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## Publications

Publications (108)

We consider the one-dimensional compressible Navier–Stokes system for a viscous and heat-conducting ideal polytropic gas when the viscosity (Formula presented.) and the heat conductivity (Formula presented.) depend on the specific volume (Formula presented.) and the temperature (Formula presented.) and are both proportional to (Formula presented.)...

In this article, we are concerned with the construction of global smooth small-amplitude solutions to the Cauchy problem of the one species Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions. Compared with the former result obtained by Duan and Liu in [12] for the two species model, we do not ask the initial perturbation t...

The two-species Vlasov–Maxwell–Boltzmann system is an important model for plasma physics describing the time evolution of dilute charged particles consisting of electrons and ions under the influence of the self-consistent internally generated Lorentz forces. In physical situations the ion mass is usually much larger than the electron mass so that...

We study the large-time behavior of solutions to the compressible
Navier-Stokes equations for a viscous and heat-conducting ideal polytropic gas
in the one-dimensional half-space. A rarefaction wave and its superposition
with a non-degenerate stationary solution are shown to be asymptotically stable
for the outflow problem with large initial pertur...

This paper is concerned with the inflow problem for the one-dimensional
compressible Navier-Stokes equations. For such a problem, F. M. Huang, A.
Matsumura and X. D. Shi showed that there exists viscous shock wave solution to
the inflow problem and both the boundary layer solution, the viscous shock
wave, and their superposition are time-asymptotic...

This paper is concerned with the global existence of classical solutions with large initial data away from vacuum to the Cauchy problem of the one-dimensional isothermal compressible fluid models of Korteweg type with density-dependent viscosity coefficient and capillarity coefficient. The case when the viscosity coefficient and the capillarity coe...

We consider the construction of global non-vacuum solutions to the
one-dimensional compressible Navier-Stokes equations for a viscous and
heat-conducting ideal polytropic gas whose transport coefficients depend on
both the density and the temperature. A global solvability result to its Cauchy
problem is obtained for general adiabatic exponent and l...

This paper is concerned with the existence and time-asymptotic nonlinear stability of traveling wave solutions to the Cauchy problem of the one-dimensional compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. The existence of traveling wave solutions is obtained by the phase pla...

Since the work [13] by Guo [Invent. Math. 153 (2003), no. 3, 593--630], how
to establish the global existence of perturbative classical solutions around a
global Maxwellian to the Vlasov-Maxwell-Boltzmann system with the whole range
of soft potentials has been an open problem. This is mainly due to the complex
structure of the system, in particular...

The classical one-species Vlasov-Poisson-Landau system describes dynamics of electrons interacting with its self-consistent electrostatic field as well as its grazing collisions modeled by the famous Landau (Fokker-Planck) collision kernel. We show in this manuscript that the Cauchy problem for the one-species Vlasov-Poisson-Landau system which inc...

Even though the system of the compressible Navier–Stokes equations is not a limiting system of the Boltzmann equation when the Knudsen number tends to zero, it is the second order approximation by applying the Chapman–Enskog expansion. The purpose of this paper is to justify this approximation rigorously in mathematics. That is, if the difference b...

The dynamics of dilute electrons can be modeled by the fundamental
one-species Vlasov-Poisson-Boltzmann system which describes mutual interactions
of the electrons through collisions in the self-consistent electrostatic field.
For cutoff intermolecular interactions, although there are some progress on the
construction of global smooth solutions to...

This paper is concerned with the inflow problem for the one-dimensional
compressible Navier-Stokes equations. For such a problem, Matsumura and
Nishihara showed in [A. Matsumura and K. Nishihara, Large-time behaviors of
solutions to an inflow problem in the half space for a one-dimensional system
of compressible viscous gas. Comm. Math. Phys. 222 (...

Le sujet de cet article porte sur l'existence, l'unicité et la stabilité non linéaire des solutions stationnaires du problème de Cauchy d'un système compressible de Navier–Stokes–Korteweg, affecté par la source de masse, la force extérieure de forme générale, et la source d'énergie dans R3R3. Basées sur la méthode L2L2 à poids et certaines estimati...

