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.     

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.     

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.     

Exact solutions for the Cauchy problem to the 3D spherically symmetric incompressible Navier-Stokes equations

In this article, we establish exact solutions to the Cauchy problem for the 3D spherically symmetric incompressible Navier-Stokes equations and further study the existence and asymptotic behavior of solution.     

Gain of regularity for the KP-I equation

In this paper we study the smoothness properties of solutions to the KP-I equation. We show that the equation's dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data ? possesses certain regularity and sufficient decay as x→∞, then the solution u(t) will be smoother than ? for 0<t?T where T is the existence time of the solution.     

Existence and regularity of the solution of a mixed boundary value problem for the Keldysh equation with a nonlinear absorption term

The Keldysh equation is a more general form of the classic Tricomi equation from fluid dynamics. Its well-posedness and the regularity of its solution are interesting and important. The Keldysh equation is elliptic in y>0 and is degenerate at the line y=0 in R². Adding a special nonlinear absorption term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain—similar to the potential fluid shock reflection problem. By means of an elliptic regularization technique, a delicate a priori estimate and compact argument, we show that the solution of a mixed boundary value problem of the Keldysh equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary under some conditions. We believe that this kind of regularity result for the solution will be rather useful.     

Ultradifferential operators in the study of Gevrey solvability and regularity

In this work we present a new representation formula for ultradistributions using the so‐called ultradifferential operators. The main difference between our representation result and other works is that here we do not break the duality of Gevrey functions of other s and their ultradistributions, i.e., we locally represent an element of by an infinite order operator acting on a function of class . Our main application was in the local solvability of the differential complex associated to a locally integrable structure in a Gevrey environment.     

Global well-posedness for the density-dependent incompressible magnetohydrodynamic flows in bounded domains

In this paper, we study the three-dimensional incompressible magnetohydrodynamic equations in a smooth bounded domains, in which the viscosity of the fluid and the magnetic diffusivity are concerned with density. The existence of global strong solutions is established in vacuum cases, provided the assumption that(|| ?μ(ρ0)|| Lp +|| ?ν(ρ0) ||Lq + ||b0|| L3+ ||ρ0|| L∞)(p, q 3) is small enough, there is not any smallness condition on the velocity.     

Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity–pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier–Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.     

Well‐posedness and unique continuation property for the generalized Ostrovsky equation with low regularity

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Ostrovsky equation as follows: with initial data in the modified Sobolev space . Using Fourier restriction norm method, Tao's [k,Z]?multiplier method and the contraction mapping principle, we show that the local well‐posedness is established for the initial data with (k = 2) and is established for the initial data with (k = 3). Using these results and conservation laws, we also prove that the IVP is globally well‐posed for the initial data with s = 0(k = 2,3). Finally, using complex variables technique and Paley–Wiener theorem, we prove the unique continuation property for the IVP benefited from the ideas of Zhang ZY. et al., On the unique continuation property for the modified Kawahara equation, Advances in Mathematics (China), http://advmath.pku.edu.cn/CN/10.11845/sxjz.2014078b . Copyright © 2015 John Wiley & Sons, Ltd.     

Homogenization of a convection–diffusion equation in perforated domains with a weak adsorption

The aim of this paper is to study the asymptotic behavior of the solution of a convection–diffusion equation in perforated domains with oscillating velocity and a Robin boundary condition which describes the adsorption on the bord of the obstacles. Without any periodicity assumption, for a large range of perforated media and by mean of variational homogenization, we find the global behavior when the characteristic size ε of the perforations tends to zero. The homogenized model, is a convection–diffusion equation but with an extra term coming from the weak adsorption boundary condition. An example is presented to illustrate the methodology.     

Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence

This paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible Navier–Stokes equations. Incompressibility and convective effects are both stabilized adding an interior penalty term giving L ²-control of the jump of the gradient of the approximate solution over the internal faces. Using continuous equal-order finite elements for both velocities and pressures, in a space semi-discretized formulation, we prove convergence of the approximate solution. The error estimates hold irrespective of the Reynolds number, and hence also for the incompressible Euler equations, provided the exact solution is smooth.     

Existence and uniqueness analysis of a detached shock problem for the potential flow

We study a problem for two-dimensional steady potential and isentropic Euler equations in a bounded domain, where an artificial detached shock interacts with a wedge. Using the stream function, we obtain a free boundary problem for the subsonic state and the detached artificial shock curve and we prove that such configuration admits a unique solution in certain weighted Hölder spaces. The proof is based on various Hölder and Schauder estimates for second-order elliptic equations and fixed point theorems. Moreover, we pose an energy principle and remark that the physical attached shock is the minimizer of the energy functional.     

On the uniqueness in critical spaces for compressible Navier-Stokes equations

We show some new uniqueness results for compressible flows with data having critical regularity. In the barotropic case, uniqueness is stated whenever the space dimension N satisfies N ≥ 2, and in the polytropic case, whenever N ≥ 3. The endpoints N = 2 in the barotropic case and N = 3 in the polytropic case were left open in [4], [5] and [6].     

A new path to the non blow-up of incompressible flows

One of the most challenging questions in fluid dynamics is whether the three-dimensional (3D) incompressible Navier-Stokes, 3D Euler and two-dimensional Quasi-Geostrophic (2D QG) equations can develop a finite-time singularity from smooth initial data. Recently, from a numerical point of view, Luo & Hou presented a class of potentially singular solutions to the Euler equations in a fluid with solid boundary [1], [2]. Furthermore, in two recent papers [3], [4], Tao indicates a significant barrier to establishing global regularity for the 3D Euler and Navier-Stokes equations, in that any method for achieving this, must use the finer geometric structure of these equations. In this paper, we show that the singularity discovered by Luo & Hou which lies right on the boundary is not relevant in the case of the whole domain ${\mathbb{R}}^3$. We reveal also that the translation and rotation invariance present in the Euler, Navier-Stokes and 2D QG equations are the key for the non blow-up in finite time of the solutions. The translation and rotation invariance of these equations combined with the anisotropic structure of regions of high vorticity allowed to establish a new geometric non blow-up criterion which yield us to the non blow-up of the solutions in all the Kerr's numerical experiments and to show that the potential mechanism of blow-up introduced in [5] cannot lead to the blow-up in finite time of solutions of Euler equations.     

The blow‐up criteria of smooth solutions to the generalized and ideal incompressible viscoelastic flow

In this paper, we prove two blow‐up criteria of smooth solution: one for the generalized incompressible Oldroyd model with fractional Laplacian velocity dissipation (?Δ)^αu in the space and one for the inviscid Oldroyd model. Assume that (u(t,x),F(t,x)) is a smooth solution to the generalized Oldroyd model in [0,T); then, the solution (u(t,x),F(t,x)) does not develop singularity until t = T provided . For the ideal impressible viscoelastic flow, it is shown that the smooth solution (u,F) can be extended beyond T if , which is an improvement of the result given by Hu and Hynd (A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15(2013), 431–437). Copyright © 2015 John Wiley & Sons, Ltd.