算子的时-频域表示与广义Gabor乘子

广义Leray-alpha方程的局部和整体低正则解

Local and Global low-regularity solutions to the Generalized Leray-alpha equations

近年来，研究Navier-斯托克斯方程的各种近似方程已成为一种普遍现象.其中之一是勒雷- $α$方程，它通过用$(1-α^2\triangle)$算子$(1-α^2\triangle)u$替换（在大多数位置）方程中的解$u$来正则化斯托克斯方程.另一个是广义斯托克斯方程，它用符号形式为$|ξ|^γ$的傅立叶乘数代替拉普拉斯算子（$γ=2$是标准的斯托克斯方程），最近在[14]中，Tao还考虑了$|ξ|^γ/g(|ξ|)$形式的乘数，其中$g$（本质上）是对数。广义Leray- $α$方程通过引入正则化项并用更一般的傅立叶乘数替换拉普拉斯算子来组合这两种修改，包括允许与[14]中使用的那些类似的$g$项。本文的目标是在低正则性和/或非$L^2$初始数据下得到解的存在唯一性。我们还将使用能量估计来将这些局部存在性结果扩展到全局存在性结果。

On multilinear distorted multiplier estimate and its applications

In this article, we investigate the multilinear distorted multiplier estimate (Coifman-Meyer type theorem) associated with the Schrödinger operator $H=-Δ+ V$ in the framework of the corresponding distorted Fourier transform. Our result is the "distorted" analog of the multilinear Coifman-Meyer multiplier operator theorem in \cite{CM1}, which extends the bilinear estimates of Germain, Hani and Walsh's in \cite{PZS} to the multilinear case for all dimensions. As applications, we give the estimate of Leibniz's law of integer order derivations for the multilinear distorted multiplier for the first…

Boundedness of spectral multipliers of generalized Laplacians on compact manifolds with boundary

Consider a second order, strongly elliptic negative semidefinite differential operator $L$ (maybe a system) on a compact Riemannian manifold $\overline{M}$ with smooth boundary, where the domain of $L$ is defined by a coercive boundary condition. Classically known results, and also recent work in \cite{DOS} and \cite{DM} establish sufficient conditions for $L^\infty-\text{BMO}_L$ continuity of $φ(\sqrt{A})$, where $φ\in S^0_1(\mathbb{R})$, and $A$ is a suitable elliptic operator. Using a variant of the Cheeger-Gromov-Taylor functional calculus due to \cite{MMV}, and short time bounds on the in…

Generalized Mehler formula for time-dependent non-selfadjoint quadratic operators and propagation of singularities

We study evolution equations associated to time-dependent dissipative non-selfadjoint quadratic operators. We prove that the solution operators to these non-autonomous evolution equations are given by Fourier integral operators whose kernels are Gaussian tempered distributions associated to non-negative complex symplectic linear transformations, and we derive a generalized Mehler formula for their Weyl symbols. Some applications to the study of the propagation of Gabor singularities (characterizing the lack of Schwartz regularity) for the solutions to non-autonomous quadratic evolution equatio…

Optimal System and Conservation Laws for the Generalized Fisher Equation in Cylindrical Coordinates

The reaction diffusion equation arises in physical situations in problems from population growth, genetics and physical sciences. We consider the generalised Fisher equation in cylindrical coordinates from Lie theory stand point. An invariance method is performed and the optimal set of nonequivalent symmetries is obtained. Finally, the conservation laws are constructed using 'multiplier method'. We determine multipliers as functions of the dependent and independent variables only. The conservation laws are computed and presented in terms of conserved vector corresponding to each multiplier.

m-耗散广义阻抗边界条件、离散谱与分数阶Sobolev空间之间的逐点乘子

M-dissipative generalized impedance boundary conditions, discrete spectra, and pointwise multipliers between fractional Sobolev spaces

本文研究了有界Lipschitz域中m-耗散广义阻抗边界条件下声学算子的性质，证明了其紧预解算子与阻抗算子紧性等价，并应用于谱离散性及波动方程。

Extension of Mikhlin Multiplier Theorem to Fractional Derivatives and Stable Processes

In this paper, we prove a new generalized Mikhlin multiplier theorem whose conditions are given with respect to fractional derivatives in integral forms with two different integration intervals. We also discuss the connection between fractional derivatives and stable processes and prove a version of Mikhlin theorem under a condition given in terms of the infinitesimal generator of symmetric stable process. The classical Mikhlin theorem is shown to be a corollary of this new generalized version in this paper.

Boussinesq和其他广义波动方程的色散-耗散估计

Dispersive-Dissipative estimates for Boussinesq and other generalized wave equations

针对一类广义波动方程（特别是粘性Boussinesq方程），推导了与相位函数几何假设及低高频退化相关的色散-耗散估计，并分析了耗散对解衰减率的影响。

广义七阶KdV方程的守恒律

Conservation laws for a generalized seventh order KdV equation

本文用乘子法完整分类了广义七阶KdV方程的低阶局部守恒律，用李方法分类了点对称性，发现无额外对称性，并确定了约化方程的积分因子。