Continuity of the flow map for symmetric hyperbolic systems and its application to the Euler-Poisson system
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UWE BRAUER∗ AND LAVI KARP Dedicated to Lawrence Zalcman Abstract. We show the continuity of the flow map for quasilinear symmetric hyperbolic systems with general right-hand sides in different functional setting, including weighted Sobolev spaces Hs,δ . An essential tool to achieve the continuity of the flow map is a new type of energy estimate, which we call a low regularity energy estimate. We then apply these results to the Euler–Poisson system which describes various systems of physical interest.
1
Introduction
The purpose of our work is to prove the continuity of the flow map for quasilinear symmetric hyperbolic systems in the topology of either the ordinary Sobolev spaces H s , or the weighted Sobolev spaces Hs,δ , and under the assumption that lower order terms have limited regularity. Once that is proven, we then apply this result to the Euler–Poisson–Makino system. The Euler–Poisson–Makino system is a modification of the Euler–Poisson system in which the density is replaced by a variable, which we denote as the Makino variable from now on. This variable is a nonlinear function of the density. This variable change allows us to include situations where the density can be zero. The Euler–Poisson–Makino system consists of quasilinear symmetric hyperbolic evolution equations coupled to an elliptic equation. The existence and the uniqueness of solutions in this setting have been proved already by Makino [Mak86] in the H s spaces, and recently by [BK18] in the weighted Sobolev spaces Hs,δ . Thus our result about the continuity of the flow map shows that these systems are well-posed in the sense of Hadamard. ∗ U. B. gratefully acknowledges support from Grant MTM2016-75465 and Grant PID2019103860GB-I00 by MINECO, Spain and UCM-GR17-920894.
113 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0125-4
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U. BRAUER AND L. KARP
The general symmetric hyperbolic system we have in mind has the following form: ⎧ ⎨A0 (U)∂ U + d Aa (U)∂ U = G(U), t a a=1 (1.1) ⎩U(0, x) = u0 (x), where A0 , A1 , . . . , Ad are N × N smooth symmetric matrices, A0 is positive definite and G : RN → Rd is a nonlinear function. It is well known that if u0 belongs to the Sobolev space H s and s > d2 + 1, then there exists a positive T and a unique solution U to (1.1) such that (1.2)
U ∈ C0 ([0, T]; H s) ∩ C1 ([0, T]; H s−1).
This result was first proved by Marsden and Fischer [FM72], and Kato [Kat75a]. The goal of this paper is to investigate the continuous dependence on the initial data, or equivalently the continuity of the flow map. Definition 1 (Continuity of the flow map). We say that the flow map is continuous if for any u0 ∈ H s , there exists a neighborhood B ⊂ H s of u0 such that for every u ∈ B the map u → U from B to C0 ([0, T]; H s) is continuous, where U denotes the solution to (1.1) with initial data u. Equivalently, let {un } ⊂ B and U n be the corresponding solution to (1.1) with initial data un , and U 0 the solution with initial data u0 . Then the flow map is continuous, if un → u0 in H s implies
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