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.     

Gevrey regularity for the supercritical quasi-geostrophic equation

In this paper, following the techniques of Foias and Temam, we establish suitable Gevrey class regularity of solutions to the supercritical quasi-geostrophic equations in the whole space, with initial data in “critical” Sobolev spaces. Moreover, the Gevrey class that we obtain is “near optimal” and as a corollary, we obtain temporal decay rates of higher order Sobolev norms of the solutions. Unlike the Navier–Stokes or the subcritical quasi-geostrophic equations, the low dissipation poses a difficulty in establishing Gevrey regularity. A new commutator estimate in Gevrey classes, involving the dyadic Littlewood–Paley operators, is established that allow us to exploit the cancellation properties of the equation and circumvent this difficulty.     

On the regularity of a class of generalized quasi-geostrophic equations

In this article we consider the following generalized quasi-geostrophic equation

     

Global well-posedness and long-time dynamics for a higher order quasi-geostrophic type equation

In this paper we study a higher order viscous quasi-geostrophic type equation. This equation was derived in [11] as the limit dynamics of a singularly perturbed Navier–Stokes–Korteweg system with Coriolis force, when the Mach, Rossby and Weber numbers go to zero at the same rate.The scope of the present paper is twofold. First of all, we investigate well-posedness of such a model on the whole space ${\mathbb{R}}^2$: we prove that it is well-posed in $H^s$ for any $s\ge 3$, globally in time. Interestingly enough, we show that this equation owns two levels of energy estimates, for which one gets existence and uniqueness of weak solutions with different regularities (namely, $H^3$ and $H^4$ regularities); this fact can be viewed as a remainder of the so called BD-entropy structure of the original system.In the second part of the paper we investigate the long-time behavior of these solutions. We show that they converge to the solution of the corresponding linear parabolic type equation, with same initial datum and external force. Our proof is based on dispersive estimates both for the solutions to the linear and non-linear problems.     

Gevrey regularity of the periodic gKdV equation

Using the multilinear estimates, which were derived for proving well-posedness of the generalized Korteweg-de Vries (gKdV) equation, it is shown that if the initial data belongs to Gevrey space Gσ, σ?1, in the space variable then the solution to the corresponding Cauchy problem for gKdV belongs also to Gσ in the space variable. Moreover, the solution is not necessarily Gσ in the time variable. However, it belongs to G3σ near 0. When σ=1 these are analytic regularity results for gKdV.     

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.     

On the continuation principles for the Euler equations and the quasi-geostrophic equation

We obtain new continuation principle of the local classical solutions of the 3D Euler equations, where the regularity condition of the direction field of the vorticiy and the integrability condition of the magnitude of the vorticity are incorporated simultaneously. The regularity of the vorticity direction field is most appropriately measured by the Triebel-Lizorkin type of norm. Similar result is also obtained for the inviscid 2D quasi-geostrophic equation.     

Solutions of quasi-geostrophic turbulence in multi-layered configurations

We consider quasi-geostrophic (QG) models in two- and three-layers that are useful in theoretical studies of planetary atmospheres and oceans. In these models, the streamfunctions are given by (1+2) partial differential systems of evolution equations. A two-layer QG model, in a simplified version, is dependent exclusively on the Rossby radius of deformation. However, the f-plane QG point vortex model contains factors such as the density, thickness of each layer, the Coriolis parameter, and the constant of gravitational acceleration, and this two-layered model admits a lesser number of Lie point symmetries, as compared to the simplified model. Finally, we study a three-layer oceanography QG model of special interest, which includes asymmetric wind curl forcing or Ekman pumping, that drives double-gyre ocean circulation. In three-layers, we obtain solutions pertaining to the wind-driven doublegyre ocean flow for a range of physically relevant features, such as lateral friction and the analogue parameters of the f-plane QG model. Zero-order invariants are used to reduce the partial differential systems to ordinary differential systems. We determine conservation laws for these QG systems via multiplier methods.     

Finite time singularities in a 1D model of the quasi-geostrophic equation

In this paper we study 1D equations with nonlocal flux. These models have resemblance of the 2D quasi-geostrophic equation. We show the existence of singularities in finite time and construct explicit solutions to the equations where the singularities formed are shocks. For the critical viscosity case we show formation of singularities and global existence of solutions for small initial data.     

Remarks on the method of modulus of continuity and the modified dissipative Porous Media Equation

We employ Besov space techniques and the method of modulus of continuity to obtain the global well-posedness of the modified Porous Media Equation.     

Macroscopic regularity for the Boltzmann equation

The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of(x, t) in the region R3×(0, +∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.     

Existence and regularity to an elliptic equation with logarithmic nonlinearity

We study the nonlinear elliptic problem −Δu=χ{u>0}(logu+λf(x,u)) in Ω⊂Rn with u=0 on ∂Ω. The function is nondecreasing, sublinear and fu is continuous. For every λ>0, we obtain a maximal solution uλ?0 and prove its global regularity . There is a constant λ^∗ such that uλ vanishes on a set of positive measure for 0<λ<λ^∗, and uλ>0 for λ>λ^∗. If f is concave, for λ>λ^∗ we characterize uλ by its stability.     

Local regularity criteria of a suitable weak solution to MHD equations

We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are H¨older continuous near boundary provided that the scaled mixed L_(x,t)~(p,q) -norm of the velocity vector field with 3/p + 2/q ≤ 2,2 q ∞ is sufficiently small near the boundary. Also, we will investigate that for this solution u ∈ L_(x,t)~(p,q) with 1≤3/p+2/q≤3/2, 3 p ∞, the Hausdorff dimension of its singular set is no greater than max{p, q}(3/p+2/q-1).     

Two regularity criteria for the 3D MHD equations

This work establishes two regularity criteria for the 3D incompressible MHD equations. The first one is in terms of the derivative of the velocity field in one direction while the second one requires suitable boundedness of the derivative of the pressure in one direction.     

On the regularity of the Navier-Stokes equation in a thin periodic domain

We address the global regularity of solutions of the Navier-Stokes equations in a thin domain Ω=[0,L₁]×[0,L₂]×[0,?] with periodic boundary conditions, where L₁,L₂>0 and ?∈(0,1/2). We prove that if

     

A study on the global regularity for a model of the 3D axisymmetric Navier-Stokes equations

This paper investigates the global regularity issue concerning a model equation proposed by Hou and Lei (2008) [9] to understand the stabilizing effects of the nonlinear terms in the 3D axisymmetric Navier-Stokes and Euler equations. We establish the global regularity of a generalized version of their model with a fractional Laplacian when the fractional power satisfies an explicit condition. This condition is exactly the same as in the case of the 3D generalized Navier-Stokes equations and is due to the balance between a more regular nonlinearity and a less effective (five-dimensional) Laplacian.     

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].     

On the low regularity of the modified Korteweg-de Vries equation with a dissipative term

We consider a dissipative version of the modified Korteweg-de Vries equation ut+uxxx−uxx+x(u³)=0. We prove global well-posedness results on the associated Cauchy problem in the Sobolev spaces Hs(R) for s>−1/4 while for s<−1/2 we prove some ill-posedness issues.