Interactions of delta shock waves for the Chaplygin gas equations with split delta functions

In this article, we study the interactions of delta shock waves for the one-dimensional Euler equations for Chaplygin gas with split delta functions. We constructively obtain the solutions when the initial data are three piecewise constant states. The global structure and large time-asymptotic behaviors of the solutions are analyzed case by case. Moreover, we obtain the stability of solutions by letting perturbed parameter ? tend to zero.     

Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions

In this article, we are concerned with the interactions of delta shock waves with contact discontinuities for the relativistic Euler equations for Chaplygin gas by using split delta functions method. The solutions are obtained constructively and globally when the initial data consists of three piecewise constant states. The global structure and large time‐asymptotic behaviors of the solutions are analyzed case by case. During the process of the interaction, the strengths of delta shock waves are computed completely. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with special initial data by letting perturbed parameter ε tends to zero. Copyright © 2014 John Wiley & Sons, Ltd.     

Interactions of delta shock waves for the Aw-Rascle traffic model with split delta functions

This paper is concerned with the interactions of $\delta$-shock waves for the Aw-Rascle traffic model with split delta functions. The solutions are obtained constructively when the initial data are three piecewise constant states. The global structure and large time-asymptotic behaviors of the solutions are analyzed case by case. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with initial data by studying the limits of the solutions when the perturbed parameter $\varepsilon\rightarrow0$.     

Solutions to quasilinear hyperbolic conservation laws with initial discontinuities

We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens only on two disjoint unit circles and the initial data are two different constant states, global solutions are constructed and some new phenomena are discovered. In the analysis, we first construct the solution for 0 ≤ t T~*.Then, when T~*≤ t T′, we get a new shock wave between two rarefactions, and then, when t T′,another shock wave between two shock waves occurs. Finally, we give the large time behavior of the solution when t →∞. The technique does not involve dimensional reduction or coordinate transformation.     

Two-dimensional Riemann problems for zero-pressure gas dynamics with three constant states

The Riemann problems for two-dimensional zero-pressure gas dynamics are solved completely when the initial data take three constant states having discontinuities on x,y-positive and x-negative axes. With the help of characteristic analysis, by studying interactions among delta-shocks, vacuums and contact discontinuities, the Riemann solutions constructed exhibit nine different explicit configurations. The Mach-reflection-like configurations appear in some solutions.     

Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term

The paper studies the global existence, asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term. It proves that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as t→+∞, respectively, in the states of large initial data and small initial energy. In particular, in the case of space dimension N=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. And an example is given.     

Structural stability of solutions to the Riemann problem for a scalar conservation law

The aim of this paper is to study the structural stability of solutions to the Riemann problem for a scalar conservation law with a linear flux function involving discontinuous coefficients. It is proved that the Riemann solution is possibly instable when one of the Riemann initial data is at the vacuum. Furthermore, we point out that the Riemann solution is also possibly instable even when the Riemann initial data stay far away from vacuum. In order to deal with it, we perturb the Riemann initial data by taking three piecewise constant states and then the global structures and large time asymptotic behaviors of the solutions are obtained constructively. It is also proved that the Riemann solutions are unstable in some certain situations under the local small perturbations of the Riemann initial data by letting the perturbed parameter ε tend to zero. In addition, the interaction of the delta standing wave and the contact vacuum state is considered which appear in the Riemann solutions.     

Wave interactions and stability of the Riemann solutions for the chromatography equations

We prove that the Riemann solutions are stable for the chromatography system under the local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the wave interactions by applying the method of characteristic analysis. It is noteworthy that both the propagation directions of the shock wave S and rarefaction wave R are unchanged when they interact with the contact discontinuity J. Moreover, the global structures and large time asymptotic behaviors of the perturbed Riemann solutions are constructed and analyzed case by case.     

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.     

Existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order

This paper is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order: ut+∇⋅(|∇Δu|^p−2∇Δu)=f(u) in Ω⊂RN with boundary condition u=Δu=0 and initial data u₀. The substantial difficulty is that the general maximum principle does not hold for it. The solutions are obtained for both the steady-state case and the developing case by the fixed point theorem and the semi-discretization method. Unlike the general procedures used in the previous papers on the subject, we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables, respectively. By a compactness argument with necessary estimates, we show that the two approximation sequences converge to the same limit, i.e., the solution to be determined. In addition, the decays of solutions towards the constant steady states are established via the entropy method. Finally, it is interesting to observe that the solutions just tend to the initial data u₀ as p→∞.     

Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equations in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L²) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time-dependent, Hamiltonian, linearized dynamics around a carefully chosen one-parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.     

Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate

We consider a prototype reaction-diffusion system which models a network of two consecutive reactions in which chemical components A and B form an intermediate C which decays into two products P and Q. Such a situation often occurs in applications and in the typical case when the intermediate is highly reactive, the species C is eliminated from the system by means of a quasi-steady-state approximation. In this paper, we prove the convergence of the solutions in L², as the decay rate of the intermediate tends to infinity, for all bounded initial data, even in the case of initial boundary layers. The limiting system is indeed the one which results from formal application of the QSSA. The proof combines the recent L²-approach to reaction-diffusion systems having at most quadratic reaction terms, with local L^∞-bounds which are independent of the decay rate of the intermediate. We also prove existence of global classical solutions to the initial system.     

Smooth solutions for a p-system of mixed elliptic-hyperbolic type

In this note we analyze smooth solutions of a p-system of the mixed, elliptic-hyperbolic type. A motivating example for this is a 2-components reduction of the Benney moments chain which appears to be connected to the theory of integrable systems. We don’t assume a-priori that the solutions in question are in the Hyperbolic region. Our main result states that the only smooth solutions of the system which are periodic in x are necessarily constants. As for the initial value problem, we prove that if the initial data are strictly hyperbolic and periodic in x, then the solution cannot extend to [t ₀;+∞) and shocks are necessarily created.     

Stability of the Riemann solutions for a Chaplygin gas

In this paper, we study the structural stability of solutions to the Riemann problem for one-dimensional isentropic Chaplygin gas. We perturb the Riemann initial data by taking three piecewise constant states and construct the global structure. By letting the perturbed parameter ε$\epsilon$ tend to zero, we prove that the Riemann solutions are stable under the local small perturbations of the Riemann initial data even when the initial perturbed density depending on the parameter but with no mass concentration limit.     

Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for a class of quasi-linear wave equations with nonlinear damping and source terms. It proves that when α?max{m,p}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, under rather mild conditions on initial data, the Cauchy problem admits a global weak solution. Especially in the case of space dimension N=1, the weak solutions are regularized and so generalized and classical solution both prove to be unique. On the other hand, if 0?α<1, and the initial energy is negative, then under certain opposite conditions, any weak solution of the Cauchy problem blows up in finite time. And an example is shown.     

Wave Interactions and Stability of Riemann Solutions of the Aw–Rascle Model for Generalized Chaplygin Gas

In this paper, we mainly study the interactions of the elementary waves of the Aw–Rascle model for the generalized Chaplygin gas. With the characteristic analysis method, the solutions are obtained constructively when the initial data are three piecewise constant states. Moreover, by studying the limits of the perturbed Riemann solutions as the perturbed parameter \(\varepsilon\) tends to zero, we find that the Riemann solutions are stable for such perturbations on the initial data.     

Analytical blowup solutions to the 2-dimensional isothermal Euler-Poisson equations of gaseous stars

We study the Euler-Poisson equations of describing the evolution of the gaseous star in astrophysics. Firstly, we construct a family of analytical blowup solutions for the isothermal case in R². Furthermore the blowup rate of the above solutions is also studied and some remarks about the applicability of such solutions to the Navier-Stokes-Poisson equations and the drift-diffusion model in semiconductors are included. Finally, for the isothermal case (γ=1), the result of Makino and Perthame for the tame solutions is extended to show that the life span of such solutions must be finite if the initial data is with compact support.     

A competition between Fujita and Strauss type exponents for blow-up of semi-linear wave equations with scale-invariant damping and mass

We obtain a blow-up result for solutions to a semi-linear wave equation with scale-invariant dissipation and mass and power non-linearity, in the case in which the model has a “wave like” behavior. We perform a change of variables that transforms our starting equation in a strictly hyperbolic semi-linear wave equation with time-dependent speed of propagation. Applying Kato's lemma we prove a blow-up result for solutions to the transformed equation under some assumptions on the initial data. The limit case, that is, when the exponent p is exactly equal to the upper bound of the range of admissible values of p yielding blow-up needs special considerations. In this critical case an explicit integral representation formula for solutions of the corresponding linear Cauchy problem in 1d is derived. Finally, carrying out the inverse change of variables we get a non-existence result for global (in time) solutions to the original model.     

Boundedness of global positive solutions of a porous medium equation with a moving localized source

In this paper a porous medium equation with a moving localized source ut=ur(Δu+af(u(x₀(t),t))) is considered. It is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data. Moreover, in one space dimension case, it is also shown that all global positive solutions of the above equation are uniformly bounded, and this differs from that of a porous medium equation with a local source.