GLOBAL SMOOTH SOLUTIONS TO CAUCHY PROBLEMS FOR MULTIDIMENSIONAL QUASILINEAR HYPERBOLIC SYSTEMS

§ 1. Introduction It is well-known that the smooth solutions to Cauchy problems for quasilinear hyperbolic systems, generally speaking, exist only locally in time and will occur singularities in finite time, even if the initial data are sufficiently smooch and small (see [1—2]). Therefore, the interesting problem is that what conditions can ensure the global existence of classical solutions for quasilinear hyperbolic systems Up to now, the most results on global existence of classical solutions are     

SOLUTIONS CONTAINING DELTA－WAVES OF CAUCHY PROBLEMS FOR A NONSTRICTLY HYPERBOLIC SYSTEM

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BREAKDOWN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying initial data, and obtains a blow-up result for C1 solution to Cauchy problem.     

TWO-DIMENSIONAL RIEMANN PROBLEM FOR A HYPERBOLIC SYSTEM OF CONSERVATION LAWS

§ 1. Introduction The Riemann problem is the most basic problem for both analytical theory and numerical computation of nonlinear hyperbolic conservation laws. B. Riemann suggested and solved it for one dimensional isentropic flow in 1860. A lot of work have been done for 1-D case since 1940's. For 2-D scalar conservation law, it has been solved by Wagner and Zhang and Zheng. For 2-D system of Euler equations in gas dynamios, after some demonstration and analysis with characteristic method in both physics and phase spaces a set of conjectures on the structure of solutions have been formulated by Zhang and Zheng. To approach the proof of the conjectures, we consider a 2×2 system first. In the present paper we discuss the following system:     

EXISTENCE AND NONEXISTENCE OF GLOBAL CONTINUOUS SOLUTIONS OF RIEMANN PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS~(**)

Consider Riemann problem(E),(R)under the condition(M).It is proved that,as0相似文献   

STUDY OF THE GLOBAL SOLUTIONS FOR A NONLINEAR HYPERBOLIC SYSTEM

We shall compare the solution of the Cauchy problem with the solution of the corresponding Riemann problem (1.1) with the initial data     

STUDY OF THE GLOBAL SOLUTIONS FOR A NONLINEAR HYPERBOLIC SYSTEM

§1 Introduction We shall compare the solution of the Caucky problem with the solution of the corresponding Riemann problem (1.1) with the initial data     

GLOBAL SMOOTH SOLUTION OF CAUCHY PROBLEMS FOR A CLASS OF QUASILINEAR HYPERBOLIC SYSTEMS

In this article we prove that certain kinds of inhomogeneous terms can smoothen thesolution,for first order quasilinear hyperbolic systems globally in time provided that theinitial data are small.     

GLOBAL SMOOTH SOLUTIONS OF DISSIPATIVE BOUNDARY VALUE PROBLEMS FOR FIRST ORDER QUASILINEAR HYPERBOLIC SYSTEMS

This paper discusses the following initial-boundary value problems for the first orderquasilinear hyperbolic systems:(u)/(t)+A(u)(u)/(x)=0,(1)u~Ⅱ=F(u~Ⅰ),as x=0,(2)u~Ⅰ=G(u~Ⅱ),as x=L,(3)u=u~0(x),as t=0,(4)where the boundary conditions(2),(3)satisfy F(0)=0,G(0)=0 and the dissipativeconditions,that is,the spectral radii of matrices B_1=(F)/(u~Ⅰ)(0)(G)/(u~Ⅱ)(0)and B_2(G)/(u~Ⅱ)(0)(F)/(u~Ⅰ)(0) are less than unit.Under certain assumptions it is proved that the initial-boundary problem (1)—(4)admits a unique global smooth solution u(x,t)and the C~1-norm丨u(t)丨σ~2of u(x,t)decaysexponentially to zero as t→∞,provided that the C~1-norm丨u~0丨σ~1of the initial data issufficiently small.     

GLOBALLY DEFINED CLASSICAL SOLUTIONS TO FREE BOUNDARY PROBLEMS WITH CHARACTERISTIC BOUNDARY FOR QUASILINEAR HYPERBOLIC SYSTEMS

In this paper, the authors prove the global existence and uniqueness of classical solutions to some free boundary problems with characteristic boundary for the reducible quasilinear hyperbolic system.     

