[家里蹲大学数学杂志]第049期2011年广州偏微分方程暑期班试题

[家里蹲大学数学杂志]第049期2011年广州偏微分方程暑期班试题

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 $calF_t$-submartingale with $N_0=0$, then for $1<p<infty$, $$bex Esez{sup_{0leq tleq T}sev{N_t}^p} leq sex{frac{p}{p-1}}^p Esez{sev{N_T}^p}. eex$$

b)        The set $calS$ of simple processes is dense in the Hilbert space $sex{calH, sen{cdot}_{calH}}$, where $$bex calS:=left{xi_t=sum_{k=0}^n xi_kchi_{[t_k,t_{k+1}]}(t): 0=t_0<t_1<cdots<t_nleq T,right.\ left.xi_kincalF_{t_k}, sup_ksen{xi_k}_infty<inftyright}, eex$$ and $$bex calH:=left{H: [0,T]timesOmega to bbR mbox{ is continuous and } calF_tmbox{-adapted}:right.\ left. sen{H}_{calH}^2 := Esez{int_0^Tsev{H(s)}^2rd s}<inftyright}. eex$$  Set $$bex calM:=left{ M=(M_t)_{tin [0,T]} mbox{ is continuous } calF_tmbox{-martingales such that } right.\ left. sen{M}_calM^2 :=sup_{0leq tleq T} Esez{sev{M_t}^2} <+infty right}. eex$$ Then $(calM,sen{cdot}_calM)$ is a Hilbert space.  Let $xi: [0,T]times Omegato bbR$ be the simple process given by $$bex xi_t=sum_{k=0}^n xi_kchi_{[t_k,t_{k+1}]}(t), eex$$ where $0=t_0<t_1<cdots<t_n=T$, and $xi_kin calF_{t_k}$ such that $dps{sup_k sev{xi_k}<infty}$. Define $$bex M_t=int_0^txi_krd W_s :=sum_{k=0}^n xi_ksex{W_{t_{k+1}wedge t-W_{t_kwedge t}}}, eex$$  

a)          Prove that $M_t$ is a continuous $calF_t$-martingale.

b)         Prove the It^o's isometry identity: $$bex Esez{sev{M_t}^2} = Esez{int_0^tsev{xi_s}^2rd s}. eex$$

c)        Using the Doob maximal inequality, prove that $$bex Esez{sup_{0leq tleq T} sev{M_t}^2} leq 4 Esez{int_0^T sev{xi_s}^2rd s}. eex$$

d)        Given $Hin calH$, let $H_nin calS$ be a sequence such that $sen{H_n-H}_{calH}to 0$ as $ntoinfty$. Prove that $dps{M_t^n =int_0^t H_n(s)rd W_s}$ is a Cauchy sequence in $sex{calM,sen{cdot}_calM}$. Let $M$ be the limit of $sed{M_n(t); tin [0,T]}$ in $sex{calM,sen{cdot}_calM}$. Prove that this limit does not depend on the choice of the sequence $H_n$ which tends to $H$ in $sex{calH,sen{cdot}_calH}$. Denote by $dps{M_t:=int_0^t H(s)rd W_s}$, i.e. $$bex int_0^t H(s)rd W_s =lim_{ntoinfty} int_0^t H_n(s)rd W_s,mbox{ in } sex{calM,sen{cdot}_calM}. eex$$

e)         Prove that $dps{M_t=int_0^t H(s)rd W_s}$ is a $calF_t$-martingale and satisfies $$bex Esez{sev{M_t}^2} = Esez{int_0^t sev{H(s)}^2rd s}, eex$$ and $$bex Esez{sup_{0leq tleq T}sev{M_t}^2} leq 4 Esez{int_0^T sev{H(s)}^2rd s}. eex$$

f)         Using the Borel-Cantelli lemma, prove that $ P$-a.s., $M=(M_t)in C([0,T];bbR)$.

