Related papers: Regularized and Approximate Equations for Sharp Fr…
We prove uniform estimates for the decay rate of the Fourier transform of measures supported on real-analytic hypersurfaces in R^3. If the surface contains the origin and is oriented such that its normal at the origin is in the direction of…
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
The $c$-curvature of a complete surface with Gauss curvature close to 1 in $C^2$ norm is almost-positive (in the sense of Kim--McCann). Our proof goes by a careful case by case analysis combined with perturbation arguments from the constant…
In this paper, we consider the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. Global existence of classical solutions is established when the dissipation…
We generalize the classical K\"onig's and B\"ottcher's Theorems in complex dynamics to certain quasiregular mappings in the plane. Our approach to these results is unified in the sense that it does not depend on the local injectivity, or…
Generalizing the well-known relations on characteristic functions on a plane to the case of a one-dimensional regular surface (curve) with compact support, we establish implicit equations for these functions. Introducing an approximation,…
This article is concerned with the analysis of the one-dimensional compressible Euler equations with a singular pressure law, the so-called hard sphere equation of state. The result is twofold. First, we establish the existence of bounded…
By applying implicit function theorem on contour dynamics, we prove the existence of co-rotating and travelling patch solutions for both Euler and the generalized surface quasi-geostrophic equation. The solutions obtained constitute a…
It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential…
In this paper, we study generalized constant ratio (GCR) hypersurfaces in Euclidean spaces. We mainly focus on the hypersurfaces in $\mathbb E^4$. First, we deal with $\delta(2)$-ideal GCR hypersurfaces. Then, we study on hypersurfaces with…
We study the critical and super-critical dissipative quasi-geostrophic equations in $\bR^2$ or $\bT^2$. Higher regularity of mild solutions with arbitrary initial data in $H^{2-\gamma}$ is proved. As a corollary, we obtain a global…
Quasiclassical generalized Weierstrass representation (GWR) for highly corrugated surfaces with slow modulation in the four-dimensional Euclidean space is proposed. Integrable deformations of such surfaces are described by the…
The momentum formulation of the surface quasi-geostrophic equations consists of two nonlinear terms, besides the pressure term, one of which cannot be written in a divergence form. When the anti-divergence operator is applied to such…
We show that for any uniformly elliptic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term one can find an approximating equation which has a unique continuous and having the second…
We show the existence of nontrivial stationary weak solutions to the surface quasi-geostrophic equations on the two dimensional periodic torus.
We obtain $C^2$ a priori estimates for solutions of the nonlinear second-order elliptic equation related to the geometric problem of finding a strictly locally convex hypersurface with prescribed curvature and boundary in a space form.…
A quasi-Newton method with cubic regularization is designed for solving Riemannian unconstrained nonconvex optimization problems. The proposed algorithm is fully adaptive with at most ${\cal O} (\epsilon_g^{-3/2})$ iterations to achieve a…
Quasi-poloidal (QP) magnetic fields have desirable properties for confining plasma: no radial drift of guiding centres (with positive implications for neoclassical transport), zero Pfirsch-Schl\"uter current, a lower level of damping for…
In this work we study the asymptotic behavior of solutions of the incompressible two-dimensional Euler equations in the exterior of a single smooth obstacle when the obstacle becomes very thin tending to a curve. We extend results by…
We consider solutions to the generalized Surface Quasi-geostrophic equation ($\gamma$-SQG) when the velocity is more singular than the active scalar function (i.e. $\gamma\in(0,1)$). In this paper we establish strong ill-posedness in…