相关论文: On the parabolic Donaldson's equation over a compa…

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.

Let $X$ be a complex four-dimensional compact Calabi-Yau manifold equipped with a K\"ahler form $\omega$ and a holomorphic four-form $\Omega$. Under certain assumptions, we define Donaldson-Thomas type deformation invariants by studying the…

In this article, we establish some uniqueness and symmetry results of self-similar solutions to curvature flows by some homogeneous speed functions of principal curvatures in some warped product spaces. In particular, we proved that any…

We study a semilinear parabolic equation that possesses global bounded weak solutions whose gradient has a singularity in the interior of the domain for all $t>0$. The singularity of these solutions is of the same type as the singularity of…

For the damped wave equation on a compact manifold with {\em continuous} dampings, the geometric control condition is necessary and sufficient for {uniform} stabilisation. In this article, on the two dimensional torus, in the special case…

We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish…

Let $(M, g)$ be a closed Riemannian manifold of dimension $5$. Assume that $(M, g)$ is not conformally equivalent to the round sphere. If the scalar curvature $R_g\geq 0$ and the $Q$-curvature $Q_g\geq 0$ on $M$ with $Q_g(p)>0$ for some…

We show that any viscosity solution to a general fully nonlinear nonlocal elliptic equation can be approximated by smooth ($C^\infty$) solutions.

We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian…

We consider the homogeneous Landau equation in $\mathbb{R}^3$ with Coulomb potential and initial data in polynomially weighted $L^{3/2}$. We show that there exists a smooth solution that is bounded for all positive times. The proof is based…

We give a proof of existence and uniqueness of viscosity solutions to parabolic quasilinear equations for a fairly general class of nonconvex Hamiltonians with superlinear growth in the gradient variable. The approach is mainly based on…

We prove the $C^{\alpha}$ regularity for weak solutions to a class of ultraparabolic equation, with measurable coefficients. The results generalized our recent $C^{\alpha}$ regularity results of Prandtl's system to high dimensional cases.

We apply the idea of using a combination of virial inequalities and Kato smoothing, previously applied to NLS and generalized KdV pure power equations to Euler-Poisson: we assume that a solution remains very close for all times to a soliton…

Let $\overline{M}$ be a compact complex manifold with smooth K\"ahler metric $\eta$, and let $D$ be a smooth divisor on $\overline{M}$. Let $M=\overline{M}\setminus D$ and let $\hat{\omega}$ be a Carlson-Griffiths type metric on $M$. We…

We study the initial value problem of fully nonlinear third-order equations on the torus. Under some conditions on the nonlinearity and the data, we prove that the equation behaves like a parabolic one: there exists a unique local solution…

It was recently proven by De Lellis, Kappeler, and Topalov that the periodic Cauchy problem for the Camassa-Holm equations is locally well-posed in the space Lip (T) endowed with the topology of H^1 (T). We prove here that the Lagrangian…

We show that the Sobolev embedding is compact on punctured manifolds with conical singularities. On the other hand, we find the Sobolev inequality does not hold on punctured manifolds with Poincar\'{e} like metric, on which one has…

We prove some basic properties of Donaldson's flow of surfaces in a hyperkahler 4-manifold. When the initial submanifold is symplectic with respect to one K\"ahler form and Lagrangian with respect to another, we show that certain kinds of…

We introduce the gradient flow of the Seiberg-Witten functional on a compact, orientable Riemannian 4-manifold and show the global existence of a unique smooth solution to the flow. The flow converges uniquely in $C^\infty$ up to gauge to a…

In this paper, we study the existence and smoothness of a density function to the solution of a Mckean-Vlasov equation with the aid of Malliavin calculus. We first show the existence of the density function under assumptions that the…