On regularity criteria for the n-dimensional Navier-Stokes equations in terms of the pressure

We study the Cauchy problem for the n-dimensional Navier-Stokes equations (n?3), and prove some regularity criteria involving the integrability of the pressure or the pressure gradient for weak solutions in the Morrey, Besov and multiplier spaces.     

Global regularity for the 2D Boussinesq equations with partial viscosity terms

In this paper, we prove the global in time regularity for the 2D Boussinesq system with either the zero diffusivity or the zero viscosity. We also prove that as diffusivity (viscosity) tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion (viscosity) equations. Our result for the zero diffusion system, in particular, solves the Problem no. 3 posed by Moffatt in [R.L. Ricca, (Ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001, pp. 3-10].     

Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces

In this paper we study the super-critical 2D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove a global well-posedness result for small initial data lying in critical Besov spaces constructed over Lebesgue spaces Lp, with p∈[1,∞]. Local results for arbitrary initial data are also given.     

Local well-posedness for the 2D non-dissipative quasi-geostrophic equation in Besov spaces

In this paper we prove the local-in-time well-posedness for the 2D non-dissipative quasi-geostrophic equation, and study the blow-up criterion in the critical Besov spaces. These results improve the previous one by Constantin et al. [P. Constantin, A. Majda, E. Tabak, Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar, Nonlinearity 7 (1994) 1495–1533].     

Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations

We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. The coupling arises from a drag force exerted by each other. We establish existence of global weak solutions for the system in two and three dimensions. Furthermore, we obtain the existence and uniqueness result of global smooth solutions for dimension two. In case of three dimensions, we also prove that strong solutions exist globally in time for the Vlasov-Stokes system.     

Blow up and grazing collision in viscous fluid solid interaction systems

We investigate qualitative properties of strong solutions to a classical system describing the fall of a rigid ball under the action of gravity inside a bounded cavity filled with a viscous incompressible fluid. We prove contact between the ball and the boundary of the cavity implies blow up of strong solutions and such a contact has to occur in finite time under symmetry assumptions on the initial data.     

On the Boussinesq/Full dispersion systems and Boussinesq/Boussinesq systems for internal waves

In this paper, we consider some asymptotic models for internal waves in the small amplitude/small amplitude regime, which were derived recently by Bona, Lannes and Saut. We first prove that the Boussinesq/Full dispersion systems and the Boussinesq/Boussinesq systems can be derived from the Full dispersion/Full dispersion systems. Then using a contraction-mapping argument and the energy method, we will prove that the derived systems that are linearly well-posed are in fact locally nonlinearly well-posed in suitable Sobolev classes. In particular, we improve and extend some known results on the well-posedness of Boussinesq systems for surface waves.     

Boundary-layer effects for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane

The main purpose of this paper is to justify the Stokes-Blasius law of boundary-layer thickness for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane, i.e., we shall prove that the boundary-layer thickness is of the value δ(ε)=εα with any α∈(0,1/2) for small diffusivity coefficient ε>0. Moreover, the convergence rates of the vanishing diffusivity limit are also obtained.     

Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order

In this paper, we investigate the existence and uniqueness of the solution to the Cauchy problem for a class of nonlinear wave equations of higher order and prove the existence and nonexistence of global solutions to this problem by a potential well method.     

Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion

We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with anisotropic viscosity acting only in the horizontal direction, which arises in ocean dynamics models. Global well-posedness for this system was proven by Danchin and Paicu; however, an additional smoothness assumption on the initial density was needed to prove uniqueness. They stated that it is not clear whether uniqueness holds without this additional assumption. The present work resolves this question and we establish uniqueness without this additional assumption. Furthermore, the proof provided here is more elementary; we use only tools available in the standard theory of Sobolev spaces, and without resorting to para-product calculus. We use a new approach by defining an auxiliary “stream-function” associated with the density, analogous to the stream-function associated with the vorticity in 2D incompressible Euler equations, then we adapt some of the ideas of Yudovich for proving uniqueness for 2D Euler equations.     

On the regularity criteria for weak solutions to the magnetohydrodynamic equations

We study the regularity criteria for weak solutions to the incompressible magnetohydrodynamic equations. Some regularity criteria are obtained for weak solutions to the magnetohydrodynamic equations, which generalize the results in [C. He, Z. Xin, On the regularity of solutions to the magneto-hydrodynamic equations, J. Differential Equations 213 (2) (2005) 235-254]. Our results reveal that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solutions to the magnetohydrodynamic equations.     

Transonic shocks for steady Euler flows with cylindrical symmetry

In this paper, we prove the existence of transonic shocks adjacent to a uniform one for the full Euler system for steady compressible fluids with cylindrical symmetry in a cylinder, and consequently show the stability of such uniform transonic shocks. Mathematically we solve a free boundary problem for a quasi-linear elliptic–hyperbolic composite system. This reveals that the boundary conditions and equations interact in a subtle way. The key point is to “separate” in a suitable way the elliptic and hyperbolic parts of the system. The approach developed here can be applied to deal with certain multidimensional problems concerning stability of transonic shocks for the full Euler system.     

On the a priori estimates for the Euler, the Navier-Stokes and the quasi-geostrophic equations

We prove new a priori estimates for the 3D Euler, the 3D Navier-Stokes and the 2D quasi-geostrophic equations by the method of similarity transforms.     

Uniform stability at permanently acting disturbances of solutions of Navier-Stokes equations for compressible isothermic fluids

We prove that the uniform stability at permanently acting disturbances of a given solution of the Navier-Stokes equations for viscous compressible isothermic fluid is a consequence of the uniform exponential stability of the zero solution of so-called linearized equations.The research was supported by the grant No. 201/93/2177 of Grant Agency of Czech Republic.     

On the nonlinear Schrödinger limit of the magnetic Zakharov system

In this paper, we study the limit behavior of a smooth solution for the magnetic type Zakharov system. As the parameters tend to infinity, we prove that solutions of the magnetic Zakharov system converge to the solution of the nonlinear Schrödinger equation. Moreover, the detailed convergence rate is also investigated.     

Unique solvability of equations of motion for ferrofluids

We study the differential system governing the flow of an incompressible ferrofluid under the action of a magnetic field. The system is a combination of the Navier-Stokes equations, the angular momentum equation, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of the unique strong solution to the system posed in a bounded domain of R³ and equipped with initial and boundary conditions.     

Behaviour of weak solutions of compressible Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor

The aim of this paper is to study the behaviour of a weak solution to Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor for time going to infinity. In an analogous way as in [18], we construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is mentioned as well.     

On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law

This paper is concerned with the existence and uniqueness of global, periodic and stationary solutions for flows of incompressible viscoelastic fluids for which the extra-stress tensor satisfies a differential constitutive law. More precisely, we prove that the results obtained by C Guillopé and J.C. Saut [5] remain true without any restriction on the smallness of the retardation parameter.     

Uniqueness of transonic shock solutions in a duct for steady potential flow

We study the uniqueness of solutions with a transonic shock in a duct in a class of transonic shock solutions, which are not necessarily small perturbations of the background solution, for steady potential flow. We prove that, for given uniform supersonic upstream flow in a straight duct, there exists a unique uniform pressure at the exit of the duct such that a transonic shock solution exists in the duct, which is unique modulo translation. For any other given uniform pressure at the exit, there exists no transonic shock solution in the duct. This is equivalent to establishing a uniqueness theorem for a free boundary problem of a partial differential equation of second order in a bounded or unbounded duct. The proof is based on the maximum/comparison principle and a judicious choice of special transonic shock solutions as a comparison solution.