2. Consider the following SDE on $bbR^m$: $$bex rd X_t=rd W_t-n V(X_t)rd t,quad X_0=x, eex$$ where $Vin C_b^2(bbR^m)$. Fix $T>0$. Suppose that $u(t,x)in C_b^{1,2}([0,T]timesbbR^m,bbR)$ is a solution of the heat equation $$bex left{ba{ll} frac{p u}{p t}(t,x) =frac{1}{2}lap u(t,x) -sef{n V(x),n u(t,x)},&mbox{in }[0,T)times bbR^m,\ u(0,x)=f(x),&xin bbR^m, earight. eex$$ where $fin C_b(bbR^m)$. Applying It^o's formula to $u(T-t,X_t)$, prove that $$bex u(t,x)= E_xsez{f(X_t)},quad forall tgeq 0, xin bbR^m. eex$$  

3. Consider the following SPDE on $[0,T]times S^1$: $$beelabel{1} frac{p}{p t}u(t,x) =lap u+dot W(t,x), eee$$ where $tin [0,infty)$ and $xin S^1=[0,2pi]$, $dps{lap=frac{p^2}{p x^2}}$ is the Laplace operator on $S^1$, and $W(t,x)$ is the space-time white noise on $[0,infty)times S^1$.  Recall that $lap$ is a compact operator on $L^2(S^1,rd x)$ and the spectral of $lap$ is given by $$bex mbox{Sp}(lap)=sed{-n^2; nin bbN}. eex$$ Indeed, let $$bex e_{2n}(x)=frac{1}{sqrt{pi}}cos(nx),quad e_{2n+1}(x)=frac{1}{sqrt{pi}} sin (nx),quad ninbbN, xin S^1. eex$$  Then $$bex lap e_{2n}=-n^2 e_{2n},quad lap e_{2n+1}=-n^2e_{2n+1},quad forall ninbbN. eex$$ The set $sed{e_n}$ consists of a complete orthonormal basis of $L^2(S^1,rd x)$. Write $$bex W(t,x)=sum_{n=1}^infty W_n(t)e_n(x), eex$$ where $W_n(t)$ are i.i.d Brownian motion on $bbR^1$.

(a) Let $$bex X_t(cdot) =u(t,cdot)in L^2(S^1,rd x). eex$$ Prove that $X_t$ satisfies the Ornstein-Uhlenbeck SDE on $L^2(S^1,rd x)$: $$bex rd X_t=lap X_t+rd W_t, eex$$ and $dps{W_t=sum_{n=0}^infty W_n(t)e_n}$ is the cylinder Brownian motion on $L^2(S^1,rd x)$.  

(b) Let $dps{u(t,x)=sum_{nin bbN} u_n(t)e_n(x)}$ be the orthogonal decomposition of $u(t,cdot)$ in $L^2(S^1,rd x)$. Prove that $u_n(t)$ satisfies the Langevin SDE on $bbR$: $$bex rd u_n(t)=-n^2 u_n(t)rd t+rd W_n(t), eex$$ and solve this Langevin SDE with initial condition $u_n(0)=u_nin bbR$.  

© Find the mild solution to the SPDE eqref{1} with initial condition $dps{u(0,x)=sum_{n=0}^infty u_ne_n(x)}$ for $dps{sum_{n=0}^infty sev{u_n}^2<+infty}$.  

(d) Recall that the domain of $lap$ is given by $$bex H_0=left{u=sum_{n=1}^infty u_ne_nin L^2(S^1,rd x); u_n=sef{u,e_n},right.\ left.mbox{ and } sum_{n=1}^infty n^2 sev{u_n}^2<inftyright} . eex$$ Let $$bex rd mu(u)=prod_{n=1}^infty frac{n}{sqrt{2pi}} mbox{exp}sez{-frac{n^2sev{u_n}^2}{2}}rd u_n. eex$$ Prove that $mu$ is a Gaussian measure on $(H,calB(H))$ with mean zero and with covariance matrix $Q=sex{q_{ij}}_{bbNtimesbbN}$ with $$bex q_{ij}=frac{1}{i^2}delta_{ij}, eex$$ i.e., $mu=calN(0,Q)$.  Formally we write $$bex Q=sex{-lap}^{-1},quad mu=calN(0,sex{-lap}^{-1}). eex$$

(e) Prove that $mu$ is an invariant measure for the Ornstein-Uhlenbeck processs $X_t$ on $L^2(S^1,rd x)$.  