The dynamics of dilute electrons can be modeled by the fundamental one-species Vlasov-Poisson-Boltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electrostatic field. For cutoff intermolecular interactions, although there are some progress on the construction of global smooth solutions to...

We first construct the global existence of the classical solutions near the absolute equilibrium to the Boltzmann equation with general collision kernels under the Navier-Stokes scaling in whole space. In particular, a global energy estimate uniform in the Knudsen number is derived. Then, by using the local conservation laws of this sequence of sol...

The Cauchy problem of the relativistic Landau–Maxwell system in R3R3 is investigated. For perturbative initial data with suitable regularity and integrability, we obtain the optimal large-time decay rates of the relativistic Landau–Maxwell system. For the proof, a new interactive instant energy functional is introduced to capture the macroscopic di...

We present a L2-stability theory of the Vlasov-Maxwell-Boltzmann system for the two-species collisional plasma. We show that in a perturbative regime of a global Maxwellian, the L2-distance between two strong solutions can be controlled by that between initial data in a Lipschitz manner. Our stability result extends earlier results [Ha, S.-Y. and X...

Nonlinear stability of stationary waves to damped wave equations has been studied by many authors in recent years, the main difficulty lies in how to control the possible growth of its solutions caused by the nonlinearity of the equation under consideration. An effective way to overcome such a difficulty is to use the smallness of the initial pertu...

This paper is concerned with nonlinear stability of planar viscous shock profiles of the generalized Benjamin–Bona–Mahony–Burgers equations in two dimensions. Our analysis is motivated by J. Goodman’s work (1989) [4] on the nonlinear stability of scalar viscous shock profiles in two dimensions and is based on some new decay estimates on the planar...

A global solvability result of the Cauchy problem of the two-species
Vlasov-Maxwell-Landau system near a given global Maxwellian is established by
employing an approach different than that of [5]. Compared with that of [5],
the minimal regularity index and the smallness assumptions we imposed on the
initial data are weaker. Our analysis does not re...

This paper is concerned with the Cauchy problem of the Vlasov–Poisson–Boltzmann system near a given global Maxwellian with angular cutoff for a class of soft potentials in three space dimensions and the main purpose here is to derive the global solvability of such a problem without the neutral condition which was imposed in Duan et al. (2013) [11]...

This paper is concerned with nonlinear stability of viscous shock profiles for the one-dimensional isentropic compressible Navier-Stokes equations. For the case when the diffusion wave introduced in [T.-P. Liu, Mem. Am. Math. Soc. 328, 108 p. (1985; Zbl 0617.35058); Commun. Pure Appl. Math. 39, 565–594 (1986; Zbl 0617.76069)] is excluded, such a pr...

This paper is concerned with the nonlinear stability of planar stationary waves of generalized Benjamin–Bona–Mahony–Burgers equations in half-plane. The planar stationary waves are shown to be globally nonlinear stable and then, by employing the space–time weighted energy method developed by Kawashima and Matsumura in 1995 and a generalized Hardy t...

This paper is concerned with the global stability of boundary layer solutions, together with the corresponding convergence rates, to the initial-boundary value problem for one-dimensional damped wave equation with a nonlinear convection term in the half space R+.

This paper is concerned with the Cauchy problem of the one-dimensional compressible Navier-Stokes equations with degenerate temperature dependent transport coefficients which satisfy conditions from the consideration in kinetic theory. A result on the existence and uniqueness of a globally smooth nonvacuum solution is obtained provided that the (ga...

This paper is concerned with the construction of global smooth solutions away from vacuum to the Cauchy problem of the one-dimensional compressible Navier-Stokes-Poisson system with large data and density dependent viscosity coefficient and density and temper-ature dependent heat conductivity coefficient. The proof is based on some detailed analysi...

Although there recently have been extensive studies on the pertur-bation theory of the angular non-cutoff Boltzmann equation (cf. [4] and [17]), it remains mathematically unknown when there is a self-consistent Lorentz force coupled with the Maxwell equations in the nonrelativistic approximation. In the paper, for perturbative initial data with sui...

