In this paper, we obtain a result about the global existence of weak solutions to the d-dimensional Boussinesq-Navier-Stokes system, with viscosity dependent on temperature. The initial temperature is only supposed to be bounded, while the initial velocity belongs to some critical Besov Space, invariant to the scaling of this system. We suppose the viscosity close enough to a positive constant, and the $L^\infty$-norm of their difference plus the Besov norm of the horizontal component of the initial velocity is supposed to be exponentially small with respect to the vertical component of the initial velocity. In the preliminaries, and in the appendix, we consider some $L^p_t L^q_x$ regularity Theorems for the heat kernel, which play an important role in the main proof of this article.

We study uniqueness of sign-changing radial solutions for the following semi-linear elliptic equation \begin{align*} \Delta u-u+|u|^{p-1}u=0\quad{\rm{in}}\ \mathbb{R}^N,\quad u\in H^1(\mathbb{R}^N), \end{align*} where $1 < p < \frac{N+2}{N-2}$, $N\geq3$. It is well-known that this equation has a unique positive radial solution. The existence of sign-changing radial solutions with exactly $k$ nodes is also known. However, the uniqueness of such solutions is open. In this paper, we show that such sign-changing radial solution is unique when $p$ is close to $\frac{N+2}{N-2}$. Moreover, those solutions are non-degenerate, i.e., the kernel of the linearized operator is exactly $N$-dimensional.

We study the three-dimensional stochastic nonhomogeneous magnetohydrodynamics system with random external forces that involve feedback, i.e., multiplicative noise, and are non-Lipschitz. We prove the existence of a global martingale solution via a semi-Galerkin approximation scheme with stochastic calculus and applications of Prokhorov's and Skorokhod's theorems. Furthermore, using de Rham's theorem for processes, we prove the existence of the pressure term.

We introduce a new optimal transport distance between nonnegative finite Radon measures with possibly different masses. The construction is based on non-conservative continuity equations and a corresponding modified Benamou-Brenier formula. We establish various topological and geometrical properties of the resulting metric space, derive some formal Riemannian structure, and develop differential calculus following F. Otto's approach. Finally, we apply these ideas to identify a model of animal dispersal proposed by MacCall and Cosner as a gradient flow in our formalism and obtain new long-time convergence results.

We consider $p$-evolution equations, for $p\geq2$, with complex valued coefficients. We prove that a necessary condition for $H^\infty$ well-posedness of the associated Cauchy problem is that the imaginary part of the coefficient of the subprincipal part (in the sense of Petrowski) satisfies a decay estimate as $|x|\to+\infty$.