(f) (Not required) Prove that $mu$ is the unique invariant measure for the Ornstein-Uhlenbeck process $X_t$ on $L^2(S^1,rd x)$.  

可压 Navier-Stokes 方程

1. Consider the compressible fluid flow with damping: $$bex left{ba{ll} p_trho+Div(rhobbu)=0,\ p_t(rhobbu)+Divsex{rhobbuotimesbbu} +n p=-rhobbu. earight. eex$$ Can this system satisfy Kawashima's condition?

2. Follow the similar analysis for Lemma 2.1 to prove (2.18) in Proposition 2.2.

3. Give the details of the proof of Lemma 3.1.

4. Give the details of the proof of Theorem 5.3.

5. Give the complete proof of Lemmas 6.4 and 6.5.

几何分析 [参考答案链接]

1.(15') 设 $R(X,Y): calX(M)to calX(M)$ 为曲率, 求证:  

(1) $R(X,Y)(fZ_1+gZ_2) =fR(X,Y)Z_1+gR(X,Y)Z_2$,  $forall X,Y,Z_1,Z_2in calX(M), f,hin C^infty (M)$;

(2) $R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0$,  $forall X,Y,Zin calX(M)$.  

2.(10') 设 $V(t), J(t)$ 是沿最短测地线 $gamma(t), tin [0,1]$ 的向量场, 它们满足 $$bex V(t)perp dotgamma(t),quad J(t)perp dotgamma(t),quad V(0)=J(0),quad V(1)=J(1), eex$$ 且 $J(t)$ 是 Jacobi 场, 求证: $$bex I(J,J)leq I(V,V), eex$$ 其中 $I$ 为 $gamma$ 上的指标形式.  

3.(10') 设 $gamma(t): (-infty,+infty)to M$ 为一条测地直线, 相应地记 $$bex gamma_+=gamma|_{[0,+infty)},quad gamma_-=gamma|_{(-infty,0]} eex$$ 及两 Busemann 函数 $$bex B_{gamma_+}(x)=lim_{tto+infty}sez{d(x,gamma(t))-t}; eex$$ $$bex B_{gamma_-}(x)=lim_{tto-infty}sez{d(x,gamma(t))+t}. eex$$ 求证: $$bex B_{gamma_+}+B_{gamma_-}=0,quadmbox{在 } gamma mbox{ 上}; eex$$ $$bex B_{gamma_+}+B_{gamma_-}geq 0,quadmbox{在 } M mbox{ 上}. eex$$  

4.(15') 设 $M$ 为紧流形, 再设 $g_{ij}(t)$ 满足 Ricci 流, 且 $f(t),tau(t)$ 满足 $$bex frac{p }{p t}f=-lap f+sev{n f}^2-R+frac{n}{2tau},quad frac{p }{p t}tau =-1. eex$$ 求证:  

(1) $$bex frac{rd}{rd t}int_M sex{4pi tau}^{-frac{n}{2}} e^{-f}, rd vol_{g_{ij}}=0; eex$$

(2) $$bex & &frac{rd }{rd x}int_M  sez{tau sex{R+sev{n f}^2} +f-n}(4pi^tau)^{-frac{n}{2}} e^{-f},rd vol_{g_{ij}}\ & &=int_M 2tau sev{R_{ij}+n_in_j f-frac{1}{2tau}g_{ij}}^2 (4pi tau)^{-frac{n}{2}} e^{-f},rd vol_{g_{ij}}. eex$$ 

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