This paper is concerned with nonlinear stability of strong planar rarefaction waves for the Jin–Xin relaxation approximation of scalar conservation laws in several dimensions. For such a problem, local stability of weak or strong planar rarefaction waves have been obtained in Luo (1997) [20] and Zhao (2000) [43] respectively. For the global stabili...

This paper is concerned with the Cauchy problem on the Vlasov–Poisson–Boltzmann system for hard potentials in the whole space. When the initial data is a small perturbation of a global Maxwellian, a satisfactory global existence theory of classical solutions to this problem, together with the corresponding temporal decay estimates on the global sol...

This paper is concerned with the existence, uniqueness and nonlinear
stability of stationary solutions to the Cauchy problem of the full
compressible Navier-Stokes-Korteweg system effected by external force of
general form in $\mathbb{R}^3$. Based on the weighted-$L^2$ method and some
elaborate $L^\infty$ estimates of solutions to the linearized pr...

This paper is concerned with the existence, uniqueness and time-asymptotic
stability of time periodic solutions to the compressible Navier-Stokes-Korteweg
system effected by a time periodic external force in $\mathbb{R}^n$. Our
analysis is based on a combination of the energy method and the time decay
estimates of solutions to the linearized system...

Based on the recent study on the Vlasov-Poisson-Boltzmann system with general
angular cutoff potentials [3, 4], we establish in this paper the global
existence of classical solutions to the Cauchy problem of the
Vlasov-Poisson-Landau system that includes the Coulomb potential. This then
provides a different approach on this topic from the recent wo...

An important physical model describing the dynamics of dilute weakly ionized
plasmas in the collisional kinetic theory is the Vlasov-Poisson-Boltzmann
system for which the plasma responds strongly to the self-consistent
electrostatic force. This paper is concerned with the electron dynamics of
kinetic plasmas in the whole space when the positive ch...

This paper is concerned with the diffusive expansion for solutions of the rescaled Boltzmann equation in the whole space

This paper is concerned with the optimal temporal decay estimates on the solutions of the Cauchy problem of the Cahn–Hilliard equation. It is shown in Liu, Wang and Zhao (2007) [11] that such a Cauchy problem admits a unique global smooth solution u(t,x) provided that the smooth nonlinear function φ(u) satisfies a local growth condition. Furthermor...

In this paper we investigate the exponential time decay rate of solutions toward traveling waves for the Cauchy problem of generalized Korteweg–de Vries–Burgers equations equation(E)ut+δuxxx−νuxx+f(u)x=0,t>0,x∈R with prescribed initial data equation(I)u(x,0)=u0(x)→u±,asx→±∞. Here δ≠0δ≠0 and ν>0ν>0 are real constants, u+≠u−u+≠u− are two given consta...

This paper is concerned with the initial–boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation in the half-space R+(I) Here u(t,x) is an unknown function of t>0 and x∈R+, u+≠ub are two given constant states and the nonlinear function f(u)∈C2(R) is assumed to be a strictly convex function of u. We first show that the corres...

The existence of stationary solution to an exterior domain of the Boltzmann equation was first studied by S. Ukai and K. Asano in [25, 27] and was recently generalized by S. Ukai, T. Yang, and H. J. Zhao in[29] to more general boundary conditions. We note, however, that the results obtained in [25, 29] require that the temperature of the far field...

The exterior problem arising from the study of a flow past an obstacle is one of the most classical and important subjects in gas dynamics and fluid mechanics. The point of this problem is to assign the bulk velocity at infinity, which is not a trivial driving force on the flow so that some non-trivial solution profiles persist. In this paper, we c...

Although the decay in time estimates of the semi-group generated by the linearized Boltzmann operator without forcing have been well established, there is no corresponding result for the case with general external force. This paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted Sobolev spaces for the...