FORMATION OF SINGULARITIES FOR A KIND OF QUASILINEAR NON－STRICTLY HYPERBOLIC SYSTEM

The author gets a blow-up result of C1 solution to the Cauchy problem for a first order quasilinear non-strictly hyperbolic system in one space dimension.     

POINTWISE ESTIMATES OF SOLUTIONS TO CAUCHY PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS＊＊＊

This paper considers the pointwise estimate of the solutions to Cauchy problem for quasilin-ear hyperbolic systems, which bases on the existence of the solutions by using the fundamental solutions. It gives a sharp pointwise estimates of the solutions on domam under consideration. Specially, the estimate is precise near each characteristic direction.     

GLOBAL SMOOTH SOLUTIONS FOR A CLASS OF QUASILINEAR HYPERBOLIC SYSTEMS WITH DISSIPATIVE TERMS

In this paper the authors prove the existence and uniqueness of global smooth solutionsto the Cauchy problem for quasilinear hyperbolic systems with some kinds of dissipativeterms.     

EXTENSION OF SOLUTIONS TO RIEMANN BOUNDARY VALUE PROBLEMS AND ITS APPLICATION

In this paper,solutions of Riemann boundary value problems with nodes are extended to the case where they may have singularties of high order at the nodes.Moreover, further extension is discussed when the free term of the problem involved also possesses singularities at the nodes.As an application,certain singular integral equation is discussed.     

GLOBAL SHOCK SOLUTIONS TO A CLASS OF PISTON PROBLEMS FOR THE SYSTEM OF ONE DIMENSIONAL ISENTROPIC FLOW

The autors apply the result obtained in [1] to consider a class of discontinuous piston problems for the system of one dimensional isentropic flow and prove that this problem admits a unique global classical discontinuous solution only containing one shock.     

GLOBAL SMOOTH SOLUTIONS FOR INHOMOGENEOUS QUASILINEAR HYPERBOLIC SYSTEMS

It is wellknown that, for the Cauchy problem of quasilinear hyperbolic systems, in general, discontinuities may occur in solutions as the time variable increses, even if the initial data are sufficiently smooth. On the other hand, there exist some examples in which the solution of the corresponding Cauchy problem for quasilinear hyperbolic systems are globally smooth. Therefore, it is of great interest to determine the conditions under which solutions are globally smooth and the conditions under which singularities of the solution     

POSITIVE SOLUTIONS TO A SEMILINEAR SYSTEM OF SECOND-ORDER TWO-POINT BOUNDARY VALUE PROBLEMS

By using Krasnosel'skii fixed point theorem of cone expansion-compression type, the results on the existence of one, two and three positive solutions are established for a semilinear second-order system of two-point boundary value problems.     

NONLINEAR INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEM

Consider the nonlinear inltial-boundary value problem for quasilinear hyperbolicsystem:Let k≥2[n/2] 6,(F,g)∈ H~k(R_ ;Ω)×H~k(R_ ;Ω),and their traces at t=0 are zeroup to the order k-1.If for u=0,the problem(*)at t=0 is a Kreiss hyperbolic system,and the boundaryconditions satisfy the uniformly Lopatinsky criteria,then there exists a T>0 such that(*)has a unique H~k soluton in(0,T).In the Appendix,for symmetric hyperbolic systems,a comparison between theuniformly Lopatinsky condition and the stable admissible condition is given.     

ON THE UNIQUENESS OF THE WEAK SOLUTIONS OF A QUASILINEAR HYPERBOLIC SYSTEM WITH A SINGULAR SOURCE TERM

This paper is a continuation of the authors'previous paper[1].In this paper the authorsprove,assuming additional conditions on the initial data,some results about the existence anduniqueness of the entropy weak solutions of the Cauchy problem for the singular hyperbolicsystem a_t+(au)_x_2au/x=0,u_t+1/2(a~2+u~2)_x=0,x>0,t≥0.     

EXISTENCE OF GLOBAL SMOOTH SOLUTIONS FOR TWO IMPORTANT NONSTRICTLY QUASILINEAR HYPERBOLIC SYSTEMS

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