The semilinear second-order hyperbolic equation with data strongly singular at one point

In this paper we study the initial value problem for the scalar semilinear strictly hyperbolic equation in multidimensional space with data strongly singular at one point. Under the assumption of the initial data being conormal with respect to one point and bounded or regular with a certain low degree, the existence of the solution to this problem is obtained; meanwhile, it is proved that the singularity of the solution will spread on the forward characteristic cone of the hyperbolic operator issuing from this point, and the solution is bounded and conormal with respect to this cone.     

An algebra of meromorphic corner symbols

Operators on manifolds with corners that have base configurations with geometric singularities can be analysed in the frame of a conormal symbolic structure which is in spirit similar to the one for conical singularities of Kondrat'ev's work. Solvability of elliptic equations and asymptotics of solutions are determined by meromorphic conormal symbols. We study the case when the base has edge singularities which is a natural assumption in a number of applications. There are new phenomena, caused by a specific kind of higher degeneracy of the underlying symbols. We introduce an algebra of meromorphic edge operators that depend on complex parameters and investigate meromorphic inverses in the parameter-dependent elliptic case. Among the examples are resolvents of elliptic differential operators on manifolds with edges.     

Convergence of the Lax–Friedrichs scheme for isothermal gas dynamics with semiconductor devices

In this paper we study the existence of global solutions to the Euler equations of compressible isothermal gas dynamics with semiconductor devices. We construct the approximate solutions by Lax–Friedrichs scheme. The convergence and consistency are obtained by using the compensated compactness framework for γ = 1. The global entropy solutions in L^∞ are obtained. We deal with the initial data containing unbounded velocity which is different from the isentropic case. Received: November 18, 2003     

The Cauchy problem for the 1-D Dirac–Klein–Gordon equation

The Cauchy problem for the Dirac–Klein–Gordon equation are discussed in one space dimension. Time local and global existence for solutions with rough data, especially the solutions for Klein–Gordon equation in the critical and super critical Sobolev norm of [4] are considered. The solutions with general propagation speeds are dealt with.      

Boundary-contact problems for domains with conical singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have conical singularities. Such problems represent continuous operators between weighted Sobolev spaces and subspaces with asymptotics. Ellipticity is formulated in terms of extra transmission conditions along the interfaces with a control of the conormal symbolic structure near conical singularities. We show regularity and asymptotics of solutions in weighted spaces, and we construct parametrices. The result will be illustrated by a number of explicit examples.     

On the singularities of solutions to 4-D semilinear dispersive wave equations

In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichartz＇s inequality with the commutator argument techniques, we show that the weak solutions stay globally conormal if the Cauchy data are conormal     

Nonuniqueness of solutions of Riemann problems

We investigate a general mechanism, utilizing nonclassical shock waves, for nonuniqueness of solutions of Riemann initial-value problems for systems of two conservation laws. This nonuniqueness occurs whenever there exists a pair of viscous shock waves forming a 2-cycle, i.e., two statesU ₁ andU ₂ such that a traveling wave leads fromU ₁ toU ₂ and another leads fromU ₂ toU ₁. We prove that a 2-cycle gives rise to an open region of Riemann data for which there exist multiple solutions of the Riemann problem, and we determine all solutions within a certain class. We also present results from numerical experiments that illustrate how these solutions arise in the time-asymptotic limit of solutions of the conservation laws, as augmented by viscosity terms.     

The self-similar solutions of the one-dimensional semilinear Tricomi-type equations

In this article we investigate the issue of existence of global in time solutions of semilinear Tricomi-type equations. We give conditions that relate the nonlinearity, the speed of propagation, and the order of singularities of initial data. These conditions guarantee existence of global in time solutions. In particular, we prove existence of solutions invariant under dilation by solving the Cauchy problem with initial data which are homogeneous functions.     

Global singularity structures of weak solutions to 4-D semilinear dispersive wave equations

In this paper, we are concerned with the global singularity structures of weak solutions to 4-D semilinear dispersive wave equations whose initial data are chosen to be discontinuous on the unit sphere. Combining Strichartz's inequality with the commutator argument techniques, we show that the weak solutions are C²−regular away from the focusing cone surface |x|=|t−1| and the outgoing cone surface |x|=t+1. This research was supported by the National Natural Science Foundation of China and the Doctoral Foundation of NEM of China.     

Higher order shadow waves and delta shock blow up in the Chaplygin gas

The introductory part of this paper contains an overview of known results about elementary and delta shock solutions to Riemann problem for well known Chaplygin gas model (nowadays used in cosmological theories for dark energy) in terms of entropic shadow waves. Shadow waves are introduced in [17] and they are represented by shocks depending on a small parameter ε with unbounded amplitudes having a distributional limit involving the Dirac delta function. In a search for admissible solutions to all possible cases of mutual interactions of waves arising from double Riemann initial data we found same cases that cannot be resolved with already known types of elementary or shadow wave solutions. These cases are resolved by introducing a sequence of higher order shadow waves depending on integer powers of ε. It is shown that such waves have a distributional limit but only until some finite time T.     

