On strong solutions of the multi-layer quasi-geostrophic equations of the ocean

In this article, we study the multi-layer quasi-geostrophic equations of the ocean. The existence of strong solutions is proved. We also prove the existence of a maximal attractor in L²(Ω)$L^2\left(\Omega \right)$ and we derive estimates of its Hausdorff and fractal dimensions in terms of the data. Our estimates rely on a new formulation that we introduce for the multi-layer quasi-geostrophic equation of the ocean, which replaces the nonhomogeneous boundary conditions (and the nonlocal constraint) on the stream-function by a simple homogeneous Dirichlet boundary condition. This work improves the results given in [C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier–Stokes equations and estimate of its dimension, Adv. Math. Sci. Appl. 4 (2) (1994) 465–489].     

Global well-posedness of the Cauchy problem of the fifth-order shallow water equation

By using the I-method, we prove that the Cauchy problem of the fifth-order shallow water equation is globally well-posed in the Sobolev space Hs(R) provided .     

Instability of solitary wave solutions for the generalized BO-ZK equation

In this paper we study the generalized BO-ZK equation in two space dimensions

ut+upux+αHuxx+εuxyy=0.     

On the Cauchy problem for a coupled system of third-order nonlinear Schrödinger equations

We investigate some well-posedness issues for the initial value problem (IVP) associated with the system

     

On the shock wave spectrum for isentropic gas dynamics with capillarity

We consider the stability problem for shock layers in Slemrod's model of an isentropic gas with capillarity. We show that these traveling waves are monotone in the weak capillarity case, and become highly oscillatory as the capillarity strength increases. Using a spectral energy estimate we prove that small-amplitude monotone shocks are spectrally stable. We also show that monotone shocks have no unstable real spectrum regardless of amplitude; this implies that any instabilities of these monotone traveling waves, if they exist, must occur through a Hopf-like bifurcation, where one or more conjugate pairs of eigenvalues cross the imaginary axis. We then conduct a systematic numerical Evans function study, which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths. In particular, we show that the transition from monotone to nonmonotone profiles does not appear to trigger any instabilities.     

On the regularity criteria of a surface quasi-geostrophic equation

We obtain regularity criteria for a quasi-geostrophic equation that depends more on one direction than the others. In particular, we show that in the critical case, the global regularity depends only on a partial derivative rather than a gradient of the solution.     

Stability of the Camassa-Holm solitons

Summary. We consider the stability problem of the solitary wave solutions of a completely integrable equation that arises as a model for the unidirectional propagation of shallow water waves. We prove that the solitary waves possess the spectral properties of solitons and that their shapes are stable under small disturbances.     

On the global regularity of generalized Leray-alpha type models

We generalize Leray-alpha type models studied in Cheskidov et al. (2005) [1] and Linshiz and Titi (2007) [4] via fractional Laplacians and employ Besov space techniques to obtain global regularity results with the logarithmically supercritical dissipation.     

Neumann problem for the Korteweg-de Vries equation

We consider Neumann initial-boundary value problem for the Korteweg-de Vries equation on a half-line

(0.1)     

The equations of potential flow of compressible viscous fluid at low Reynolds number

The existence, uniqueness, and stabilization of solutions are investigated for two approximate models of viscous compressible fluid.     

Stability for steady states of Navier-Stokes-Poisson equations

In this paper, we study the stationary solution and nonlinear stability of Navier-Stokes-Poisson equations. Using variational method, we construct steady states of the N-S-P system as minimizers of a suitably defined energy functional, then show their dynamical stability against general, i.e. not necessarily spherically symmetric perturbation.     

Existence of the universal attractor for the 3-D viscous primitive equations of large-scale moist atmosphere

In this paper, we consider the initial-boundary value problem for the three-dimensional viscous primitive equations of large-scale moist atmosphere which are used to describe the turbulent behavior of long-term weather prediction and climate changes. By obtaining the existence and uniqueness of global strong solutions for the problem and studying the long-time behavior of strong solutions, we prove the existence of the universal attractor for the dynamical system generated by the primitive equations of large-scale moist atmosphere.     

On mathematical foundation of the Brownian motor theory

The paper contains mathematical justification of basic facts concerning the Brownian motor theory. The homogenization theorems are proved for the Brownian motion in periodic tubes with a constant drift. The study is based on an application of the Bloch decomposition. The effective drift and effective diffusivity are expressed in terms of the principal eigenvalue of the Bloch spectral problem on the cell of periodicity as well as in terms of the harmonic coordinate and the density of the invariant measure. We apply the formulas for the effective parameters to study the motion in periodic tubes with nearly separated dead zones.     

The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation

A generalization of the Camassa-Holm equation, a model for shallow water waves, is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space Hs(R) with is established via a limiting procedure. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space Hs with is developed.     

A non-autonomous two-phase flow model with oscillating external force and its global attractor

In this article, we consider a non-autonomous diffuse interface model for an isothermal incompressible two-phase flow in a two-dimensional bounded domain. Assuming that the external force is singularly oscillating and depends on a small parameter ?, we prove the existence of the uniform global attractor A^?. Furthermore, using the method similar to that of Chepyzhov and Vishik (2007) [22] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of A^? as ? goes to zero. Let us mention that the nonlinearity involved in the model considered in this article is slightly stronger than the one in the two-dimensional Navier-Stokes system studied in Chepyzhov and Vishik (2007) [22].     

The periodic Cauchy problem of the modified Hunter-Saxton equation

We prove that the periodic initial value problem for the modified Hunter-Saxton equation is locally well-posed for initial data in the space of continuously differentiable functions on the circle and in Sobolev spaces when s > 3/2. We also study the analytic regularity (both in space and time variables) of this problem and prove a Cauchy-Kowalevski type theorem. Our approach is to rewrite the equation and derive the estimates which permit application of o.d.e. techniques in Banach spaces. For the analytic regularity we use a contraction argument on an appropriate scale of Banach spaces to obtain analyticity in both time and space variables.     

On spherically symmetric solutions of the relativistic Euler equation

This work investigates the spherical symmetric solutions of more realistic equation of states. We generalize the method of Hsu et al. (Methods Appl. Anal. 8 (2001) 159) to show the existence of spherical symmetric weak solution of the relativistic Euler equation with initial data containing the vacuum state.     

Yudovich type solution for the 2D inviscid Boussinesq system with critical and supercritical dissipation

In this paper we consider the Yudovich type solution of the 2D inviscid Boussinesq system with critical and supercritical dissipation. For the critical case, we show that the system admits a global and unique Yudovich type solution; for the supercritical case, we prove the local and unique existence of Yudovich type solution, and the global result under a smallness condition of _θ0${\theta }_0$. We also give a refined blowup criterion in the supercritical case.