Fractional Laplacians and Nilpotent Lie Groups

The aim of this short article is to generalize, with a slighthly different point of view, some new results concerning the fractional powers of the Laplace operator to the setting of Nilpotent Lie Groups and to study its relationship with the solutions of a partial differential equation in the spirit of the articles of Caffarelli & Silvestre and Stinga & Torrea.Comment: 10

On the existence, regularity and uniqueness of LpL^p-solutions to the steady-state 3D Boussinesq system in the whole space

We consider the steady-state Boussinesq system in the whole three-dimensional space, with the action of external forces and the gravitational acceleration. First, for $3<p\leq +\infty$ we prove the existence of weak $L^p$-solutions. Moreover, within the framework of a slightly modified system, we discuss the possibly non-existence of $L^p-$solutions for $1\leq p \leq 3$. Then, we use the more general setting of the $L^{p,\infty}-$spaces to show that weak solutions and their derivatives are H\"older continuous functions, where the maximum gain of regularity is determined by the initial regularity of the external forces and the gravitational acceleration. As a bi-product, we get a new regularity criterion for the steady-state Navier-Stokes equations. Furthermore, in the particular homogeneous case when the external forces are equal to zero; and for a range of values of the parameter $p$, we show that weak solutions are not only smooth enough, but also they are identical to the trivial (zero) solution. This result is of independent interest, and it is also known as the Liouville-type problem for the steady-state Boussinesq system.Comment: 28 page

On the regularity of very weak solutions for an elliptic coupled system of liquid crystal flows

We consider here an elliptic coupled system describing the dynamics of liquid crystals flows. This system is posed on the whole n-dimensional space. We introduce first the notion of very weak solutions for this system. Then, within the fairly general framework of the Morrey spaces, we derive some sufficient conditions on the very weak solutions which improve their regularity. As a bi-product, we also prove a new regularity criterium for the time-independing Navier-Stokes equations.Comment: 14 page

Asymptotic behavior of a generalized Navier-Stokes-Bardina's model and applications to related models

We consider here a general theoretical equation on the whole three-dimensional space, which contains as a particular case some relevant equations of the fluid dynamics as the Navier-Stokes-Bardina's model, the fractional and the classical Navier-Stokes equations with an additional drag/friction term. These equations arise from ocean and atmospheric models. For the general equation, we study first the existence and in some cases the uniqueness of finite energy solutions. Then, we use a general framework to study their long behavior with respect to the weak and the strong topology of the phase space. We thus prove the existence of a weak global attractor and in some cases the existence of a strong global attractor. Moreover, we study some sufficient conditions to insure the weak global attractor becomes a strong global attractor. As a bi-product, we obtain some new results on the long time description of the fractional and classical Navier-Stokes models with a damping term.Comment: 29 page

Frequency decay for Navier-Stokes stationary solutions

We consider stationary Navier-Stokes equations in R 3 with a regular external force and we prove exponential frequency decay of the solutions. Moreover, if the external force is small enough, we give a pointwise exponential frequency decay for such solutions according to the K41 theory. If a damping term is added to the equation, a pointwise decay is obtained without the smallness condition over the force

From non-local to local Navier-Stokes equations

Inspired by some experimental (numerical) works on fractional diffusion PDEs, we develop a rigorous framework to prove that solutions to the fractional Navier-Stokes equations, which involve the fractional Laplacian operator $(-\Delta)^{\frac{\alpha}{2}}$, converge to a solution of the classical case, with the classical Laplacian operator, when $\alpha$ goes to $2$. Precisely, in the setting of mild solutions, we prove a uniform convergence in both the time and spatial variables and derive a convergence rate.Comment: 12 page

Existence of infinite-energy and discretely self-similar global weak solutions for 3D MHD equations

This paper deals with the existence of global weak solutions for 3D MHD equations when the initial data belong to the weighted spaces $L^2_{w_\gamma}$, with $w_\gamma(x)=(1+\vert x\vert)^{-\gamma}$ and $0 \leq \gamma \leq 2$. Moreover, we prove the existence of discretely self-similar solutions for 3D MHD equations for discretely self-similar initial data which are locally square integrable. Our methods are inspired of a recent work of P. Fern\'aidez-Dalgo and P.G. Lemarié-Rieusset for the 3D Navier-Stokes equations

Some Liouville theorems for stationary Navier-Stokes equations in Lebesgue and Morrey spaces

Uniqueness of Leray solutions of the 3D Navier-Stokes equations is a challenging open problem. In this article we will study this problem for the 3D stationary Navier-Stokes equations and under some additional hypotheses, stated in terms of Lebesgue and Morrey spaces, we will show that the trivial solution U = 0 is the unique solution. This type of results are known as Liouville theorems