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Acta Mathematica Scientia 2012年第5期32卷 1727-1742页

作者：王志刚李亚纯 Department of Mathematics Shanghai Jiaotong University Department of Mathematics Fuyang Normal College

The aim of this paper is to prove the well-posedness（existence and uniqueness） of the L p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L p initial *** use the device of doubling variables and some technical analysis to prove the uniqueness *** we can prove that the L p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.

关键词： degenerate parabolic-hyperbolic equation L p entropy solution device of doubling variables vanishing viscosity method

THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA

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Acta Mathematica Scientia 2023年第4期43卷 1668-1674页

作者：吴菲王泽军陈芳启 Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing 211106China

In this paper,we study the global existence of BV solutions of the initial value problem for the isentropic p-system,where the state equation of the gas is given by P=Av^(-γ).Forγ>1,the general existence result for large initial data has not been *** using the Glimm scheme,Nishida,Smoller and Diperna successively obtained the global existence results for(γ-1)TV(v_(0)(x),u_(0)(x))being *** the present paper,by adopting a rescaling technique,we improve these results and obtain the global existence result under the condition that(γ-1)^(γ+1)(TV(v_(0)(x)))~(γ-1)(TV(u_(0)(x)))^(2) is small,which implies that,for fixedγ>1,either TV(v_(0)(x))or TV(u_(0)(x))can be arbitrarily large.

关键词： conservation laws p-system vanishing viscosity method BV solutions

INITIAL BOUNDARY VALUE PROBLEM FOR THE 3D MAGNETIC-CURVATURE-DRIVEN RAYLEIGH-TAYLOR MODEL

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Acta Mathematica Scientia 2020年第2期40卷 529-542页

作者： Xueke PU Boling GUO School of Mathematics and Information Science Guangzhou UniversityGuangzhou 510006China Institute of Applied Physics and Computational Mathematics P.O.Box 8009Beijing 100088China

This article studies the initial-boundary value problem for a three dimensional magnetic-curvature-driven Rayleigh-Taylor *** first obtain the global existence of weak solutions for the full model equation by employing the Galerkin’s approximation ***,for a slightly simplified model,we show the existence and uniqueness of global strong solutions via the Banach’s fixed point theorem and vanishing viscosity method.

关键词： Magnetic-curvature-driven Rayleigh-Taylor model weak solutions strong solutions Banach fixed point theorem vanishing viscosity method

Generalized Persistence of Entropy Weak Solutions for System of Hyperbolic Conservation Laws

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Chinese Annals of Mathematics,Series B 2022年第4期43卷 499-508页

作者： Yi ZHOU School of Mathematical Sciences Fudan UniversityShanghai 200433China

Let u(t,x)be the solution to the Cauchy problem of a scalar conservation law in one space *** is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution can best lie in the space of bounded total *** is impossible that the solutions belong to,for example,H^(1) because by Sobolev embedding theorem H^(1) functions are Holder ***,the author notes that from any point(t,x),he can draw a generalized characteristic downward which meets the initial axis at y=α(t,x).If he regards u as a function of(t,y),it indeed belongs to H^(1) as a function of y if the initial data belongs to H^(1).He may call this generalized persistence(of high regularity)of the entropy weak *** main purpose of this paper is to prove some kinds of generalized persistence(of high regularity)for the scalar and 2×2 Temple system of hyperbolic conservation laws in one space dimension.

关键词： Quasilinear hyperbolic system Cauchy problem Entropy weak solution vanishing viscosity method

Dual‑Wind Discontinuous Galerkin methods for Stationary Hamilton‑Jacobi Equations and Regularized Hamilton‑Jacobi Equations

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Communications on Applied Mathematics and Computation 2022年第2期4卷 563-596页

作者： Xiaobing Feng Thomas Lewis Aaron Rapp Department of Mathematics The University of TennesseeKnoxvilleTN 37996USA Department of Mathematics and Statistics The University of North Carolina at GreensboroGreensboroNC 27402USA Department of Mathematical Sciences University of the Virgin IslandsKingshillUSVI 00850-9781US Virgin Islands

This paper develops and analyzes a new family of dual-wind discontinuous Galerkin(DG)methods for stationary Hamilton-Jacobi equations and their vanishing viscosity *** new DG methods are designed using the DG fnite element discrete calculus framework of[17]that defnes discrete diferential operators to replace continuous differential operators when discretizing a partial diferential equation(PDE).The proposed methods,which are non-monotone,utilize a dual-winding methodology and a new skewsymmetric DG derivative operator that,when combined,eliminate the need for choosing indeterminable penalty *** relationship between these new methods and the local DG methods proposed in[38]for Hamilton-Jacobi equations as well as the generalized-monotone fnite diference methods proposed in[13]and corresponding DG methods proposed in[12]for fully nonlinear second order PDEs is also *** and stability are established for the proposed dual-wind DG *** stability results are shown to hold independent of the scaling of the stabilizer allowing for choices that go beyond the Godunov barrier for monotone *** experiments are provided to gauge the performance of the new methods.

关键词： Hamilton-Jacobi equations Discontinuous Galerkin methods vanishing viscosity method