We prove that the initial value problem for the semi-linear Schrödinger and wave equations is well-posed in the Besov space ${\dot{B}}_2^{\frac{n}{2}-\frac{2}{p},\infty }\left({𝐑}^n\right)$, when the nonlinearity is of type $u^p$, for $p\in 𝐍$. This allows us to obtain self-similar solutions, as well as to recover previously known results for the solutions under weaker smallness assumptions on the data.

The most important result stated in this paper is a theorem on the existence of global solutions for the Navier-Stokes equations in Rn when the initial velocity belongs to the space weak Ln(Rn) with a sufficiently small norm. Furthermore, this fact leads us to obtain self-similar solutions if the initial velocity is, besides, an homogeneous function of degree -1. Partial uniqueness is also discussed.

We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form $120$ degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.

We continue the study started by the first author of the semiclassical Kac Operator. This kind of operator has been obtained for example by M. Kac as he was studying a 2D spin lattice by the so-called “transfer operator method”. We are interested here in the thermodynamical limit $\Lambda \left(h\right)$ of the ground state energy of this operator. For Kac’s spin model, $\Lambda \left(h\right)$ is the free energy per spin, and the semiclassical regime corresponds to the mean-field approximation. Under suitable assumptions, which are satisfied...

In this paper, we study the semiclassical limit of the cubic nonlinear Schrödinger equation with the Neumann boundary condition in an exterior domain. We prove that before the formation of singularities in the limit system, the quantum density and the quantum momentum converge to the unique solution of the compressible Euler equation with the slip boundary condition as the scaling parameter approaches $0.$

In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying...

We consider a singularly perturbed elliptic equation ${ϵ}^2\Delta u-V\left(x\right)u+f\left(u\right)=0,u\left(x\right)>0$ on ${ℝ}^N$, ${\mathrm{𝚕𝚒𝚖}}_{\left|x\right|\to \infty }u\left(x\right)=0$, where $V\left(x\right)>0$ for any $x\in {ℝ}^N$. The singularly perturbed problem has corresponding limiting problems $\Delta U-cU+f\left(U\right)=0,U\left(x\right)>0$ on ${ℝ}^N$, ${\mathrm{𝚕𝚒𝚖}}_{\left|x\right|\to \infty }U\left(x\right)=0,c>0$. Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity $f$ for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential $V$ under possibly general conditions on $f$. In...

We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.

Using some perturbation results in critical point theory, we prove that a class of nonlinear Schrödinger equations possesses semiclassical states that concentrate near the critical points of the potential $V$.

We consider the Cauchy problem for the three-dimensional Navier-Stokes equations, and provide an optimal regularity criterion in terms of $u_3$ and ${\omega }_3$, which are the third components of the velocity and vorticity, respectively. This gives an affirmative answer to an open problem in the paper by P. Penel, M. Pokorný (2004).