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[偏微分方程] (190703) By taking the inner product of the Navier-Stokes equations with $-t\lap \bbu$ show that weak solutions of the 2D Navier-Stokes equations on $\mathbb{T}^2$ satisfy $$\bex \sqrt{t} \bbu\in L^2(0,T;H^2). \eex$$ zhangzujin 2019-4-17 039 zhangzujin 2019-4-17 10:17
[偏微分方程] (190702) Show that if $M$ and $A$ are $3\times 3$ matrices and $\f{\rd M}{\rd t}=A(t)M$ then $$\bex \f{\rd}{\rd t} \det M=\tr A\cdot \det M. \eex$$ zhangzujin 2019-4-17 036 zhangzujin 2019-4-17 10:17
[偏微分方程] (190512) Show that $\calP^s(X)=0$ if and only if for every $\ve>0$, the set $X$ can be covered by a collection $\sed{Q_{r_i}^*}$ such that $\sum_i r_i^s<\ve$. zhangzujin 2019-4-9 055 zhangzujin 2019-4-9 21:21
[偏微分方程] (190511) Suppose that there exist $r_0,\al>0$ and $0<\tt,\be<1$ such that $$\hj{ E(\tt r)\leq \al+\be E(r),\ \forall\ 0<r<r_0. }$$ Show that $E(\tt^k r)\leq\f{2\al}{1-\be}$ for all $k$ sufficiently... zhangzujin 2019-4-9 039 zhangzujin 2019-4-9 21:20
[偏微分方程] (190510) Show that for any $1\leq q<\infty$, $$\hj{ \int_U |\bar p|^q\rd x \leq \int_U |p|^q\rd x, }$$ where $\bar p=\f{1}{|U|}\int_U p\rd x$. zhangzujin 2019-4-9 037 zhangzujin 2019-4-9 19:04
[偏微分方程] (190509) Suppose that $q\geq 1$ and $a,b\geq 0$. Show that $$\hj{ (a+b)^q\leq 2^{q-1}(a^q+b^q). }$$ zhangzujin 2019-4-9 032 zhangzujin 2019-4-9 19:04
[偏微分方程] (190508) Let $\psi\in C_c^\infty(\bbR^3)$ is standard ... Show that for $s\in\bbN$, $$\hj{ \sen{\psi_\ve*\bbu}_{\dot H^s}\leq \ve^{-s}\sen{\bbu}_{L^2}. }$$ zhangzujin 2019-4-9 036 zhangzujin 2019-4-9 19:03
[偏微分方程] (190507) Let $\psi\in C_c^\infty(\bbR^3)$ is standard mollifier, i.e., a function with $0\leq \psi\leq 1$ such that ... Show that for $p\leq q\leq\infty$, $$\hj{ \sen{\psi_\ve*\bbu}_{L^q}\leq \ve^{-\sex{\f{3}{p}-\f{3}{q}}} \sen{\bbu}_{L^p}. }$$ zhangzujin 2019-4-9 040 zhangzujin 2019-4-9 19:02
[偏微分方程] (190506) Let $U\subset \bbR^3$ be ... Using the Agmon inequality and the interpolation inequality to show $$\hj{ \sen{f}_{L^\infty(U)} \leq C_{k,r}\sen{f}_{L^r(U)}^\tt\sen{f}_{H^k(U)}^\tt, }$$ zhangzujin 2019-4-9 043 zhangzujin 2019-4-9 19:01
[偏微分方程] (190505) Suppose that $U\subset \bbR^3$ is a smooth bounded domain and that $\p_tg\in L^q(0,T;L^r(U))$. Show that $$\hj{ \sen{g(t_2)-g(t_1)}_{L^r(U)}\leq C|t_2-t_1|^{1-\f{1}{q}}. }$$ zhangzujin 2019-4-9 037 zhangzujin 2019-4-9 19:00
[偏微分方程] (190504) The second local regularity theorem for the Navier-Stokes equations says that: if $$\hj{ \limsup_{r\to 0}\f{1}{r}\iint_{Q_r(a,s)}\tk \leq \ve_1 }$$ for some $r>0$, then $\bbu$ is regular on $Q_\f{r}{2}(a,s)$. zhangzujin 2019-4-9 038 zhangzujin 2019-4-9 19:00
[偏微分方程] (190503) The first local regularity theorem for the Navier-Stokes equations says that: if $$\hj{ \f{1}{r^2}\iint_{Q_r(a,s)}\tk \leq \ve_0 }$$ for some $r>0$, then $\bbu$ is regular on $Q_\f{r}{2}(a,s)$. zhangzujin 2019-4-9 031 zhangzujin 2019-4-9 18:58
[偏微分方程] (190502) A sufficient condition for $f^\ve\to f\in L^2(\bbR^3)$ in $L^2(\bbR^3)$ is the following: ... (2) for every $\eta>0$, there exists $R(\eta)$ such that $\tk<\eta$ for all $\ve>0$. zhangzujin 2019-4-9 029 zhangzujin 2019-4-9 18:58
[偏微分方程] (190501) The energy of the solutions of the Leray regularised Navier-Stokes equations do not ``escape to $|x|=\infty$'' ... for any $\eta>0, T>0$, there exists $R=R(\bbu_0,T,\eta)$, such that $\tk\leq \eta$, for every $t\in [0,T]$. zhangzujin 2019-4-9 029 zhangzujin 2019-4-9 18:57
[偏微分方程] (190430) If the quantity $\f{1}{r^a}\int_{Q_r}|\n\bbu|^2$ is a scaling invariant quantity for the Navier-Stokes flow, then $a=\tk$. zhangzujin 2019-4-9 032 zhangzujin 2019-4-9 18:56
[偏微分方程] (190429) If the quantity $\f{1}{r^a}\sup_{-r^2<t<0}\int_{B_r}|\bbu(t)|^2$ is a scaling invariant quantity for the Navier-Stokes flow, then $a=\tk$. zhangzujin 2019-4-9 036 zhangzujin 2019-4-9 18:55
[偏微分方程] (190428) Leray proposed a regularised version of the Navier-Stokes equation, which reads $\tk$, $\n\cdot\bbu=0$, where $\psi_\ve$ is a mollifier in the space variables. zhangzujin 2019-4-9 028 zhangzujin 2019-4-9 18:55
[偏微分方程] (190427) The localised form of the energy inequality for the Navier-Stokes equations on $\Om \subset \bbR^3$ reads as $\tk$, for any smooth scalar function $\varphi \geq 0$ with compact support in the space-time domain $\Om\times (0,T)$. zhangzujin 2019-4-9 038 zhangzujin 2019-4-9 18:55
[偏微分方程] (190426) Given $(x_0,t_0)\in\bbR^3\times \bbR$, and $R>0$, the parabolic cylinder $Q_R(x_0,t_0)=\tk$. Moreover, $(x_0,t_0)\tk Q_R^*(x_0,t_0)$. zhangzujin 2019-4-9 031 zhangzujin 2019-4-9 18:54
[偏微分方程] (190425) Given $(x_0,t_0)\in\bbR^3\times \bbR$, and $R>0$, the centred parabolic cylinder $Q_R^*(x_0,t_0)=\tk$. Moreover, $(x_0,t_0)\tk Q_R^*(x_0,t_0)$. zhangzujin 2019-4-9 033 zhangzujin 2019-4-9 18:54
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