This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space ℝ + u tt -u xx +u t +f(u) x =0,t>0,x∈ℝ + ,u(0,x)=u 0 (x)→u + ,asx→+∞,u t (0,x)=u 1 (x),u(t,0)=u b ·(I) For the non-degenerate case f ' (u + )<0, it is shown in the paper by Y. Ueda [Asymptotic convergence towa...

This paper is concerned with the Cauchy problem of the Cahn–Hilliard equation{∂u∂t+Δφ(u)+Δ2u=0,x∈RN,t>0,u|t=0=u0(x),x∈RN. First, we construct a local smooth solution u(t,x) to the above Cauchy problem, then by combining some a priori estimates, Sobolev's embedding theorem and the continuity argument, the local smooth solution u(t,x) is extended ste...

This paper is concerned with the global stability of strong rarefaction waves for the generalized KdV–Burgers equation. In contrast to former results obtained by Z.A. Wang and C.J. Zhu [Stability of the rarefaction wave for the generalized KdV–Burgers equation, Acta Math. Sci. 22B (3) (2002) 319–328], ours do not require the strength of the rarefac...

The time evolution of the distribution function for the charged particles in a dilute gas is governed by the Vlasov–Poisson–Boltzmann system when the force is self-induced and its potential function satisfies the Poisson equation. In this paper, we give a satisfactory global existence theory of classical solutions to this system when the initial da...

The dynamics of dilute electrons can be modelled by the fundamental Vlasov–Poisson–Boltzmann system which describes mutual
interactions of the electrons through collisions in the self-consistent electric field. In this paper, it is shown that any
smooth perturbation of a given global Maxwellian leads to a unique global-in-time classical solution wh...

This paper is concerned with the global stability of strong rarefaction waves for a class of 2×2 hyperbolic conservation laws with artificial viscosity, i.e., the p-system with artificial viscosity where εi(i=1,2) are positive constants and p(v) is a smooth function defined on v>0 satisfying p′(v)<0, p″(v)>0 for v>0.Let (V(t,x),U(t,x)) be the smoot...

For the compressible Navier-Stokes equations with a stationary potential force, the stability of the stationary solutions was studied by Matsumura and Nishida. The convergence rate to the stationary solutions in time was later studied by Deckelnick which was improved by Shibata and Tanaka for more general external forces. This paper deals with the...

For the Boltzmann equation with an external force in the form of the gradient
of a potential function in space variable, the stability of its stationary solutions as local
Maxwellians was studied by S. Ukai et al. (2005) through the energy method. Based on
this stability analysis and some techniques on analyzing the convergence rates to station-
ar...

We consider the Cauchy problem for the damped wave equation with absorption(∗) The behavior of u as t→∞ is expected to be same as that for the corresponding heat equation ϕt−Δϕ+|ϕ|ρ−1ϕ=0, (t,x)∈R+×RN. In the subcritical case 1<ρ<ρc(N):=1+2/N there exists a similarity solution wb(t,x) with the form depending on b=lim|x|→∞|x|2/(ρ−1)f(|x|)⩾0. Our firs...

We consider the Cauchy problem for the damped wave equation with absorption The behavior of u as t→∞ is expected to be the same as that for the corresponding heat equation which has the similarity solution wa(t,x) with the form depending on a=lim|x|→∞|x|2/(p−1)f(x)⩾0 provided that p is less than the Fujita exponent pc(N):=1+2/N. In this paper, as a...

On the half-line R+=(0,∞) the initial-boundary value problems with null-Dirichlet boundary for both the semilinear heat equation and damped wave equation are considered. The critical exponent ρc(N,k) of semilinear term for the existence and nonexistence about the semilinear heat equation on the halved space is given by ρc(N,k)=1+2/(N+k) (J. Appl. M...

For the Boltzmann equation with an external potential force depending only on the space variables, there is a family of stationary solutions, which are local Maxwellians with space-dependent density, zero velocity and constant temperature. In this paper, we study the nonlinear stability of these stationary solutions by using the energy method. The...

We study stability of subsonic phase boundary solutions in the Suliciu model for phase transitions under tri-linear structural relation. With the help of Laplace transform, the evolution of perturbation is described by a linear dynamical system, and explicit solution is obtained in terms of inverse Laplace transform. Stability is established throug...

