We prove the existence of a compact global attractor for the dynamics of the forced critical surface quasi-geostrophic equation (SQG) and prove that it has finite fractal (box-counting) dimension. In order to do so we give a new proof of global regularity for critical SQG. The main ingredient is the nonlinear maximum principle in the form of a nonlinear lower bound on the fractional Laplacian, which is used to bootstrap the regularity directly from L ∞ to C α , without the use of De Giorgi techniques. We prove that for large time, the norm of the solution measured in a sufficiently strong topology becomes bounded with bounds that depend solely on norms of the force, which is assumed to belong merely to L ∞ ∩ H 1 . Using the fact that the solution is bounded independently of the initial data after a transient time, in spaces conferring enough regularity, we prove the existence of a compact absorbing set for the dynamics in H 1 , obtain the compactness of the linearization and the continuous differentiability of the solution map. We then prove exponential decay of high yet finite dimensional volume elements in H 1 along solution trajectories, and use this property to bound the dimension of the global attractor.

We consider the Cauchy problem for the spatially inhomogeneous Landau equation with soft potentials in the case of large (i.e. non-perturbative) initial data. We construct a solution for any bounded, measurable initial data with uniform polynomial decay in the velocity variable, and that satisfies a technical lower bound assumption (but can have vacuum regions). For uniqueness in this weak class, we have to make the additional assumption that the initial data is Hölder continuous. Our hypotheses are much weaker, in terms of regularity and decay, than previous large-data well-posedness results in the literature. We also derive a continuation criterion for our solutions that is, for the case of very soft potentials, an improvement over the previous state of the art. Résumé Nous considérons le problème de Cauchy pour l'équation de Landau non-homogène en espace avec potentiels mous dans le cas de grandes (i.e. non perturbatrices) données initiales. Nour construisons une solution pour toute donnée initiale bornée et mesurable à décroissance polynomiale uniforme dans la variable de vitesse, et qui satisfait une hypothèse technique de borne inférieure (il est toujours permis d'avoir des régions de vide). Pour être unique dans cette famille générale, nous devons supposer que les données initiales sont Hölder continues. Nos hypothèses sont beaucoup plus faibles, en termes de régularité et de décroissance, que les résultats de la littérature sur le caractère bien-posé dans le cas de grandes données. Nous dérivons également un critère de continuation pour nos solutions qui est, pour les potentiels très mous, une amélioration par rapport à l'état de l'art.

We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth order Sobolev space by working in a mixed L 2 and L ∞ space that allows to compensate for potential moment generation and obtaining new estimates on the collision operator that are well-adapted to this space. Our results improve the range of parameters for which the Boltzmann equation is well-posed in this decay regime, as well as relax the restrictions on the initial regularity. As an application, we can combine our existence result with the recent conditional regularity estimates of Imbert-Silvestre (arXiv:1909.12729 [math.AP]) to conclude solutions can be continued for as long as the mass, energy, and entropy densities remain under control. This continuation criterion was previously only available in the restricted range of parameters of previous well-posedness results for polynomially decaying initial data.

We consider the spatially inhomogeneous Landau equation with soft potentials, including the case of Coulomb interactions. First, we establish the existence of solutions for a short time, assuming the initial data is in a fourth-order Sobolev space and has Gaussian decay in the velocity variable (no decay assumptions are made in the spatial variable). Next, we show that the evolution instantaneously spreads mass to every point in its domain. The resulting pointwise lower bounds have a sub-Gaussian rate of decay, which we show is optimal. The proof of mass-spreading is based on a stochastic process associated to the equation, and makes essential use of nonlocality. By combining this theorem with prior regularity results, we derive two important applications: C ∞ smoothing in all three variables, even for initial data with vacuum regions, and a continuation criterion that states the solution can be extended for as long as the mass and energy densities stay bounded from above. This is the weakest condition known to prevent blow-up. In particular, it does not require a lower bound on the mass density or an upper bound on the entropy density.

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