A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems

This paper is concerned with the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. On the basis of the existence result for the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions, provided that the C¹ norm and the BV norm of the initial data are bounded but possibly large. In contrast to former results obtained by Liu and Zhou [J. Liu, Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Methods Appl. Sci. 30 (2007) 479-500], ours do not require their assumption that the system is rich in the sense of Serre. Applications include that to the one-dimensional Born-Infeld system arising in string theory and high energy physics.     

Full asymptotics and Laurent series of layer potentials for Laplace's equation on the half‐space

We combine the calculus of conormal distributions, in particular the Pull‐Back and Push‐Forward Theorems, with the method of layer potentials to solve the Dirichlet and Neumann problems on half‐spaces. We obtain full asymptotic expansions for the solutions, show that boundary layer potential operators are elements of the full b‐calculus and give a new proof of the classical jump relations. En route, we improve Siegel and Talvila's growth estimates for the modified layer potentials in the case of polyhomogeneous boundary data. The techniques we use here can be generalised to geometrically more complex settings, as for instance the exterior domain of touching domains or domains with fibred cusps. This work is intended to be a first step in a longer program aiming at understanding the method of layer potentials in the setting of certain non‐Lipschitz singularities that can be resolved in the sense of Melrose using manifolds with corners and at applying a matching asymptotics ansatz to singular perturbations of related problems.     

Global structure and asymptotic behavior of weak solutions to flood wave equations

The present paper concerns with the global structure and asymptotic behavior of the discontinuous solutions to flood wave equations. By solving a free boundary problem, we first obtain the global structure and large time behavior of the weak solutions containing two shock waves. For the Cauchy problem with a class of initial data, we use Glimm scheme to obtain a uniform BV estimate both with respect to time and the relaxation parameter. This yields the global existence of BV solution and convergence to the equilibrium equation as the relaxation parameter tends to 0.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

A probabilistic model associated with the pressureless gas dynamics

Using a method of stochastic perturbation of a Langevin system associated with the non-viscous Burgers equation we introduce a system of PDE that can be considered as a regularization of the pressureless gas dynamics describing sticky particles. By means of this regularization we describe how starting from smooth data a δ-singularity arises in the component of density. Namely, we find the asymptotics of solution at the point of the singularity formation as the parameter of stochastic perturbation tends to zero. Then we introduce a generalized solution in the sense of free particles (FP-solution) as a special limit of the solution to the regularized system. This solution corresponds to a medium consisting of non-interacting particles. The FP-solution is a bridging step to constructing solutions to the Riemann problem for the pressureless gas dynamics describing sticky particles. We analyze the difference in the behavior of discontinuous solutions for these two models and the relations between them. In our framework we obtain a unique entropy solution to the Riemann problem in 1D case.     

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C¹ solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.     

Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws

In this paper we provide a generalized version of the Glimm scheme to establish the global existence of weak solutions to the initial-boundary value problem of 2×2 hyperbolic systems of conservation laws with source terms. We extend the methods in [J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. Dissertation, Stanford University, 1982; J.M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem, J. Differential Equations 222 (2006) 515-549] to construct the approximate solutions of Riemann and boundary Riemann problems, which can be adopted as the building block of approximate solutions for our initial-boundary value problem. By extending the results in [J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697-715] and showing the weak convergence of residuals, we obtain stability and consistency of the scheme.     

On the Cauchy problem for a generalized Camassa–Holm equation

The local well-posedness of a generalized Camassa–Holm equation is established by means of Kato's theory for quasilinear evolution equations and two types of results for the blow-up of solutions with smooth initial data are given.     

半线性弹性动力学方程组余法奇性传播

该文在余法分布理论框架下研究半线性弹性动力学方程组原点初值奇性的传播. 建立了适当的余法分布空间,利用方程组解的Stokes-Helmholtz分解方法, 讨论了余法分布空间中函数的分解性质; 如果初值函数在原点有适当余法奇性且L^∞有界, 通过构造解序列的方法证明初值问题存在仅在方程组特征面上有余法奇性,且正则性更高的唯一L^∞有界解.     

Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation

The initial boundary value problem for non-linear wave equations of Kirchhoff type with dissipation in a bounded domain is considered. We prove the blow-up of solutions for the strong dissipative term -Δ_ut$-\mathrm{\Delta }{u}_t$ and the linear dissipative term _ut$u_t$ by the energy method and give some estimates for the life span of solutions. We also show the nonexistence of global solutions with positive initial energy for non-linear dissipative term by Vitillaro's argument.