相关论文: Existence for the $\al$-patch model and the QG sha…
In this work we investigate the statistical mechanics of a family of two dimensional (2D) fluid flows, described by the generalized Euler equations, or $\alpha$-models. These models describe both nonlocal and local dynamics, with one…
We study both divergence and non-divergence form parabolic and elliptic equations in the half space $\{x_d>0\}$ whose coefficients are the product of $x_d^\alpha$ and uniformly nondegenerate bounded measurable matrix-valued functions, where…
We reexamine from first principles the classical Goldberg-Sachs theorem from General Relativity. We cast it into the form valid for complex metrics, as well as real metrics of any signature. We obtain the sharpest conditions on the…
We show that there exists a family of analytic convex global rotating solutions for the vortex patch equations, bifurcating from ellipses. As a byproduct, the analyticity proof can also be adapted to the rotating patch solutions bifurcating…
In this paper, we prove the existence of self-similar algebraic spiral solutions for 2-D incompressible Euler equations for the initial vorticity of the form $|y|^{-\frac1\mu}\ \mathring{\omega}(\theta)$ with $\mu>\frac12$ and…
We prove linear in time filamentation for perturbations of the Lamb dipole, which is a traveling wave solution to the incompressible Euler equations in $\mathbb{R}^2$. The main ingredient is a recent nonlinear orbital stability result by…
We provide rigorous evidence of the fact that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics describes the geodesic flow on the subgroup of orientation-preserving diffeomorphisms fixing one point,…
In this paper, we study the convergence of solutions of the $\alpha$-Euler equations to solutions of the Euler equations on the $2$-dimensional torus. In particular, given an initial vorticity $\omega_0$ in $L^p_x$ for $p \in (1,\infty)$,…
This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface…
In this paper we study the Cauchy problem for one multidimensional compressible nonlocal model of the dissipative quasi-geostrophic equations. First, we obtain the local existence and uniqueness of the smooth non-negative solution or the…
In this paper, we give necessary and sufficient conditions for the existence of Ulrich bundles on cubic fourfold $X$ of given rank $r$. As consequences, we show that for every integer $r\ge 2$ there exists a family of indecomposable rank…
We rigorously establish the formal asymptotics of Neu for Gross-Pitaevskii vortex dynamics in the plane. Given any integer $n\geq2$, we construct a family of $n$-vortex solutions with vortices of degree $\pm1$, and describe precisely the…
Solutions of the Navier-Stokes and Euler equations with initial conditions for 2D and 3D cases were obtained in the form of converging series, by an analytical iterative method using Fourier and Laplace transforms \cite{TT10,TT11}. There…
We establish short-time existence of solutions to the surface quasi-geostrophic equation in both the H\"{o}lder spaces $C^r(\mathbb{R}^2)$ for $r>1$ and the uniformly local Sobolev spaces $H^s_{ul}(\mathbb{R}^2)$ for $s\geq 3$. Using…
In this paper, we show the existence of a family of analytic stationary patch solutions of the SQG and gSQG equations. This answers an open problem in [F. de la Hoz, Z. Hassainia, T. Hmidi. Arch. Ration. Mech. Anal., 220(3):1209-1281,…
We prove a characterization of the Sobolev spaces $H^\alpha$ on the unit sphere $\mathbb{S}^{d-1}$, where the smoothness index $\alpha$ is any positive real number and $d\geq 2$. This characterization does not use differentiation and it is…
In this paper, we prove the existence of locally non-radial solutions to the stationary 2D Euler equations with compact support but non-concentrated around one or several points. Our solutions are of patch type, have analytic boundary,…
In this paper, we show the existence of a family of compactly supported smooth vorticities, which are solutions of the 2D incompressible Euler equation and rotate uniformly in time and space.
This work is devoted to the study of the obstacle problem associated to the Kolmogorov-Fokker-Planck operator with rough coefficients through a variational approach. In particular, after the introduction of a proper anisotropic Sobolev…
In this paper for either the sharp front Surface Quasi-Geostrophic equation or the Muskat problem we rule out the "splash singularity" blow-up scenario; in other words we prove that the contours evolving from either of these systems can not…