This paper is concerned with global stability of strong rarefaction waves of the Jin–Xin relaxation model for the p-system. The proofs are given by an elementary energy method and the existence of a positively invariant region obtained by Serre [Serre, D. (2000). Relaxations semi-lineaire et cinetique des systemes de Lois de conservation. Ann. Inst...

In this article, we give a uniform BV estimates and L1-stability of solutions to the Lax-Friedrichs' scheme for a model of radiating gas when the strict C-F-L (Courant-Friedrichs-Levy) condition is satisfied. This result implies that the approximate solutions generated by the Lax-Friedrichs' scheme converge to the solution given by the method of va...

In this paper, we prove that the Cauchy problem to a hyperbolic conservation laws with relaxation with singular initial data admits a unique global entropy solution in the sense of Definition 1.1. Compared with former results in this direction, the main ingredient of this paper lies in the fact that it contains a uniqueness result and we do not ask...

This paper concerns the time asymptotic behavior toward large rarefaction waves of the solution to general systems of 2 × 2 hyperbolic conservation laws with positive viscosity coefficient B(u) 8 < ut + F(u)x = (B(u)ux)x, u ∈ R2, : u(0,x) = u 0(x) → u± as x → ±∞. Assume that the corresponding Riemann problem 8 ut + F(u)x = 0, ( u -1, x < 0, u(0,x)...

This paper is concerned with the large time behavior of global smooth solutions to the Cauchy problem of the p-system with relaxation. Former results in this direction indicate that such a problem possesses a global smooth solution provided that the first derivative of the solutions with respect to the space variable x are sufficiently small. Under...

This paper is concerned with the large time behaviour of solutions to the Cauchy problem of the following nonlinear parabolic equations:ut=Δu+F(u,Dxu,Dx2u),u∈Rn,u(t,x)|t=0=u0(x),x∈RN,N⩾1.Under the optimal growth conditions on the smooth nonlinear function F(u,Dxu,Dx2u), we obtain the global existence results to the above Cauchy problem. The influen...

We give uniform BV estimates and L<sup>1</sup>-stability of Lax-Friedrichs' scheme for a class of n× n systems of strictly hyperbolic conservation laws whose integral curves of the eigenvector fields are straight lines, i.e., Temple class, under the assumption of small total variation. This implies that the approximate solutions generated via the L...

The asymptotic behavior of the solutions toward the contact disconti- nuity for the one-dimensional compressible Navier-Stokes equations with a free boundary is investigated. It is shown that the viscous contact discontinu- ity introduced in (3) is asymptotic stable with arbitrarily large initial pertur- bation if the adiabatic exponent g is near 1...

We are concerned with the Cauchy problem of general nonlinear parabolic equations. Under the optimal local growth conditions on the smooth nonlinear function F(u,D x u,D x 2 u), a global existence result is obtained. Furthermore, we also show that the smallness condition we imposed on the initial data is, in some sense, also optimal.

This paper is concerned with the free boundary problem for the one-dimensional compressible Navier–Stokes equations with density-dependent viscosity. A local (in time) existence result is established when the initial density is of compact support and connects to the vacuum continuously.

In this paper, we study asymptotic behaviour of the global smooth solutions to the multidimensional hydrodynamic model for semiconductors. We prove that the solution of the problem converges to a stationary solution time asymptotically exponentially fast. Copyright © 2002 John Wiley & Sons, Ltd.

This paper is concerned with the convergence rates to viscous shock profile for general scalar viscous conservation laws. Compared with former results in this direction, the main novelty in this paper lies in the fact that the initial disturbance can be chosen arbitrarily large. This answers positively an open problem proposed by A. Matsumura in [1...

In this paper, we consider the inverse problem of the scattering of a plane acoustic wave by a multilayered scatterer; especially we study the properties of the corresponding far field operator. This problem models time-harmonic acoustic or electromagnetic scattering by a penetrable homogeneous medium with an impenetrable core. The discussion is es...

