Navier-Stokes equations 1 Let $\omega$ be a domain in $\bbR^3$, complement of a compact set $\mathcal{B}$. Consider the following boundary value problem in $\omega$: $$\bee\label{NS:1} \left. \ba{cc} \left.\ba{ll} \nu \lap v=(v-\xi-\omega\times x) \c…
随机偏微分方程 Throughout this section, let $(\Omega, \calF, \calF_t,\ P)$ be a complete filtered probability space satisfying the usual conditions. 1. Recall the following results: a) The Doob maximal inequality: if $(N_t)$ is a non-negative $\calF…
1.Introduction 2.First-order Differential Equations Exercise2.1. Find solutons of the following intial-value problems in $\bbR^2$: (1)$2u_y-u_x+xu=0$ with $u(x,0)=2xe^{x^2/2}$; (2)$u_y+(1+x^2)u_x-u=0$ with $u(x,0)=\arctan x$. Solution: (1)Since $(-1,…
Copied from: http://www.elsevier.com/journals/applied-mathematics-letters/0893-9659/guide-for-authors The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief…
Problem. Suppose $x(t)\in C[0,T]$, and satisfies $$\bex t\in [0,T]\ra 1\leq x(t)\leq C_1+C_2\int_0^t x(\tau)[1+\log x(\tau)]\rd \tau. \eex$$ Prove: (1) $x(t)$ is bounded on $[0,T].$ (2) This is in stark contrast to the estimates like: $$\bex x(…
坚持每天刷一道题的小可爱还没有疯,依旧很可爱! 题目:There are two sorted arrays nums1 and nums2 of size m and n respectively.Find the median of the two sorted arrays. The overall run time complexity should be O(log (m+n)). Example 1: nums1 = [1, 3],nums2 = [2], The median is…
