Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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Compactness of a sequence

Let $\theta_n(x,t)$ a sequence such that $$\theta_n\rightarrow\theta\;\;\mbox{in}\;\;C((0,T],H^s)\;\;\mbox{where}\;\;s>1.$$ Consider $\phi \in C^{\infty}$ and ...
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Compactness of solution space of semi-linear parabolic PDE

Under what conditions a closed and bounded subset of solution space of following parabolic PDE is compact? $$x_{t}=x_{zz}+f(x,z)$$ Thank you!
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The order of a differential operator on a manifold is well-defined

In these notes, the author defines a differential operator of order at most $n$ on a manifold $M$ as an element of $\operatorname{Diff}^n(M) := \operatorname{span}_{0\leq j \leq n} ...
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Range of Influence of the Wave Equation?

Suppose $u$ is a solution of the two-dimensional wave equation and that at $t=0, u=u_t=0$ outside the disc $x_1^2+x_2^2 \le 1$. Up to what time can you be sure that $u=0$ at the point ...
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99 views

Weak subsolution and composition with convex smooth function

I'm trying to solve a problem in Partial Differential Equations by Lawrence C. Evans. It goes like this Assume $u\in H^1(U)$ is a bounded weak solution of \begin{equation}-\sum_{i,j=1}^n ...
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Banach spaces involving time

Let's suppoe $u\in L^2(0,T;H_0^1(\Omega))$ with $u'\in L^2(0,T;H^{-1}(\Omega))$. We know that $$u\in C([0,T];L^2(\Omega))$$. In this result can the set $\Omega$ be the whole $\mathbb{R}^n$ or we need ...
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An interesting proof using Green's representation formula?

Let $B_R(0)$ be a ball in $\mathbb R^3 $ and define $$u(x)=\int_{B_R(0)}\frac{1}{|y-x|}dy$$ Prove that $$ u(x) = \begin{cases} \frac{2}{3}\pi(3R^2-|x|^2) & \quad \text{for $0 \le|x|\le R$ ...
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Inverse continuous

According Dehman, Gérard and Lebeau in "Stabilization and control for the nonlinear Schrödinger equation on a compact surface" (Math. Z. (2006) 254:729–749, DOI 10.1007/s00209-006-0005-3) is claimed ...
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Existence of a harmonic function?

I really have no clue about this problem. Should I represent $u(x)$ using $\varphi(x')$? But how to represent that? Thanks for any help!
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A Free Boundary Problem

Is there any special way to solve such a problem. Any idea would be appreciated. At least does anybody know which method is useful to solve this problem numerically? Is it even solvable numerically? ...
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Is solving Poisson's equation in Polars different from Cartesian?

I'm having trouble figuring out how to separate variables in polar coordinates in 2D. In cartesian coordinates it is fairly simple to use eigenfunction ideas because I can group together the x, y ...
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How to derive the cigar soliton solution to the Ricci flow equation?

I am trying to derive the cigar soliton solution to the Ricci flow equation. Such solution has the form $$ {\frac {{{\it dx}}^{2}+{{\it dy}}^{2}}{{{\rm e}^{4\,t}}+{x}^{2}+{y}^{2 }}} $$ I am ...
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102 views

Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question. I need some regularity results for the simple and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental ...
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Integral form of Bessel's (n = 0)

While studying for a comprehensive exam, I have come across this old problem: Consider the Helmholtz equation in the $\mathbb{R}^2$ plane $$u_{xx} + u_{yy} + \omega^2u = 0$$ Derive an ...
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Form-invariant solution to PDEs

I'm trying to understand how to create form-invariant solutions to PDEs. Let $u(x,t): \mathbb{R}^2 \to \mathbb{C}$, which solves the differential equation $\hat{L}u(x,t)=0$. $u(x,t)$ is ...
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quasilinear partial differential equation

Given a PDE $ f e^2 \frac{\partial f}{\partial x}- e f^2 \frac{\partial f}{\partial y} + M_1 f^4 + M_2 f^2 + M_3=0 $ Note that $M_1$ , $M_2$ and $M_3$ are functions of $\cos (x-y)$ and $\sin (x-y)$ ...
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Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
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Solving 1D telegrapher's equation by reduction to two-dimensional wave equation

