since f is strictly convex and smooth: indeed Oleinik s entropy condition and the fact that u is admissible imply that a = f (u) satisfies (6) with α = constant and x u C. Next, we notice that w λ is a L distributional solution to the equation t w λ + x (a λ w λ ) = 0, w λ (x, 0) = δu (x), (9) f(u λ ) f(u) if u λ u, where a λ = u λ u f (u) if u λ = u. 4
The key point now is to prove that w λ is actually a duality solution to (9). Since w λ is a distributional solution, by Remark 2.3 we only have to verify that a λ satisfies (6) with some α L 1 loc (]0, T [). Once again we use Oleinik s entropy condition and the strict convexity of f, to obtain x u λ 1/(γt). A direct computation and the convexity of f lead to x a λ f /(γt). Finally, a λ is bounded in L (]0, T [ R) and converges almost everywhere to a = f (u), so that a λ a in L w as λ tends to 0. We are thus in position to apply the weakened version of the stability theorem 2.4 (see Remark 2.5), which gives exactly the expected convergence result, since the limit equation (8) has a unique solution. We actually proved here that µ can be interpreted as a weak directional derivative. Indeed, consider the operator J : L S M which to the initial data u associates the solution u to (1). The above convergence result states exactly that the duality solution µ to (3) is in a weak sense the derivative DJ(u ; δu ) of J at u in the direction δu. Moreover, (7) in Theorem 2.2 asserts exactly that DJ(u ; δu ) is a continuous linear operator on M loc (R) with respect to δu, which means that, also in some weak sense, J is Gâteaux differentiable in u, provided u is an admissible initial datum, and f is convex. We illustrate this result on a simple example: consider for u an admissible shock u = u l 1I x<0 + u r 1I x>0, with u l > u r, and assume that δu is a smooth compactly supported function. Denote by u R the solution to the unperturbed Riemann problem. Our result states that the solution u λ to this problem satisfies u λ = u R + λµ + λν λ, (10) where ν λ 0 in S M and µ solves (8). Actually, µ can be computed explicitely by integrating (8). This leads to a transport equation with discontinuous coefficient, which can be solved in the sense of duality solutions (see [2]). Differentiating the solution, we get µ(t, x) = δu (x f (u l )t)1i x<σt + δu (x f (u r )t)1i x>σt + + [ v 1 ((σ f (u r ))t) v 1 ((σ f (u l ))t) ] δ x σt, (11) where σ is the shock velocity, σ = [f(δu )]/[δu ], and v 1 a primitive of δu. Another approach to this problem, followed by Majda [5], is to prove directly that, for λ small enough, the solution u λ is a function with a smooth discontinuity line x = ξ λ (t). One can perform an asymptotic expansion of u λ and ξ λ in terms of λ. Quite straightforward computations, which are detailed for instance in [3], lead to the following expressions, up to higher order terms in λ. First, concerning the perturbed shock speed, σ λ = dξ λ /dt = σ 0 + λσ 1, with σ 0 (t) σ, σ 1 (t) = (f (u r ) σ)δu ((σ f (u r ))t) (f (u l ) σ)δu ((σ f (u l ))t) u r u l. Next, the solution is given by { ul + λδu u λ (t, x) = (x f (u l )t) for x < σt + λξ 1 (t), u r + λδu (x f (u r )t) for x > σt + λξ 1 (t), 5
where ξ 1 (t) = t 0 σ1 (s) ds. The last approach thus gives an approximation of the solution as a discontinuous function, as well as an approximation of the position of the perturbed shock. Our framework may seem strange at first sight, since we approximate a function by a measure concentrated on the original shock. However, the information on the perturbed shock position is also contained in formulæ (10)-(11), since the coefficient of the Dirac mass in (11) equals exactly (u r u l )ξ 1 (t). References [1] F. Bouchut and F. James, Équations de transport unidimensionnelles à coefficients discontinus, C.R. Acad. Sci. Paris, Série I, 320 (1995), 1097-1102. [2] F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Analysis TMA, 32 (1998), n 7, 891-933. [3] E. Godlewski, M. Olazabal and P.-A. Raviart, On the linearization of hyperbolic systems of conservation laws. Application to stability, Équations aux dérivées partielles et applications, articles dédiés à J.-L. Lions, Gauthier- Villars, Paris, (1998), 549-570. [4] S.N. Kružkov, First-order quasilinear equations in several independant variables, Math. USSR Sb., 10 (1970), 217-243. [5] A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 275 (1982). [6] M. Olazabal Résolution numérique du système des perturbations linéaires d un écoulement MHD, Thèse université Paris 6, 1998. [7] O.A. Oleinik, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. (2), 26 (1963), 95-172. 6