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

This paper develops and analyzes a new family of dual-wind discontinuous Galerkin(DG)methods for stationary Hamilton-Jacobi equations and their vanishing viscosity regularizations.The 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 constants.The 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 examined.Admissibility and stability are established for the proposed dual-wind DG methods.The stability results are shown to hold independent of the scaling of the stabilizer allowing for choices that go beyond the Godunov barrier for monotone schemes.Numerical experiments are provided to gauge the performance of the new methods.

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

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 obtained.By 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 small.In 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.

THE HOMOGENEOUS DIRICHLET PROBLEM FOR QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC-HYPERBOLIC EQUATION WITH L^p INITIAL VALUE

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 value.We use the device of doubling variables and some technical analysis to prove the uniqueness result.Moreover 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.

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

This article studies the initial-boundary value problem for a three dimensional magnetic-curvature-driven Rayleigh-Taylor model.We first obtain the global existence of weak solutions for the full model equation by employing the Galerkin’s approximation method.Secondly,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.

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

Let u(t,x)be the solution to the Cauchy problem of a scalar conservation law in one space dimension.It 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 variations.It is impossible that the solutions belong to,for example,H^(1) because by Sobolev embedding theorem H^(1) functions are Holder continuous.However,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 solutions.The 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.