The solution $w : \mathbb R \times \mathbb R_{+} \to \mathbb R$ of the Cauchy problem for the telegrapher's equation $$ w_{tt} - c^2 w_{xx} + c^2 \lambda^2 w = 0 $$ with $$ w(x,0) = 0, \qquad ...
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Regularity of a Weak Solution to Fokker-Planck Equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
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eigen value problem with Robin Boundary Conditions at both ends

This is a problem from the book Partial Differential Equations by Walter.A.Strauss. Consider the eigen value problem with Robin Boundary Conditions at both ends: $-X''=\lambda X$ $X'(0)-a_0X(0)=0$ ...
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Missing step in Evan's PDE?

In the attached proof, I am having trouble justifying the presence of "$x_n^{p-1}$" in eh line immediately below "Thus" and "Letting $m\to\infty$". I understand he uses Minkowski's inequality, and ...
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$L^2(S;W^{1,p})$-regularity for solution of parabolic pde in Hilbert space setting

I have a question regarding the regularity theory for parabolic pdes. I am considering the following "minimal working example": Let $\Omega\subset\mathbb{R}^n$, $n\in\mathbb{N}$, be and bounded ...
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Anyone help me with this PDE using Fourier Transform?

I have this: $$\frac{\partial c}{\partial t} + p\frac{\partial c}{\partial z}+\lambda p\frac{\partial^{2} c}{\partial z\partial t}-\frac{\partial^{2} c}{\partial z^2}=0\quad(1)$$ $$c(z,0)=\delta(z)$$ ...
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Hölder estimate of solution to linear parabolic PDE

Consider a standard Cauchy-Dirichlet problem for linear parabolic PDE: \begin{cases} u_t + \mathcal{L}u=-g, \ (t,x) \in [0,T)\times \Omega \\ u(T,x)=0 , \ x\in \Omega\\ u|_{\partial \Omega}=0 ...
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Tensor differential equations

I am reading Ringstrom's book The Cauchy problem in General Relativity, But I don't really understand Chapter 12 associating to tensor equations. I want to read some other material about this. Could ...
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Are there any results linking hyperbolic operators and Pseudo-Riemaniann geometry?

In the case of a $n$-dimensional pseudo-riemannian manifold one has a symmetric bilinear form, $h$, with signature $(1,n)$ and the analogous operator to the Laplace -Beltrami operator,$\Delta$, is the ...
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180 views

Conservation of momentum for nonlinear Schrodinger equation

I am having trouble proving the following momentum conservation law. Given smooth compactly supported solution $u(x,t)\in \mathbb{C}$ where $(x,t)\in \mathbb{R}^n\times \mathbb{R}$ of $iu_t+\Delta u= ...
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Variational Problem on a manifold - bounding my functional from below

Let $(M, g_{ij})$ be a compact Riemannian manifold. Let $f \in C^{0,\alpha}(M)$ be a given function. I have the linear operator $L = \Delta h - 12u^2h$, where $u \in C^{2,\alpha}$. I need to show ...
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Passing to the limit in a PDE; problem with subsequence (please check my answer)

For $w \in L^2(0,T;H^1)$, consider the PDE $$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$ where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and ...
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A regularity result for a parabolic PDE? Want $u' \in L^\infty((0,T)\times \Omega)$

Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times ...
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Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$

Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which ...
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$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

Suppose $f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$ Then, clearly, $|f|\in L^{2}(\mathbb R)$ and therefore by Plancheral theorem, we get, $\widehat{|f|} \in L^{2}(\mathbb R).$ Also, assume, $f, ...
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What does $\cos(n,t)$ mean?

In the book by JL Lions "Quelques methodes...", (Chapter 2, Section 3.3, page 197), he uses the notation $$\cos(n,t)$$ in a boundary condition on a domain $\Omega(t)$, where $n$ denotes the normal ...
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106 views

Crank Nicolson Method PDE

I have the following PDE $0=\partial _t u+\frac{1}{2}\partial_{xx} u$, now I assume that $t\in[0,T], x\in[0,L]$ and initial data $u(T,x)=g(x), u(t,0)=a(t), u(t,L)=b(t)$ The grid $\{(ik,jh): ...
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What is compensated compactness?

