2017-2018-2偏微分方程复习题

这些题目都是在上课过程中不断增加的. 希望每次上课都能增加一些题目. 不断地积累出许多有意思的问题及其解答. 更多的请见 (我的教学): http://www.cnblogs.com/zhangzujin/p/3527416.html

Problem: Let $v(t,x)$ solve the following initial-value problem $$ex seddm{ p_tv-lap v=0,\ v|_{t=0}=u. } eex$$ Show that $v(t,x)$ has the representation $v(t,x)=e^{tlap}u(x)$. Here, $e^{tlap}u(x)$ is defined through the Fourier transform by $e^{tlap}u=calF^{-1}(e^{-t|xi|^2} calF u)$. 2017-2018-2偏微分方程复习题解析1

Problem: Show the Bony decomposition $$ex uv=dot T_uv+dot T_vu+dot R(u,v), eex$$ where $$ex dot T_uv=sum_j dot S_{j-1} udot lap_jv,quad dot R(u,v)=sum_{|k-j|leq 1} dot lap_k udot lap_j v. eex$$ 2017-2018-2偏微分方程复习题解析2

Problem: Suppose that the function $f:bR^d obR$ is radial, that is, for any $x,yinbR^d$ with $|x|=|y|$, we have $f(x)=f(y)$. Show that the Fourier transform $calF(xi)$ is also radial. 2017-2018-2偏微分方程复习题解析3

Problem: For any positive $s$, we have $$ex sup_{t>0}sum_{jinbZ} t^s2^{2js} e^{-ct2^{2j}}<infty. eex$$ 2017-2018-2偏微分方程复习题解析4

Problem: Let $X,Y$ be Banach spaces, $T:X o Y$ be a linear map. $T$ is said to be bounded, if $exists M>0$, such that $forall xin X, sen{Tx}leq Msen{x}$. Show that $T$ is bounded iff (if and only if) for any bounded subset $Bsubset X$, $T(B)$ is a bounded subset of $Y$. 2017-2018-2偏微分方程复习题解析5

Problem: If $a$ is a smooth homogeneous function of degree $m$, show that $$ex |dot lap_ju(x)|leq C2^{jm}(Mu)(x), eex$$ where $$ex (Mf)(x)=sup_{r>0}f{1}{|B(x,r)|} int_{B(x,r)}|u(y)| d y eex$$ is the Hardy-Littlewood maximal function. 2017-2018-2偏微分方程复习题解析6

Problem: (1) Give the definition of the semi-norm $sen{u}_{dot H^s}$ and $sen{u}_{dot B^s_{p,q}}$, where $sinbR$, $1leq p,qleqinfty$. (2) Show that $sen{u}_{dot H^s}$ and $sen{u}_{dot B^s_{2,2}}$ are equivalent. 2017-2018-2偏微分方程复习题解析7

Problem: (1) Narrate the resonance theorem. (2) Let $X$ be a Banach space, and denote by $C_w([0,T];X)$ be all the maps $$ex a{cccc} u:&[0,T]& o& X\ &t&mapsto &u(t) ea eex$$ such that for any functional $phiin X'$, the function $[0,T] i tmapsto sef{phi,u(t)}$ is continuous. Utilize (1) to show $C_w([0,T];X)subset L^infty([0,T];X)$. 2017-2018-2偏微分方程复习题解析8

Problem: Let $K,f,g$ be in $calD(bR^d)$, and $K$ is radial (for definition, see Problem 2). Show that $$ex int (K*f)(x)g(x) d x=int f(x)(K*g)(x) d x. eex$$ 2017-2018-2偏微分方程复习题解析9

Problem: Consider the three-dimensional Navier-Stokes equations $$ee ag{*} seddm{ p_tu+(ucdot )u-lap u+ P=0,\ cdot u=0,\ u|_{t=0}=u_0. } eee$$ Let $u_0in L^2(bR^3)$. Then by the Leray's famous work, there exists at least one weak solution of (*). Suppose that $u_0in H^1(bR^3)$ and $$ee ag{**} uin L^p(0,T;L^q(bR^3)),quad f{2}{p}+f{3}{q}=1,quad 3<qleqinfty. eee$$ Show that $u$ is a strong solution, i.e., $uin L^infty(0,T;L^2(bR^3))cap L^2(0,T;H^1(bR^3))$. This is the classical Ladyzhenskaya-Prodi-Serrin condition. 2017-2018-2偏微分方程复习题解析10

Problem: Let $v=(v_1,v_2,v_3)$ be smooth vector field. Show that $-lap v=curlcurl v- Div v$. 2017-2018-2偏微分方程复习题解析11