The semi-geostrophic system is widely used in the modelling of large-scale atmospheric flows. In this paper, we prove existence of solutions of the incompressible semi-geostrophic equations in a fully three-dimensional domain with a free upper boundary condition. We show that, using methods similar to those introduced in the pioneering work of Benamou and Brenier, who analysed the same system but with a rigid boundary condition, we can prove the existence of solutions for the incompressible free boundary problem. The proof is based on optimal transport results as well as the analysis of Hamiltonian ODEs in spaces of probability measures given by Ambrosio and Gangbo. We also show how these techniques can be modified to yield the same result also for the compressible version of the system.
In this paper, we investigate pointwise time analyticity of solutions to fractional heat equations in the settings of $mathbb{R}^d$ and a complete Riemannian manifold $mathrm{M}$. On one hand, in $mathbb{R}^d$, we prove that any solution $u=u(t,x)$ to $u_t(t,x)-mathrm{L}_alpha^{kappa} u(t,x)=0$, where $mathrm{L}_alpha^{kappa}$ is a nonlocal operator of order $alpha$, is time analytic in $(0,1]$ if $u$ satisfies the growth condition $|u(t,x)|leq C(1+|x|)^{alpha-epsilon}$ for any $(t,x)in (0,1]times mathbb{R}^d$ and $epsilonin(0,alpha)$. We also obtain pointwise estimates for $partial_t^kp_alpha(t,x;y)$, where $p_alpha(t,x;y)$ is the fractional heat kernel. Furthermore, under the same growth condition, we show that the mild solution is the unique solution. On the other hand, in a manifold $mathrm{M}$, we also prove the time analyticity of the mild solution under the same growth condition and the time analyticity of the fractional heat kernel, when $mathrm{M}$ satisfies the Poincare inequality and the volume doubling condition. Moreover, we also study the time and space derivatives of the fractional heat kernel in $mathbb{R}^d$ using the method of Fourier transform and contour integrals. We find that when $alphain (0,1]$, the fractional heat kernel is time analytic at $t=0$ when $x eq 0$, which differs from the standard heat kernel. As corollaries, we obtain sharp solvability condition for the backward fractional heat equation and time analyticity of some nonlinear fractional heat equations with power nonlinearity of order $p$. These results are related to those in [8] and [11] which deal with local equations.
We are concerned with the existence of periodic travelling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation(including incompressible Euler equation), known as von Karman vortex street. These solutions are of $C^1$ type, and are obtained by studying a semilinear problem on an infinite strip whose width equals to the period. By a variational characterization of solutions, we also show the relationship between vortex size, travelling speed and street structure. In particular, the vortices with positive and negative intensity have equal or unequal scaling size in our construction, which constitutes the regularization for Karman point vortex street.