Problem: Let $v(t,x)$ solve the following initial-value problem $$\bex \seddm{ \p_tv-\lap v=0,\\ v|_{t=0}=u. } \eex$$ Show that $v(t,x)$ has the representation $v(t,x)=\e^{t\lap}u(x)$. Here, $\e^{t\lap}u(x)$ is defined through the Fourier transform by $\e^{t\lap}u=\calF^{-1}(\e^{-t|\xi|^2} \calF u)$. 2017-2018-2偏微分方程复习题解析1
Problem: Show the Bony decomposition $$\bex uv=\dot T_uv+\dot T_vu+\dot R(u,v), \eex$$ where $$\bex \dot T_uv=\sum_j \dot S_{j-1} u\dot \lap_jv,\quad \dot R(u,v)=\sum_{|k-j|\leq 1} \dot \lap_k u\dot \lap_j v. \eex$$ 2017-2018-2偏微分方程复习题解析2
Problem: Suppose that the function $f:\bbR^d\to\bbR$ is radial, that is, for any $x,y\in\bbR^d$ 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 $$\bex \sup_{t>0}\sum_{j\in\bbZ} t^s2^{2js} \e^{-ct2^{2j}}<\infty. \eex$$ 2017-2018-2偏微分方程复习题解析4
Problem: Let $X,Y$ be Banach spaces, $T:X\to Y$ be a linear map. $T$ is said to be bounded, if $\exists\ M>0$, such that $\forall\ x\in X,\ \sen{Tx}\leq M\sen{x}$. Show that $T$ is bounded iff (if and only if) for any bounded subset $B\subset X$, $T(B)$ is a bounded subset of $Y$.
Problem: If $a$ is a smooth homogeneous function of degree $m$, show that $$\bex |\dot \lap_ju(x)|\leq C2^{jm}(Mu)(x), \eex$$ where $$\bex (Mf)(x)=\sup_{r>0}\f{1}{|B(x,r)|} \int_{B(x,r)}|u(y)|\rd y \eex$$ is the Hardy-Littlewood maximal function.
Problem: (1) Give the definition of the semi-norm $\sen{u}_{\dot H^s}$ and $\sen{u}_{\dot B^s_{p,q}}$, where $s\in\bbR$, $1\leq p,q\leq\infty$. (2) Show that $\sen{u}_{\dot H^s}$ and $\sen{u}_{\dot B^s_{2,2}}$ are equivalent.
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 $$\bex \ba{cccc} u:&[0,T]&\to& X\\ &t&\mapsto &u(t) \ea \eex$$ such that for any functional $\phi\in X'$, the function $[0,T]\ni t\mapsto \sef{\phi,u(t)}$ is continuous. Utilize (1) to show $C_w([0,T];X)\subset L^\infty([0,T];X)$.
Problem: Let $K,f,g$ be in $\calD(\bbR^d)$, and $K$ is radial (for definition, see Problem 2). Show that $$\bex \int (K*f)(x)g(x)\rd x=\int f(x)(K*g)(x)\rd x. \eex$$
Problem: Consider the three-dimensional Navier-Stokes equations $$\bee\tag{*} \seddm{ \p_tu+(u\cdot\n)u-\lap u+\n P=0,\\ \n\cdot u=0,\\ u|_{t=0}=u_0. } \eee$$ Let $u_0\in L^2(\bbR^3)$. Then by the Leray's famous work, there exists at least one weak solution of (*). Suppose that $u_0\in H^1(\bbR^3)$ and $$\bee\tag{**} u\in L^p(0,T;L^q(\bbR^3)),\quad \f{2}{p}+\f{3}{q}=1,\quad 3<q\leq\infty. \eee$$ Show that $u$ is a strong solution, i.e., $u\in L^\infty(0,T;L^2(\bbR^3))\cap L^2(0,T;H^1(\bbR^3))$. This is the classical Ladyzhenskaya-Prodi-Serrin condition.
Problem: Let $v=(v_1,v_2,v_3)$ be smooth vector field. Show that $-\lap v=\curl\curl v-\n \Div v$.