We study the Cauchy problem for the quasi-geostrophic equations in a unit ball of the two dimensional space with the homogeneous Dirichlet boundary condition. We show the existence, the uniqueness of the strong solution in the framework of Besov spaces. We establish a spectral localization technique and commutator estimates.
This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the advected scalar by up to one order of derivative. We also consider a non-degenerate modification of the endpoint case in which the velocity is less smooth than the advected scalar by slightly more than one order. The existence and uniqueness theory of these equations in the borderline Sobolev spaces is addressed, as well as the instantaneous smoothing effect of their corresponding solutions. In particular, it is shown that solutions emanating from initial data belonging to these Sobolev classes immediately enter a Gevrey class. Such results appear to be the first of its kind for a quasilinear parabolic equation whose coefficients are of higher order than its linear term; they rely on an approximation scheme which modifies the flux in such a way that preserves the underlying commutator structure lost by having to work in the critical space setting, as well as delicate adaptations of well-known commutator estimates to Gevrey classes.
In this paper, we present an extension of Uzawas algorithm and apply it to build approximating sequences of mean field games systems. We prove that Uzawas iterations can be used in a more general situation than the one in it is usually used. We then present some numerical results of those iterations on discrete mean field games systems of optimal stopping, impulse control and continuous control.
In this paper, we study the existence of global classical solutions to the generalized surface quasi-geostrophic equation. By using the variational method, we provide some new families of global classical solutions for to the generalized surface quasi-geostrophic equation. These solutions mainly consist of rotating solutions and travelling-wave solutions.
We construct examples of solutions to the conservative surface quasi-geostrophic (SQG) equation that must either exhibit infinite in time growth of derivatives or blow up in finite time.
We study the homogenization of elliptic systems of equations in divergence form where the coefficients are compositions of periodic functions with a random diffeomorphism with stationary gradient. This is done in the spirit of scalar stochastic homogenization by Blanc, Le Bris and P.-L. Lions. An application of the abstract result is given for Maxwells equations in random dissipative bianisotropic media.