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Explicit Traveling Wave Solutions to Nonlinear Evolution Equations

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Chinese Annals of Mathematics,Series B 2011年第6期32卷 929-964页

作者： linghai zhang Department of Mathematics Lehigh University 14 East Packer Avenue Bethlehem Pennsylvania 18015 USA

First of all,some technical tools are *** the author studies explicit traveling wave solutions to nonlinear dispersive wave equations,nonlinear dissipative dispersive wave equations,nonlinear convection equations,nonl... 详细信息

关键词： Explicit traveling wave solutions Nonlinear partial differential eauations Reduction of order

Solutions to Some Open Problems in Fluid Dynamics

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Chinese Annals of Mathematics,Series B 2008年第2期29卷 179-198页

作者： linghai zhang Department of Mathematics Lehigh University 14 East Packer Avenue Bethlehem Pennsylvania 18015

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are *** equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.

关键词： Exact limits Sharp rates of decay Fluid dynamics equation Global smooth solutions

Decay of Solutions of Generalized Benjamin-Bona-Mahony Equations

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Acta Mathematica Sinica,English Series 1994年第4期10卷 428-438页

作者： zhang linghai (1) 1. Department of Mathematics The Ohio State University 43210 Columbus Ohio USA

This paper is a discussion of global solutions to the initial value problems for general- ized Banjamin-Bona-Mahony *** long time behaviors of the solutions are presented with the initial data in some certain Sobolev ... 详细信息

关键词： zhang Decay of Solutions of Generalized Benjamin-Bona-Mahony Equations Math

LONG-TIME UNIFORM STABILITY OF SOLUTION TO MAGNETOHYDRODYNAMICS EQUATION

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Applied Mathematics:A Journal of Chinese Universities 1999年第1期 47-52页

作者： Guo Boling\ zhang linghai\ Huang Haiyang

§１ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭａｉｎＲｅｓｕｌｔｓＷｅｓｔｕｄｙｔｈｅｌｏｎｇ－ｔｉｍｅｕｎｉｆｏｒｍｓｔａｂｉｌｉｔｙｏｆｓｏｌｕｔｉｏｎｓｔｏｔｈｅｉｎｉｔｉａｌｖａｌｕｅｐｒｏｂｌｅｍｓｆｏｒ２－ｄｉｍｅ...

§１ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭａｉｎＲｅｓｕｌｔｓＷｅｓｔｕｄｙｔｈｅｌｏｎｇ－ｔｉｍｅｕｎｉｆｏｒｍｓｔａｂｉｌｉｔｙｏｆｓｏｌｕｔｉｏｎｓｔｏｔｈｅｉｎｉｔｉａｌｖａｌｕｅｐｒｏｂｌｅｍｓｆｏｒ２－ｄｉｍｅｎｓｉｏｎａｌＭａｇｎｅｔｏｈｙｄ...

关键词：

THE IMPROVED FOURIER SPLITTING METHOD AND DECAY ESTIMATES OF THE GLOBAL SOLUTIONS OF THE CAUCHY PROBLEMS FOR NONLINEAR SYSTEMS OF FLUID DYNAMICS EQUATIONS

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Annals of Applied Mathematics 2016年第4期32卷 396-417页

作者： linghai zhang Dept.of Math. Lehigh University

Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations. The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems. We will couple togeth- er the elementary uniform energy estimates of the global weak solutions and a well known Gronwall＇s inequality to improve the Fourier splitting method. This method was initiated by Maria Schonbek in the 1980＇s to study the op- timal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations. As applications, the decay esti- mates with sharp rates of the global weak solutions of the Cauchy problems for n-dimensional incompressible Navier-Stokes equations, for the n-dimensional magnetohydrodynamics equations and for many other very interesting nonlin- ear evolution equations with dissipations can be established.

关键词： nonlinear systems of fluid dynamics equations global weaksolutions decay estimates uniform energy estimates Fourier transformation Plancherel's identity Gronwall's inequality improved Fourier splitting method

DECAY OF GLOBAL SOLUTIONS OF TWO NONLINEAR EVOLUTION EQUATIONS

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Acta Mathematicae Applicatae Sinica 1997年第1期13卷 23-32页

作者： GUO BOLING zhang linghai(Institute of Applied Physics and Computational Mathematics, Beijing 100088, China) Institute of Applied Physics and Computational Mathematics Beijing China

We study the decay of solutions of two nonlinear evolution equations: the Benjamin-OnoBurgers and the Schrodinger-Burgers equations. We establish sharp rates of L2 decay of global solutions to these problems, with initial data Uo(x)∈L1∩L2. The decay results of the solutions follow from the a priori L2 integral estimstes and the Fourier transform. The standard argument relies on a technique that involves the splitting of the phase space into two time-dependent subdomains.