We are concerned with the large time decay estimates of solutions to the Cauchy problem of nonlinear parabolic equations. Under the optimal growth conditions on the smooth nonlinear function F(u,D x u) as (u,D x u→(0,0), a global existence result is obtained and the influence of the nonlinear function F(u,D x u) on the large time behavior of the co...

This paper is concerned with the p-system with frictional damping and our main purpose is two-fold: First, we show that for a certain class of given large initial data (v0(x), u0(x)), the Cauchy problem (1.1), (1.2) admits a unique global smooth solution (v(t, x), u(t, x)) and such a solution tends time-asymptotically, at the optimal Lp(2⩽p⩽∞) deca...

We study the existence problem for the following nonstrictly hyperbolic system: u(t) + 1/2(3u(2) + v(2))(x) = 0, v(t) + (uv)(x) = 0, with singular initial data, i.e., (u(t, x), v(t, x))\(t=0) = (u(0)(x),v(0)(x)) is an element of L-4(R, R-2). A strong convergence result of the L-4(R+ x R, R-2) bounded approximating sequences generated by the method...

This paper, as a follow-up to the recent work of K. Nishihara (1997, J. Differential Equations133, 384–395), is concerned with the asymptotic behaviors of the solution of quasilinear hyperbolic equation with linear damping which satisfies the following prescribed initial condition:Compared with the results obtained by K. Nishihara, the main novelti...

This work is concerned with the Cauchy problem for a given class of nonlinear dispersive equations. Results for this problem are derived by employing the theory of Fourier analysis.

We study the Cauchy problem for a generalized Kuramoto–Sivashinsky system in multidimension. The zero equilibrium is linearly stable for coefficients within a certain range. We further establish the global existence, nonlinear stability, and optimal decay rate of the solution for coefficients within the same range, though with suitable restriction...

This paper considers the optimal temporal decay estimates on the solutions to the following multidimensional generalized BBM-Burgers equations with dissipative term u t +∑ j=1 N f j (u) x j -Δu t -Δu+Δ 2 u=0,x∈ℝ N ,t>0(1) with initial data u(t,x)| t=0 =u 0 (x),x∈ℝ N ·(2) Here u(t,x)=(u 1 (t,x),⋯,u n (t,x)) T is the unknown vector-valued function, f...

This paper is a continuation of our previons paper H.-J. Zhao and B.-J. Xuan [Nonlinear Anal., Theory Methods Appl. 28, 1835–1849 (1997; Zbl 0873.35087)]. It is concerned with the existence and convergence of the global smooth solutions for the multidimensional generalized BBM-Burgers equations with dissipative term u t +∑ j=1 N f j (u) x j -αΔu t...

In this paper, we study the initial boundary value problem of the following hyperbolic system with relaxation[formula]on the half line R+ with the boundary conditions v(0, t)=v−. When the asymptotic states are stationary wave or rarefaction wave or superposition of these two kind waves, we prove the stability of these wave patterns for small pertur...

In this paper, we show that a strong planar rarefaction wave is nonlinear stable, namely it is an attractor for the relaxation approximation of the scalar conservation laws in several space dimensions. Compared with former results obtained by T. P. Liu (1987, Comm. Math. Phys.108, 153–175) and T. Luo (1997, J. Differential Equations133, 255–279), o...

This paper considers the global existence and optimal temporal decay—estimates of solutions to a class of multidimensional nonlinear evolution equations whose dispersive and dissipative terms have the same order p(p > 1). Such a class includes the multidimensional generalized Benjamin—Ono—Burgers equation and the multidimensional generalized Schrod...

In memory of a good friend and a fine mathematician Riassunto: Il lavoro concerne le stime di decadimento temporale per le soluzioni del problema di Cauchy di un sistema di leggi di conservazione paraboliche apì u dimen-sioni. I risultati ottenuti migliorano quelli ottenuti dagli autori in un lavoro precedente. Abstract: This paper is concerned wit...

