Stability of stationary solutions of parabolic equations and of the Navier-Stokes system in the whole space

Rostov on Don. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 1, pp. 151–158, January–February, 1988.     

Zero dielectric constant limit to the non-isentropic compressible Euler-Maxwell system

We investigate the zero dielectric constant limit to the non-isentropic compressible Euler-Maxwell system.We justify this singular limit rigorously in the framework of smooth solutions and obtain the nonisentropic compressible magnetohydrodynamic equations as the dielectric constant tends to zero.     

The diffusive relaxation limit of non-isentropic Euler-Maxwell equations for plasmas

This paper concerns the non-isentropic Euler-Maxwell equations for plasmas with short momentum relaxation time. With the help of the Maxwell-type iteration, it is obtained that, as the relaxation time tends to zero, periodic initial-value problem of certain scaled non-isentropic Euler-Maxwell equations has unique smooth solutions existing in the time interval where the corresponding classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the corresponding drift-diffusion model from the non-isentropic Euler-Maxwell equations.     

Multiple stationary solutions of euler-poisson equations for non-isentropic gaseous stars

The motion of the self-gravitational gaseous stars can be described by the Euler-Poisson equations. The main purpose of this paper is concerned with the existence of stationary solutions of Euler-Poisson equations for some velocity fields and entropy functions that solve the conservation of mass and energy. Under different restriction to the strength of velocity field, we get the existence and multiplicity of the stationary solutions of Euler-Poisson system.     

Rigorous derivation of incompressible type Euler equations from non-isentropic Euler-Maxwell equations

In this paper, we investigate the convergence of the time-dependent and non-isentropic Euler-Maxwell equations to incompressible Euler equations in a torus via the combined quasi-neutral and non-relativistic limit. For well prepared initial data, the convergences of solutions of the former to the solutions of the latter are justified rigorously by an analysis of asymptotic expansions and energy method.     

Stability of stationary solutions to the Navier–Stokes equations in the Besov space

We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space ${\mathbb{R}}^n$ for $n\ge 3$. It is clarified that if w is small in ${\dot{B}}_{p_{*},q^{\prime }}^{-1+\frac{n}{p_{*}}}$ for and , then for every small initial disturbance $a\in {\dot{B}}_{p_0,q}^{-1+\frac{n}{p_0}}$ with and ($1/q+1/q^{\prime }=1$), there exists a unique solution $v(t)$ of the nonstationary Navier–Stokes equations on (0, ∞) with $v(0)=w+a$ such that ${\parallel v(t)-w\parallel }_{L^r}=O(t^{-\frac{n}{2}(\frac{1}{n}-\frac{1}{r})})$ and ${\parallel v(t)-w\parallel }_{{\dot{B}}_{p,q}^s}=O(t^{-\frac{n}{2}(\frac{1}{n}-\frac{1}{p})-\frac{s}{2}})$ as $t\to \infty$, for , , and small $s>0$.     

Diffusive expansion for solutions of the Boltzmann equation in the whole space

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

(0.1)     

Stability of stationary solutions in models of the Calvin cycle

In this paper results are obtained concerning the number of positive stationary solutions in simple models of the Calvin cycle of photosynthesis and the stability of these solutions. It is proved that there are open sets of parameters in the model of Zhu et al. (2009) for which there exist two positive stationary solutions. There are never more than two isolated positive stationary solutions but under certain explicit special conditions on the parameters there is a whole continuum of positive stationary solutions. It is also shown that in the set of parameter values for which two isolated positive stationary solutions exist there is an open subset where one of the solutions is asymptotically stable and the other is unstable. In related models derived from the work of Grimbs et al. (2011), for which it was known that more than one positive stationary solution exists, it is proved that there are parameter values for which one of these solutions is asymptotically stable and the other unstable. A key technical aspect of the proofs is to exploit the fact that there is a bifurcation where the centre manifold is one-dimensional.     

Symmetry results for decay solutions of elliptic systems in the whole space

In this paper, by using the Alexandrov-Serrin method of moving planes combined with maximum principles, we prove that the decaying positive solutions of a semi-linear elliptic system in the whole space are radially symmetric about some point. The system under our consideration includes the important physical interesting case, the stationary Schrödinger system for Bose-Einstein condensate.     

Stability estimates for the solutions of control problems for the stationary magnetohydrodynamic equations

We study control problems for the stationary magnetohydrodynamic equations. In these problems, one has to find an unknown vector function occurring in the boundary condition for the magnetic field and the solution of the boundary value problem in question by minimizing a performance functional depending on the velocity and pressure. We derive new a priori estimates for the solutions of the original boundary value problem and the extremal problem and prove theorems on the local uniqueness and stability of solutions for specific performance functionals.     

Stability and instability of stationary solutions for sublinear parabolic equations

In the present paper, we study the initial boundary value problem of the sublinear parabolic equation. We prove the existence of solutions and investigate the stability and instability of stationary solutions. We show that a unique positive and a unique negative stationary solutions are exponentially stable and give the exact exponent. We prove that small stationary solutions are unstable. For one space dimensional autonomous equations, we elucidate the structure of stationary solutions and study the stability of all stationary solutions.     

Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space

We consider the Cauchy problem for a parabolic system of chemotaxis in RN(N?1), and give the decay rates and asymptotic profiles of bounded solutions.     

Regularizing effects for the classical solutions to the Landau equation in the whole space

The Landau equation describes the binary collisional effects (through long range coulombian interaction) in a plasma. In this paper, we prove that the known classical solutions to the Landau equation near Maxwellian in the whole space have a regularizing effect in all (time, space and velocity) variables, that is, become immediately smooth with respect to all variables.     

Generic hyperbolicity of stationary solutions for a reaction-diffusion system

In this paper, we study the generic hyperbolicity of equilibria of a reaction-diffusion system with respect to nonlinear terms in the set of C²-functions equipped with the Whitney Topology. To accomplish this, we combine Baire’s Lemma and the usual Transversality Theorem.     

Existence of stationary solutions to the Vlasov-Poisson-Boltzmann system

In this paper, we study the existence of stationary solutions to the Vlasov-Poisson-Boltzmann system when the background density function tends to a positive constant with a very mild decay rate as |x|→∞. In fact, the stationary Vlasov-Poisson-Boltzmann system can be written into an elliptic equation with exponential nonlinearity. Under the assumption on the decay rate being (ln(e+|x|))^−α for some α>0, it is shown that this elliptic equation has a unique solution. This result generalizes the previous work [R. Glassey, J. Schaeffer, Y. Zheng, Steady states of the Vlasov-Poisson-Fokker-Planck system, J. Math. Anal. Appl. 202 (1996) 1058-1075] where the decay rate is assumed.     

Asymptotic limits of compressible Euler-Maxwell system in plasma physics

In this talk we will discuss asymptotic limit of compressible Euler-Maxwell system in plasma physics. Some recent results about the convergence of compressible Euler-Maxwell system to the incompressible Euler equations or incompressible e-MHD equations will be given via the quasi-neutral regime. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of stationary and periodic solutions of nonlinear parabolic systems

One investigates the problem of the stability of the solutions of nonlinear parabolic initial-boundary-value problems relative to small perturbations of the class C with any0 under suitable restrictions (depending on ) on the structure of the nonlinear terms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 84, pp. 286–304, 1979.The author expresses his thanks to his scientific advisor, V. A. Solonnikov, for his help in the completion of this paper.