关键词： Decay estimates nonlinear evolution equations

PROPERTIES OF SOLUTIONS OF n-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

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Annals of Applied Mathematics 2019年第4期35卷 392-448页

作者： linghai zhang Dept.of Math. Lehigh University 17 East Memorial Drive

Consider the n-dimensional incompressible Navier-Stokes equations δ/(δt)u-α△u +(u ·△↓)u + △↓p = f(x, t), △↓· u = 0,△↓· f = 0,u(x, 0) = u0(x), △↓·u0=*** exists a global weak solution under some assumptions on the initial function and the external force. It is well known that the global weak solutions become sufficiently small and smooth after a long time. Here are several very interesting questions about the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations.· Can we establish better decay estimates with sharp rates not only for the global weak solutions but also for all order derivatives of the global weak solutions?· Can we accomplish the exact limits of all order derivatives of the global weak solutions in terms of the given information?· Can we use the global smooth solution of the linear heat equation, with the same initial function and the external force, to approximate the global weak solutions of the Navier-Stokes equations?· If we drop the nonlinear terms in the Navier-Stokes equations, will the exact limits reduce to the exact limits of the solutions of the linear heat equation?· Will the exact limits of the derivatives of the global weak solutions of the Navier-Stokes equations and the exact limits of the derivatives of the global smooth solution of the heat equation increase at the same rate as the order m of the derivative increases? In another word, will the ratio of the exact limits for the derivatives of the global weak solutions of the Navier-Stokes equations be the same as the ratio of the exact limits for the derivatives of the global smooth solutions for the linear heat equation?The positive solutions to these questions obtained in this paper will definitely help us to better understand the properties of the global weak solutions of the incompressible Navier-Stokes equations and hopefully to discover new special structures of the Navier-Stokes equations.

关键词： the n-dimensional incompressible Navier-Stokes equations decay estimates with sharp rates exact limits appropriate coupling of existing ideas and results Fourier transformation Parseval's identity Lebesgue's dominated convergence theorem Gagliardo-Nirenberg's interpolation inequality

EVANS FUNCTIONS AND INSTABILITY OF A STANDING PULSE SOLUTION OF A NONLINEAR SYSTEM OF REACTION DIFFUSION EQUATIONS

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Annals of Applied Mathematics 2016年第1期32卷 79-101页

作者： linghai zhang Dept.of Math. Lehigh University

In this paper, we consider a nonlinear system of reaction diffusion equa- tions arising from mathematical neuroscience and two nonlinear scalar reaction diffusion equations under some assumptions on their coefficients. The main purpose is to couple together linearized stability criterion （the equivalence of the nonlinear stability, the linear stability and the spectral sta- bility of the standing pulse solutions） and Evans functions to accomplish the existence and instability of standing pulse solutions of the nonlinear system of reaction diffusion equations and the nonlinear scalar reaction diffusion equa- tions. The Evans functions for the standing pulse solutions are constructed explicitly.

关键词： nonlinear system of reaction diffusion equations standing pulse solutions existence instability linearized stability criterion Evans func- tions

NEW PROOFS OF THE DECAY ESTIMATE WITH SHARP RATE OF THE GLOBAL WEAK SOLUTION OF THE n-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

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Annals of Applied Mathematics 2018年第4期34卷 416-438页

作者： linghai zhang Dept.of Math. Lehigh University

Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u_0(x) and with the incompressible conditions ? · u = 0, ? · f = 0 and ? · u_0= 0. The spatial dimension n ≥ *** that the initial function u_0∈ L1(Rn) ∩ L^2(Rn) and the external force f ∈ L^1(Rn× R+) ∩ L^1(R+, L^2(Rn)). It is well known that there holds the decay estimate with sharp rate:(1 + t)1+n/2∫Rn|u(x, t)|2 dx ≤ C, for all time t > 0, where the dimension n ≥ 2, C > 0 is a positive constant, independent of u and(x, t).The main purpose of this paper is to provide two independent proofs of the decay estimate with sharp rate, both are complete, systematic, simplified proofs, under a weaker condition on the external force. The ideas and methods introduced in this paper may have strong influence on the decay estimates with sharp rates of the global weak solutions or the global smooth solutions of similar equations, such as the n-dimensional magnetohydrodynamics equations, where the dimension n ≥ 2.

关键词： n-dimensional incompressible Navier-Stokes equations global weak solution decay estimate with sharp rate Fourier transformation Parseval's identity Gronwall's inequality