Remarks on the Rayleigh-Benard Convection on Spherical Shells

The main objective of this article is to study the effect of spherical geometry on dynamic transitions and pattern formation for the Rayleigh-Benard convection. The study is mainly motivated by the importance of spherical geometry and convection in geophysical flows. It is shown in particular that the system always undergoes a continuous (Type-I) transition to a $2l_c$-dimensional sphere $S^{2lc}$, where lc is the critical wave length corresponding to the critical Rayleigh number. Furthermore, it has shown in [12] that it is critical to add nonisotropic turbulent friction terms in the momentum equation to capture the large-scale atmospheric and oceanic circulation patterns. We show in particular that the system with turbulent friction terms added undergoes the same type of dynamic transition, and obtain an explicit formula linking the critical wave number (pattern selection), the aspect ratio, and the ratio between the horizontal and vertical turbulent friction coefficients

Random data Cauchy theory for supercritical wave equations II : A global existence result

We prove that the subquartic wave equation on the three dimensional ball $\Theta$, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in $\cap_{s<1/2} H^s(\Theta)$. We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work \cite{BT2} and invariant measure considerations which allow us to obtain also precise large time dynamical informations on our solutions

On the approximate controllability for some explosive parabolic problems

We consider in this paper distributed systems governed by parabolic evolution equations which can blow up in finite time and which are controlled by initial conditions. We study here the following question : Can one choose the initial condition in such a way that the solution does not blow up before a given time T and which is, at time T, as close as we wish from a given state ? Some general results along these lines are presented here for semilinear second order parabolic equations as well as for a non local nonlinear problem. We also give some results proving that "the more the system will blow up" the "cheaper" it will be the control

Homogenization of nonlinear stochastic partial differential equations in a general ergodic environment

In this paper, we show that the concept of sigma-convergence associated to stochastic processes can tackle the homogenization of stochastic partial differential equations. In this regard, the homogenization problem for a stochastic nonlinear partial differential equation is studied. Using some deep compactness results such as the Prokhorov and Skorokhod theorems, we prove that the sequence of solutions of this problem converges in probability towards the solution of an equation of the same type. To proceed with, we use a suitable version of sigma-convergence method, the sigma-convergence for stochastic processes, which takes into account both the deterministic and random behaviours of the solutions of the problem. We apply the homogenization result to some concrete physical situations such as the periodicity, the almost periodicity, the weak almost periodicity, and others.Comment: To appear in: Stochastic Analysis and Application

A geometric condition implying energy equality for solutions of 3D Navier-Stokes equation

We prove that every weak solution $u$ to the 3D Navier-Stokes equation that belongs to the class $L^3L^{9/2}$ and \n u belongs to $L^3L^{9/5}$ localy away from a 1/2-H\"{o}lder continuous curve in time satisfies the generalized energy equality. In particular every such solution is suitable.Comment: 10 page

Mathematical results for some α\displaystyle{\alpha} models of turbulence with critical and subcritical regularizations

In this paper, we establish the existence of a unique "regular" weak solution to turbulent flows governed by a general family of $\alpha$ models with critical regularizations. In particular this family contains the simplified Bardina model and the modified Leray-$\alpha$ model. When the regularizations are subcritical, we prove the existence of weak solutions and we establish an upper bound on the Hausdorff dimension of the time singular set of those weak solutions. The result is an interpolation between the bound proved by Scheffer for the Navier-Stokes equations and the regularity result in the critical case

On thin plate spline interpolation

We present a simple, PDE-based proof of the result [M. Johnson, 2001] that the error estimates of [J. Duchon, 1978] for thin plate spline interpolation can be improved by $h^{1/2}$. We illustrate that ${\mathcal H}$-matrix techniques can successfully be employed to solve very large thin plate spline interpolation problem

Blow up criterion for compressible nematic liquid crystal flows in dimension three

In this paper, we consider the short time strong solution to a simplified hydrodynamic flow modeling the compressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of velocity gradient and the square of maximum norm of gradient of liquid crystal director field.Comment: 22 page

Distributed optimal control of a nonstandard system of phase field equations

We investigate a distributed optimal control problem for a phase field model of Cahn-Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been recently introduced by the same authors in arXiv:1103.4585v1 [math.AP] and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.Comment: Key words: distributed optimal control, nonlinear phase field systems, first-order necessary optimality condition