$$L^{1}$$ L 1 convergences and convergence rates of approximate solutions for compres
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RESEARCH
L1 convergences and convergence rates of approximate solutions for compressible Euler equations near vacuum Hsin-Yi Lee1 , Jay Chu2* , John M. Hong1 and Ying-Chieh Lin3 * Correspondence:
[email protected] Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan Full list of author information is available at the end of the article 2
Abstract In this paper, we study the rarefaction wave case of the regularized Riemann problem proposed by Chu, Hong and Lee in SIMA MMS, 2020, for compressible Euler equations with a small parameter ν. The solutions ρν and vν of such problems stand for the density and velocity of gas flow near vacuum, respectively. We show that as ν approaches 0, the solutions ρν and vν converge to the solutions ρ and v, respectively, of pressureless compressible Euler equations in L1 sense. In addition, the L1 convergence rates of these physical quantities in terms of ν are also investigated. The L1 convergences and convergence rates are proved by two facts. One is to invent an a priori estimate coupled with the iteration method to the high-order derivatives of Riemann invariants so that we obtain the uniform boundedness of ∂xi ρν (i = 0, 1, 2) and j ∂x vν (j = 0, 1, 2, 3) on the requisite regions. The other is about convexity of characteristic curves, which is used to estimate the distances among characteristic curves in terms of ν. These theoretic results are also supported by numerical simulations. Keywords: Compressible Euler equations, Vacuum, Hyperbolic systems of conservation laws, Riemann invariants, Regularized Riemann problem, Convergence rate, Method of characteristics, A priori estimate Mathematics Subject Classification: 35L45, 35L65, 35L67, 35L81
1 Introduction The study of multiple-scales solutions to Cauchy problems of n-by-n hyperbolic conservation laws has been an important and challenging subject in the fluid dynamics and related fields. Such Cauchy problems can be modeled as
Ut (x, t, ν) + (fν (U (x, t, ν)))x = 0, U (x, 0, ν) = (x, ν),
(1.1)
where ν is a sufficiently small positive parameter, U (x, t, ν), denoted by Uν , is an unknown n-vector function, fν is a smooth n-vector function of U and ν, and (x, ν) is a n-vector function with bounded total variation. The limiting problem of (1.1) is given by setting
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ν ≡ 0 in (1.1), which is
Ut (x, t) + (f0 (U (x, t)))x = 0, U (x, 0) = 0 (x),
(1.2)
where f0 ≡ limν→0 fν and 0 (x) ≡ (x, 0). Note that fν converges to f0 uniformly with respect to ν in any compact set of U due to the smoothness of fν . Problem (1.1) can be considered as a perturbed problem of (1.2) in terms of ν. Whether (1.1) is a regular or singular perturbed problem of (1.2) has been an interesting issue in conservation laws, but under the best of our knowledge, only limited results were established. One of the important topics related to this subject is whether Uν converges to U in some function spa
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