Search

Abstract: In this paper, we study the stability of a single transonic shock wave solution to the hyperbolic conservation laws with a resonant moving source. Compared with the previous results [W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075–1098; T.P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys. 83 (2) (1982) 243–260] on this stability problem, in this paper, the transonic ith shock is assumed to be relatively strong and stable in the sense of Majda. Then the framework of [M. Lewicka, stability of patterns of non-interacting large shock waves, Indiana Univ. Math. J. 49 (4) (2000) 1515–1537; M. Lewicka, Stability conditions for patterns of noninteracting large shock waves, SIAM J. Math. Anal. 32 (5) (2001) 1094–1116 (electronic)] can be applied. A new criterion is obtained to test whether such a shock is time asymptotically stable or not. And by constructing the Liu–Yang functional, one can prove the stability of the shock under the stability condition. This is an extension of the result [S.-Y. Ha, T. Yang, stability for systems of hyperbolic conservation laws with a resonant moving source, SIAM J. Math. Anal. 34 (5) (2003) 1226–1251 (electronic); W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075–1098] to a more general case. [Copyright &y& Elsevier]

Abstract: To study the non-linear stability of a non-trivial profile for a multi-dimensional systems of gas dynamics, the combination of the Green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential. In this paper, we study the stability of a planar diffusion wave for the isentropic Euler equations with damping in two-dimensional space. By introducing an approximate Green function for the linearized equations around the planar diffusion wave and by applying the energy method, we prove the global existence and the convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave. The decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the norm. Furthermore, the constructed approximate Green function in this paper can be used for the pointwise and the estimates of the solutions concerned. In fact, the approach by combining of the Green function and energy method can be applied to other system especially when the derivatives of the coefficients in the system have certain time decay properties. [Copyright &y& Elsevier]

Abstract: In a previous paper we introduced various definitions of stability and instability for non-autonomous differential equations, and applied these to investigate the bifurcations in some simple models. In this paper we present a more systematic theory of local bifurcations in scalar non-autonomous equations. [Copyright &y& Elsevier]

Abstract: In this paper, we shall estimate the growing speed for higher Sobolev norms of the solutions to Schrödinger equations on Riemannian manifolds (), under some bilinear Strichartz estimate assumptions. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive Kolmogorov systems via the skew-product flows approach. It is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skew-product flow associated with a nonautonomous (random) competitive Kolmogorov system. [Copyright &y& Elsevier]

Abstract: Consider in this paper a linear skew-product system where or , and is a topological dynamical system on a compact metrizable space W, and where satisfies the cocycle condition based on θ and is continuously differentiable in t if . We show that ‘semi λ-exponential dichotomy’ of implies ‘λ-exponential dichotomy.’ Precisely, if Θ has no Lyapunov exponent λ and is almost uniformly λ-contracting along the λ-stable direction and if is constant a.e., then Θ is almost λ-exponentially dichotomous. To prove this, we first use Liao''s spectrum theorem, which gives integral expression of the Lyapunov exponents, and then use the semi-uniform ergodic theorem by Sturman and Stark, which allows one to derive uniform estimates from nonuniform ones. As a consequence, we obtain the open-and-dense hyperbolicity of eventual -cocycles based on a uniquely ergodic endomorphism, and of -cocycles based on a uniquely ergodic equi-continuous endomorphism, respectively. On the other hand, in the sense of -topology we obtain the density of -cocycles having positive Lyapunov exponent based on a minimal subshift satisfying the Boshernitzan condition. [Copyright &y& Elsevier]

Abstract: We construct a single transonic shock wave pattern in an infinite nozzle asymptotically converging to a cylinder, which is close to a uniform transonic shock wave. In other words, suppose there is a uniform transonic shock wave in an infinite cylinder nozzle which can be constructed easily, if we perturbed the supersonic incoming flow and the infinite nozzle a little bit, we can obtain a transonic wave near the uniform one. As a consequence, we can show that the uniform transonic wave is stable with respect to the perturbation of the incoming flow and nozzle wall. Based on the theory of [G.Q. Chen, M. Feldman, Existence and stability of multi-dimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal. 184 (2007) 185–242], the crucial parts of this paper are to derive the uniform Schauder estimates of the linear elliptic equation for the infinite nozzle asymptotically converging to a cylinder. [Copyright &y& Elsevier]

Abstract: In this paper we consider the stationary Navier–Stokes equations in a bounded domain with a variable viscosity. We prove that one can uniquely determine the viscosity function from the knowledge of boundary data. [Copyright &y& Elsevier]

Abstract: In this paper we study the limit cycles of the Liénard differential system of the form , or its equivalent system , . We provide sufficient conditions in order that the system exhibits at least n or exactly n limit cycles. [Copyright &y& Elsevier]

Abstract: In this paper a half space problem for the one-dimensional Boltzmann equation with specular reflective boundary condition is investigated. It is shown that the solution of the Boltzmann equation time-asymptotically converges to a global Maxwellian under some initial conditions. Furthermore, a time-decay rate is also obtained. [Copyright &y& Elsevier]

Abstract: In this paper we are interested in establishing up-to boundary uniform estimates for the one phase singular perturbation problem involving a nonlinear singular/degenerate elliptic operator. Our main result states: if is a domain, for some and verifies where , and with some positive constants B and M, then there exists a constant independent of ε such that . [Copyright &y& Elsevier]

Abstract: In this paper, we study the asymptotic behaviour as of the radial solution of the problem where is an annulus of , . [Copyright &y& Elsevier]

Abstract: In this paper, we study mathematical properties of an integro-differential equation that arises as a particular limit case in the study of individual cell-based model. We obtain global well-posedness for some classes of interaction potentials and finite time blow-up for others. The existence of space homogeneous steady states as well as long-time asymptotics for the solutions of the problem is also discussed. [Copyright &y& Elsevier]