In this talk we will first review various known regularity criteria and partial regularity theory for 3D incompressible Navier-Stokes equations. I will then present two generalizations of partial regularity theory of Caffarelli, Kohn and Nirenberg to the weak solutions of MHD equations. The first one is based on the framework of parabolic Morrey spaces. We will show parabolic Hölder regularity for the "suitable weak solutions" to the MHD system in small neighborhoods. This type of parabolic generalization using Morrey spaces appears to be crucial when studying the role of the pressure in the regularity theory and makes it possible to weaken the hypotheses on the pressure. The second one is a regularity result relying on the notion of "dissipative solutions". By making use of the first result, we will show the regularity of the dissipative solutions to the MHD system with a weaker hypothesis on the pressure.

This is a joint work with Diego Chamorro (Université Paris-Saclay, site Evry).

In this talk, we will consider the magneto-hydrodynamics (MHD) equations placed on the whole three-dimensional space. These equations write down as a nonlinear coupled system of two Navier-Stokes type equations, where the unknowns are the velocity field, the magnetic field and the pressure term. In the first part of this talk, within the framework of a kind of weighted L^2 spaces, we expose some new energy controls which allow us to prove the existence of global in time weak solutions. These solutions are also called the infinite energy weak solutions, in contrast with the classical theory of finite energy weak solutions in the L^2 space. The uniqueness of both finite energy and infinite energy weak solutions remains a very challenging open question. Thus, in the second part of this talk, we study a a priori condition, known as weak-strong uniqueness criterion, to ensure the uniqueness of the infinite energy weak solutions. This result is given in a fairly general multipliers type space, which contains some well-known functional spaces previously used to prove some weak-strong uniqueness criteria.

This is a joint work with Pedro Fernandez-Dalgo (Université Paris-Saclay).

This talk addresses the spectral stability of monotone traveling front solutions for reaction diffusion equations where the reaction function is of Nagumo (or bistable) type and with diffusivities which are density dependent and degenerate at zero (one of the equilibrium points of the reaction). Spectral stability is understood as the property that the spectrum of the linearized operator around the wave, acting on an exponentially weighted space, is contained in the complex half plane with non-positive real part. The degenerate fronts studied in this paper travel with positive speed above a threshold value and connect the(diffusion-degenerate) zero state with the unstable equilibrium point of the reaction function. In this case, the degeneracy of the diffusion coefficient is responsible of the loss of hyperbolicity of the asymptotic coefficient matrices of the spectral problem at one of the end points, precluding the application of standard techniques to locate the essential spectrum. This difficulty is overcome with a suitable partition of the spectrum, a generalized convergence of operators technique, the analysis of singular (or Weyl) sequences and the use of energy estimates. The monotonicity of the fronts, as well as detailed descriptions of the decay structure of eigenfunctions on a case by case basis, are key ingredients to show that all traveling fronts under consideration are spectrally stable in a suitably chosen exponentially weighted L2 energy space.

In this talk, we are going to discuss dispersive properties for the Fermi-Pasta-Ulam (FPU) system with infinitely many oscillators. Precisely, we see that FPU systems are reformulated and their solutions satisfies Strichartz, local smoothing, and maximal function estimates in comparison with linear Korteweg–de Vries (KdV) flows. With these properties, we finally show that the infinite FPU system can be approximated by counter-propagating waves governed by the KdV equation as the lattice spacing approaches zero.

Smooth periodic travelling waves in the Camassa–Holm (CH) equation are revisited in this talk. We show that these periodic waves can be characterized in two different ways by using two different Hamiltonian structures. The standard formulation, common to the Korteweg–de Vries (KdV) equation, has several disadvantages, e.g., the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We explore the nonstandard formulation common to evolution equations of CH type and prove that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a precise condition for the spectral and orbital stability of the smooth periodic travelling waves and show numerically that this condition is satisfied in the open region of three parameters where the smooth periodic waves exist.

This is a joint work with Dmitry E. Pelinovsky (McMaster University), AnnaGeyer (Delft University of Technology) and Renan H. Martins (State University of Maringa).

In this talk we will discuss the scattering of non-radial solutions in the energy space to coupled system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five without the mass-resonance condition. Our approach is based on the recent ideas introduced by Dodson and Murphy, which relies on an interaction Morawetz estimate.

We prove that a cone of nuclear operators, whose eigenfunctions belong to a p-Sobolev space and that have finite total energy, is compactly embedded in the trace norm. This result is analogous to the classical Sobolev embeddings but at operators level. In the path we prove regularity properties for the density function of any operator living in the cone. Also, departing from Lieb-Thirring type conditions, we obtain some Gagliardo-Nirenberg inequalities. By using the compactness property, several free-energy functionals for operators are shown to have a minimizer. The entropy term of these free-energy functionals is generated by a Casimir-class function related to the eigenvalue problem of the Schrödinger operator.

This talk concerns the nonlinear (massive) Dirac equation with a nonlinearity taking the form of a space-dependent mass, known as the (generalized) Soler model. The equation has standing wave solutions for frequencies w in (0,m), where m is the mass in the Dirac operator. These standing waves are generally expected to be stable (i.e., small perturbations in the initial conditions stay small) based on numerical simulations, but there are very few results in this direction.

The results that I will discuss concern the simpler question of spectral stability (and instability), i.e., the absence (or presence) of exponentially growing solutions to the linearized equation around a solitary wave. As in the case of the nonlinear Schrödinger equation, this is equivalent to the presence or absence of "unstable eigenvalues" of a non-self-adjoint operator with a particular block structure. I will present some partial results for the one-dimensional case, highlight the differences and similarities with the Schrödinger case, and discuss (a lot of) open problems.

We consider finite energy solutions for a damped Navier-Stokes-Bardina’s model and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension. Furthermore, we examine the long-term behavior of solutions of the damped Navier-Stokes-Bardina’s equation in the energy space.

In this talk I will discuss stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models \(\partial_t2\phi -\partial_x2\phi + W'(\phi) = 0, \quad (t,x)\in\mathbb{R}\times\mathbb{R}\). The orbital stability of kinks under general assumptions on the potential \(W\) is a consequence of energy arguments. The main result I will present is the derivation of a simple and explicit sufficient condition on the potential \(W\) for the asymptotic stability of a given kink. This condition applies to any static or moving kink, in particular no symmetry assumption is required. Applications of the criterion to the \(P(\phi)_2\) theories and the double sine-Gordon theory will be discussed.

This is a joint work with Y. Martel, C. Muñoz and H. Van Den Bosch.

We consider the inhomogeneous nonlinear Schrödinger (INLS) equation $$i u_t +\Delta u+|x|^{-b}|u|^{2\sigma} u = 0, \,\,\, x\in \mathbb{R}^N,$$ with \(N\geq 3\) and \(0 \lt b \lt \min\{\frac{N}{2},2\}\). We focus on the intercritical case, where the scaling invariant Sobolev index \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma}\) satisfies \(0 \lt s_c \lt 1\). In this talk, for initial data in \(\dot H^{s_c}\cap \dot H^1\), we discuss the existence of blow-up solutions and also a lower bound for the blow-up rate in the radial and non-radial settings.

This is a joint work with Mykael Cardoso (Universidade Federal do Piauí, Brasil).