In this paper, we prove the existence of global smooth solution for the Cauchy problem of nonlinearly damped p-system with large initial data. The analysis is based on several key a prioriestirmates. which are obtained by the tnaxirnum principle. Our results extend the corresponding results in [l0,11].

In this paper, the weakly nonlinear limit for the relaxation approximation of conservation laws in several space dimensions is derived through asymptotic expansions and justified by employing the energy estimates. Compared with the work of G. Q. Chen, C. D. Levermore, and T. P. Liu (1994,Comm. Pure Appl. Math.47, 787–830), the main difficulty we co...

We study the Cauchy problem for a set of nonlinear evolution equations, which has been proposed for the investigation of nonlinear interaction between ellipticity and dissipation. The zero equilibrium is linearly stable for coefficients within certain range. We further establish the global existence, nonlinear stability and optimal decay rate of th...

This paper is closely related to the work of D. Hoff and J. A. Smoller [8], M. Schonbek [l8] and is concerned with the global existence and the optimal temporal decay estimates for the following one-dimensional parabolic conservation lawsWhere is the unknown vector is an arbitrary n × 1 smooth vector—valued function defined in a ball of radius r ce...

In this paper, we study the strong convergence of a sequence of uniformLploc(R×R+) bounded approximate solutions {uϵ(x,t)} to the following scalar conservation laws[formula]with initial data[formula]Without the convexity assumption and growth condition at infinity forf(x,t,u), we prove strong convergence of a subsequence of {uε(x,t)}. Under a more...

This paper is a continuation of our previous paper. It is concerned with the global existence and the optimal temporal decay estimates for the Cauchy problem of the following multidimensional parabolic conservation laws[formula] Hereu(t,x)=(u1(t,x),…,un(t,x))tis the unknown vector,fj(u)=(fj1(u),…,fjn(u))t(j=1,2,…,N) are arbitraryn×1 smooth vector-v...

This paper considers the Cauchy problem to the system of one-dimensional isentropic/isothermal compressible flow where ε; 0 is a constant, is the adiabatic exponent. A global existence result with large initial data is established by employing Chueh, Conley, and smoller's theory of positively invariant regions, the method of energy integral, and so...

We prove some results on the global existence of smooth solutions for certain nonlinear parabolic systems of the form Ut + A(U)Ux = DUxx. Here U is a vector and A(U), D are matrices with D a constant, positive matrix. We show how to use our results to study the global continuous (or generalised) solutions to the corresponding nonlinear hyperbolic c...

This paper examines the Cauchy problem for a viscoelastic model with relaxation
with discontinuous, large initial data, where ½ ≦ μ <1, δ > 0 are constants. We first give a definition of admissible (or entropic) solutions to the system. Under this definition, we prove the existence, uniqueness and continuous dependence of the global admissible sol...

We prove the global existence of smooth solutions for certain nonlinear systems of the form is a vector, are n x n smooth matrixes and D is a constant positive matrix. We assume the Cauchy data u0 satisfies , where a is a fixed vector, Aj(u)(j =1,2,. . .,N) defined in an r-ball about sufficiently small. The main techniques we used in this paper are...

This paper considers the Cauchy problem of the following convection diffusion system uv t +1 2au 2 +2buv+v 2 bu 2 +2uv x +f(u,v)g(u,v)=εuv xx · A global existence result for the Cauchy problem is established by employing the techniques of F. B. Weissler [Isr. J. Math. 38, 29-40 (1981; Zbl 0476.35043)] and the energy method. Here a,b,ε>0 are constan...

We study the Cauchy problem for two important nonstrictly quasilinear hyperbolic systems, the isentropic gas dynamics system in Euler coordinates and the rotational degeneracy of hyperbolic systems of conservation laws. Sufficient conditions which guarantee the existence of global smooth solutions of the Cauchy problems are obtained by employing th...

We consider the globally smooth solutions of diagonalizable systems consisting of n equations. We give a sufficient condition which guarantees the global existence of smooth solutions. The techniques used in this paper can be applied to study the globally smooth (or continuous) solutions of diagonalizable nonstrict hyperbolic conversation laws.