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.     

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.     

On the solvability of one multidimensional version of the first Darboux problem for some nonlinear wave equations

In the present paper, for wave equations with power nonlinearity we investigate the problem of the existence or nonexistence of global solutions of a multidimensional version of the first Darboux problem in the conic domain.     

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.     

Global solutions for initial-boundary value problem of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of initial-boundary value problem of quasilinear wave equations. Applying the Lax's method and generalized Glimm's method, we construct the approximate solutions of initial-boundary Riemann problem near the boundary layer and perturbed Riemann problem away from the boundary layer. By showing the weak convergence of residuals for the approximate solutions, we establish the global existence for the derivatives of solutions and obtain the existence of global Lipschitz continuous solutions of the problem.     

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.     

Global conservative solutions of a modified two-component Camassa-Holm shallow water system

We first prove the existence of global conservative solutions to the Cauchy problem for a modified two-component Camassa-Holm shallow water system. Then, we show that these global solutions, which depend continuously on the initial date, construct a semigroup.     

Blow-up and pattern formation in hyperbolic models for chemotaxis in 1-D

In this paper we study finite time blow-up of solutions of a hyperbolic model for chemotaxis. Using appropriate scaling this hyperbolic model leads to a parabolic model as studied by Othmer and Stevens (1997) and Levine and Sleeman (1997). In the latter paper, explicit solutions which blow-up in finite time were constructed. Here, we adapt their method to construct a corresponding blow-up solution of the hyperbolic model. This construction enables us to compare the blow-up times of the corresponding models. We find that the hyperbolic blow-up is always later than the parabolic blow-up. Moreover, we show that solutions of the hyperbolic problem become negative near blow-up. We calculate the zero-turning-rate time explicitly and we show that this time can be either larger or smaller than the parabolic blow-up time. The blow-up models as discussed here and elsewhere are limiting cases of more realistic models for chemotaxis. At the end of the paper we discuss the relevance to biology and exhibit numerical solutions of more realistic models.     

Existence theory for nonclassical entropy solutions of scalar conservation laws

We establish a general existence theory for the Cauchy problem associated with a scalar conservation law in one-space dimension. The flux-function is assumed to be nonconvex and we consider nonclassical entropy solutions selected by a kinetic relation. To solve the Cauchy problem, we construct a sequence of approximate solutions using a wave-front tracking scheme. The main difficulty is deriving a uniform estimate on the total variation of the approximate solutions. This is achieved here by introducing a generalized total variation functional, which is decreasing in time and, additionally, reduces to the standard total variation functional when the solutions contain only classical shocks. This functional seems sufficiently robust to be useful for systems as well.Received: June 3, 2002; revised: November 12, 2002     

The forward problem for the electromagnetic Helmholtz equation with critical singularities

We study the forward problem of the magnetic Schrödinger operator with potentials that have a strong singularity at the origin. We obtain new resolvent estimates and give some applications on the spectral measure and on the solutions of the associated evolution problem.     

Globally Lipschitz continuous solutions to a class of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of Cauchy problem of quasilinear wave equations. Applying Lax's method and generalized Glimm's method, we construct the approximate solutions of the corresponding perturbed Riemann problem and establish the global existence for the derivatives of solutions. Then, the existence of global Lipschitz continuous solutions can be carried out by showing the weak convergence of residuals for the source term of equation.     

General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source

In this paper we consider a quasilinear viscoelastic wave equation in canonical form with the homogeneous Dirichlet boundary condition. We prove that, for certain class of relaxation functions and certain initial data in the stable set, the decay rate of the solution energy is similar to that of the relaxation function. This result improves earlier ones obtained by Messaoudi and Tatar [S.A. Messaoudi, N.-E. Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Methods Appl. Sci. 30 (2007) 665-680] in which only the exponential and polynomial decay rates are considered. Conversely, for certain initial data in the unstable set, there are solutions that blow up in finite time. The last result is new, since it allows a larger class of initial energy which may take positive values.     

The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors

In this paper we study the asymptotic behavior of globally smooth solutions of the Cauchy problem for the multidimensional isentropic hydrodynamic model for semiconductors in R^d. We prove that smooth solutions (close to equilibrium) of the problem converge to a stationary solution exponentially fast as t→+∞.     

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.     

Global structure of Riemann solutions to a system of two-dimensional hyperbolic conservation laws

The Riemann problem for a two-dimensional nonstrictly hyperbolic system of conservation laws is considered. Without the restriction that each jump of the initial data projects one planar elementary wave, ten topologically distinct solutions are obtained by applying the method of generalized characteristic analysis. Some of these solutions involve the nonclassical waves, i.e., the delta shock wave and the delta contact discontinuity, for which we explicitly give the expressions of their strengths, locations and propagation speeds. Moreover, we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one-dimensional Cauchy problem, which provides a verification that our construction produces the correct unique global solutions.     

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.     

Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type

We consider a hyperbolic-parabolic singular perturbation problem for a quasilinear equation of Kirchhoff type, and obtain parameter-dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation. For this purpose we show time decay estimates for hyperbolic-parabolic singular perturbation problem for linear equations with a time-dependent coefficient.     

Global existence and blow-up of solutions for some hyperbolic systems with damping and source terms

This paper deals with the global existence and blow-up of solutions to some nonlinear hyperbolic systems with damping and source terms in a bounded domain. By using the potential well method, we obtain the global existence. Moreover, for the problem with linear damping terms, blow-up of solutions is considered and some estimates for the lifespan of solutions are given.     

Global existence of shock front solutions in 1-dimensional piston problem in the relativistic Euler equations

The paper studies the 1-D piston problem of the relativistic Euler equations when the speed of the piston is a perturbation of a constant. A sequence of approximate solutions constructed by a modified Glimm scheme is proved to be convergent to the weak solution (which includes a strong leading shock) to the piston problem. In particular, we give the precise estimates on the reflection of the perturbed waves on the piston and the leading shock. The paper is supported by the National Natural Science Foundation of China (Grant 10626034) and the Special Research Fund for Selecting Excellent Young Teachers of the Universities in Shanghai.     

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.