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Existence of global solutions to isentropic gas dynamics equations with a source term

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Science China Mathematics 2010年第1期53卷 115-124页

作者： klingenberg christian Institut fr angewandte Mathematik Universitt WrzburgAm Hubland97074 WrzburgGermany

In this paper we prove existence of isentropic gas dynamic equations with a source term (1.2). To this end we construct a sequence of regular hyperbolic systems (1.1) to approximate the inhomogeneous system of isentropic gas dynamics (1.2). First,for each fixed approximation parameter δ and very general condition on P (ρ),we establish the existence of entropy solutions for the Cauchy problem (1.1) with bounded initial date (1.4). Second,letting=o(δ),we obtain a complete proof of the H-1loc compactness of weak entropy pairs of system (1.2) in the form η(ρ,u) =ρH(ρ,u) given in Chen-LeFloch (2003). Finally,for the conditions of P(ρ) given in Chen-LeFloch (2003),applied to the results in Theorems 1 and 2,we obtain the global existence of entropy solutions for the Cauchy problem (1.2) with bounded initial date (1.4).

关键词： isentropic gas dynamics system global entropy solution compactness in H-loc1 compensated compactness method

GLOBAL WEAK SOLUTIONS FOR A NONLINEAR HYPERBOLIC SYSTEM

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Acta Mathematica Scientia 2020年第5期40卷 1185-1194页

作者： Qingyou SUN Yunguang LU christian klingenberg K.K.Chen Institute for Advanced Studies Hangzhou Normal UniversityHangzhou 311121China Department of Mathematics Wuerzburg UniversityWuerzburg 97070Germany

In this paper, we study the global existence of weak solutions for the Cauchy problem of the nonlinear hyperbolic system of three equations(1.1) with bounded initial data(1.2). When we fix the third variable s, the system about the variables ρ and u is the classical isentropic gas dynamics in Eulerian coordinates with the pressure function P(ρ, s) = ese-1/ρ,which, in general, does not form a bounded invariant region. We introduce a variant of the viscosity argument, and construct the approximate solutions of(1.1) and(1.2) by adding the artificial viscosity to the Riemann invariants system(2.1). When the amplitude of the first two Riemann invariants(w1(x, 0), w2(x, 0)) of system(1.1) is small,(w1(x, 0), w2(x, 0)) are nondecreasing and the third Riemann invariant s(x, 0) is of the bounded total variation, we obtained the necessary estimates and the pointwise convergence of the viscosity solutions by the compensated compactness theory. This is an extension of the results in [1].

关键词： global weak solutions viscosity method compensa ted compac tn ess

SINGULAR LIMITS FOR INHOMOGENEOUS EQUATIONS OF ELASTICITY

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Acta Mathematica Scientia 2009年第3期29卷 645-649页

作者：陆云光 christian klingenberg Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 210016 China Department of Mathematics National University of ColombiaBogota Colombia Mathematisches Institut Universitt Würzburg Am Hubland 97074 Würzburg Germany

Based on the framework introduced in [4] or [5], the singular limits of stiff relaxation and dominant diffusion for the Cauchy problem of inhomogeneous equations of elasticity is studied. We are able to reach equilibr... 详细信息

关键词： relaxation limit equations of elasticity compensated compactness invariantregions theory

Single-Step Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for 1-D Euler Equations

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Communications on Applied Mathematics and Computation 2020年第4期2卷 541-579页

作者： Jayesh Badwaik Praveen Chandrashekar christian klingenberg Department of Mathematics University of WürzburgWürzburgGermany TIFR Center for Applicable Mathematics BangaloreIndia

We propose an explicit,single-step discontinuous Galerkin method on moving grids using the arbitrary Lagrangian-Eulerian approach for one-dimensional Euler *** grid is moved with the local fluid velocity modified by some smoothing,which is found to con-siderably reduce the numerical dissipation introduced by Riemann *** scheme preserves constant states for any mesh motion and we also study its positivity preservation *** grid refinement and coarsening are performed to maintain the mesh qual-ity and avoid the appearance of very small or large ***,higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.

关键词： Discontinuous Galerkin method Moving meshes Arbitrary Lagrangian-Eulerian Euler equations

Well-Balanced Central Scheme for the System of MHD Equations with Gravitational Source Term

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Communications in Computational Physics 2022年第8期32卷 878-898页

作者： Farah Kanbar Rony Touma christian klingenberg Institute for Mathematics University of WuerzburgGermany Department of Computer Science and Mathematics Lebanese American UniversityLebanon

A well-balanced second order finite volume central scheme for the magnetohydrodynamic(MHD)equations with gravitational source term is developed in this *** scheme is an unstaggered central scheme that evolves the numerical solution on a single grid and avoids solving Riemann problems at the cell interfaces using ghost staggered cells.A subtraction technique is used on the conservative variables with the support of a known steady state in order to manifest the well-balanced property of the *** divergence-free constraint of themagnetic field is satisfied after applying the constrained transport method(CTM)for unstaggered central schemes at the end of each time-step by correcting the components of the magnetic *** robustness of the proposed scheme is verified on a list of numerical test cases from the literature.