As the title says, what is compensated compactness? I see people talk about it in the books and papers I am reading but I can only find hand wavy definitions when I look online. Is there a definition ...
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Finite Difference Methods for arbitrary elliptic PDE

I am looking for textbook references that describe lattice numerical methods for arbitrary elliptic PDEs, particularly finite difference schemes and particularly in 2d. The few references that I have ...
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Convergence of the series $\sum_{\xi\in\mathbb Z^n} e^{2\pi ix\cdot \xi} a(x, \xi)\hat{f}(\xi)$?

I need some help with the following problem: let $a:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb C$ be a smooth function and suppose there are constantes $C_{\alpha, \beta}$ and $M(\alpha, \beta)$ ...
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In which space does my (weak) solution exist?

I am reading through some numerical analysis papers, and was wondering what characteristics of a problem define which function space the weak solution of a problem belongs to. For example, for the ...
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$W^2_p$ regularity of solutions of linear elasticity

I want to prove the following statement: Let $\Omega$ be bounded, polygonal, and convex, then for the solution of the linear elasticity (elliptic) equation with inhomogenous Dirichlet boundary ...
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wave equation $C^2$ solution

If we consider the wave equation on the half line $\mathbb{R}_+$ such that $u_{tt} -u_{xx}=0$ in $\mathbb{R_+}\times(0,\infty)$ $u(x,0)=g(x)$ and $u_t(x,0)=h(x)$ for $x\in \mathbb{R_+}$ $u(0,t)=0$ ...
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Verifying the fundamental solution of Stokes flow in R^3?

Recall that Stokes flow in $\mathbb{R}^3$ is the solution $(\textbf{u}, p)$ of $$\begin{align} -\nabla p + \mu \Delta \textbf{u} &= \textbf{f} \\ \nabla \cdot \textbf{u} &= 0, \end{align} ...
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existence theorem of eliptic equation

Consider $\Omega \subset R^n$ a bounded and open set with $\partial \Omega$ smooth . Consider the problem: $$ - \Delta u + au = f \text{ in } \Omega $$ $$ u = 0 \text{ in } \partial \Omega $$ ...
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PDE- Method of Characteristics

I am given the following equation: $ (y+u)u_x + (x+u)u_y =(x+y)u$ , when: $ u(x,2x)=3x$ . and I want to solve it using the method of characteristics. The equations are: $ x_t = y+u , y_t=x+u , u_t ...
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Are Navier-Stokes equations used in molecular biology research?

I am wondering whether Navier-Stokes equations have been used in some molecular biology research papers in the past. A quick Google search revealed that such papers exist but I want to know if there ...
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Verifying an integral identity related to asymptotic homogenization of an elliptic partial differential equation

Background I'm reading Hornung (1997)'s Homogenization and porous media, pg 3: We study a family of [1D] problems, indexed by the scale parameter $\epsilon=\frac{1}{n}$, namely, ...
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Bounds on solutions of elliptic pde's in the whole plane

My question is rather simple. Given $f$ a $L^2(\mathbb{R}^2)$ function with zero mean but supported in the whole plane, is there a bound of the form $$ \|\nabla(\Delta)^{-1} ...
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678 views

Help with a non-linear partial differential equation

I am wondering whether someone can help me with a non-linear PDE: $$\frac{\partial^2\phi}{\partial t^2} = c\frac{\partial^2\phi}{\partial x^2} \left(\frac{\partial\phi}{\partial x} \right)^{n-1}$$ ...
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A probable inspiring proof to Poincare lemma

Poincare lemma says if a smooth $p$-form $\omega$ is closed, then $\omega$ must be exact. Let's put it in another way, it says the solution of $d\omega=0$ is $\omega=d\eta$ for some $(p-1)$-form ...
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Trace spaces of Orlicz-Sobolev spaces.

Recently I have the need of study trace spaces of Orlicz-Sobolev spaces. By looking in google I have discovered in this PDF page 14 (not only in the PDF), that the main contributions come from the ...
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Ergodic mean for Schrodinger flow

Let us consider the linear Schrodinger equation in $\mathbb{R}^N$ $$ (i\partial _t+\Delta)\,u=0\mbox{ ,}\quad u(0,x)=f$$ with $f\in L^2(\mathbb{R}^N)$, and let $u(t,x)=e^{it\Delta}f$ its solution. ...