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Abstract: In this article we study a transport-diffusion equation in the framework of the stratified Lie groups. For this equation we will study the existence of the solutions, a maximum principle, a positivity principle and Hölder regularity. [Copyright &y& Elsevier]

Abstract: In this article we give a straightforward proof of refined inequalities between Lorentz spaces and Besov spaces and we generalize previous results of H. Bahouri and A. Cohen [2]. Our approach is based in the characterization of Lorentz spaces as real interpolation spaces. We will also study the sharpness and optimality of these inequalities. [Copyright &y& Elsevier]

Abstract: We study in this article the improved Sobolev inequalities with Muckenhoupt weights within the framework of stratified Lie groups. This family of inequalities estimate the norm of a function by the geometric mean of two norms corresponding to Sobolev spaces and Besov spaces . When the value p which characterizes Sobolev space is strictly larger than 1, the required result is well known in and is classically obtained by a Littlewood–Paley dyadic blocks manipulation. For these inequalities we will develop here another totally different technique. When , these two techniques are not available anymore and following M. Ledoux (2003) , in , we will treat here the critical case for general stratified Lie groups in a weighted functional space setting. Finally, we will go a step further with a new generalization of improved Sobolev inequalities using weak-type Sobolev spaces. [Copyright &y& Elsevier]

We generalize here the celebrated Partial Regularity Theory of Caffarelli, Kohn and Nirenberg to the MHD equations in the framework of parabolic Morrey spaces. This type of parabolic generalization using Morrey spaces appears to be crucial when studying the role of the pressure in the regularity theory for the classical Navier-Stokes equations as well as for the MHD equations. [ABSTRACT FROM AUTHOR]