关键词： MHD equations unstaggered central schemes well-balanced schemes steady states divergence-free constraint constrained transport method

DECAY RATE FOR DEGENERATE CONVECTION DIFFUSION EQUATIONS IN BOTH ONE AND SEVERAL SPACE DIMENSIONS

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Acta Mathematica Scientia 2015年第2期35卷 281-302页

作者：陆云光 christian klingenberg Ujjwal KOLEY Xuezhou LU Department of Mathematics Hangzhou Normal University Institut fur Mathematik Julius-Maximilians-Universitat Wiirzburg Laboratoire Ondes and Milieux Complexes UMR 6294 CNRS-Universite du Havre

We consider degenerate convection-diffusion equations in both one space dimension and several space dimensions. In the first part of this article, we are concerned with the decay rate of solutions of one dimension convection diffusion equation. On the other hand, in the second part of this article, we are concerned with a decay rate of derivatives of solution of convection diffusion equation in several space dimensions.

关键词： Degenerate convection-diffusion equations regularity decay rate Lax-Oleinik type inequality

Moving-Water Equilibria Preserving HLL-Type Schemes for the Shallow Water Equations

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Communications in Mathematical Research 2020年第3期36卷 247-271页

作者： christian klingenberg Alexander Kurganov Yongle Liu Markus Zenk Institute for Mathematics Wurzburg UniversityWurzburg 97074Germany Department of Mathematics and SUSTech International Center for Mathematics Southern University of Science and TechnologyShenzhen 518055China Department of Mathematics Harbin Institute of TechnologyHarbin 150001China.

We construct new HLL-type moving-water equilibria preserving upwind schemes for the one-dimensional Saint-Venant system of shallow water equations with nonflat bottom *** designed first-and secondorder schemes are tested on a number of numerical examples,in which we verify the well-balanced property as well as the ability of the proposed schemes to accurately capture small perturbations of moving-water steady states.

关键词： Shallow water equations Harten-Lax-Van Leer(HLL)scheme well-balanced method steady-state solutions(equilibria) moving-water and still-water equilibria

GLOBAL L~∞ SOLUTIONS TO SYSTEM OF ISENTROPIC GAS DYNAMICS IN A DIVERGENT NOZZLE WITH FRICTION

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Acta Mathematica Scientia 2019年第5期39卷 1213-1218页

作者： Qingyou SUN Yunguang LU K. K. Chen Institute for Advanced Studies Hangzhou Normal University Hangzhou 311121 China Deptartment of Mathematics Wuerzburg University Wuerzburg 97070 Germany

In this article, we study the global L^∞ entropy solutions for the Cauchy problem of system of isentropic gas dynamics in a divergent nozzle with a friction. Especially when the adiabatic exponent γ=3, we apply for the maximum principle to obtain the L^∞ estimates w(ρ^δ,ε, u^δ,ε)≤ B(t) and z(ρ^δ,ε, u^δ,ε)≤ B(t) for the viscosity solutions (ρ^δ,ε, u^δ,ε), where B(t) is a nonnegative bounded function for any finite time t. This work, in the special case γ≥ 3, extends the previous works, which provided the global entropy solutions for the same Cauchy problem with the restriction w(ρ^δ,ε, u^δ,ε)≤ 0 or z(ρ^δ,ε, u^δ,ε)≤ 0.

关键词： global L^∞ solution isentropic gas dynamics source terms flux approximation compensated compactness

Regularity of Viscous Solutions for a Degenerate Non-linear Cauchy Problem

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Journal of Partial Differential Equations 2016年第1期29卷 14-21页

作者： SASTOQUE Eric Hernandez klingenberg christian RENDON Leonardo JUAJIBIOY Juan C. Departamento de Matematicas Universidad de Magdalena Santa Marta Colombia. Department of Mathematics Wurzburg University Germany. Departamento de Matematicas Universidad Nacional de Colombia Bogota. Departamento de Ciencias Naturales Y Exactas Fundacion Universidad Autonoma de Colombia Bogota.

We consider the Cauchy problem for a class of nonlinear degenerate parabolic equation with forcing. By using the vanishing viscosity method it is possible to construct a generalized solution. Moreover, this solution i... 详细信息

关键词： Viscosity solution Holder estimates Holder continuity.