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

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3
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1answer
82 views

Is $e^{it\Delta} g(x)$ continuous in $\mathcal{S}(\mathbb{R}^d)$?

In a paper, I see the following inequality: $$\vert e^{it\Delta} g(x)- e^{it'\Delta} g(x') \vert\leqslant C (\vert t-t'\vert + \vert x-x'\vert),$$ where $C$ depends on $d$ and $\Vert g ...
0
votes
1answer
38 views

Could the functions in larger space than $L^2$ be approximated by finite element basis functions?

Let $u \in V:=\{v \in L^{1+\alpha}(\Omega): \nabla v \in \mathbf{L}^2(\Omega)\}$, where $0<\alpha<1$. Clearly, $H^1(\Omega):=\{v \in L^2(\Omega): \nabla v \in \mathbf{L}^2(\Omega)\} \subset ...
0
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1answer
95 views

A conceptual doubt on method of characteristic for solving a first order partial differential equation.

This may be a very easy question of first order partial differential equation. I have a doubt. We use method of characteristic in solving for first order linear and quasi-linear differential ...
2
votes
2answers
96 views

Coupled nonlinear PDE, how to find solution, is this a well-known problem?

During my weekly meeting with a student we stumbled upon a curious system of PDEs. Here $u,v$ are functions of $x,y$ and $$ \frac{\partial u}{\partial x} = u^2+v^2 \qquad \frac{\partial v}{\partial ...
0
votes
2answers
98 views

A question about a trace operator (is this right?)

Suppose I have proven that for $u \in H^1(\Omega) \cap C^1(\bar \Omega)$ that $$|u|_{L^2(\partial\Omega)} \leq f|u|_{H^1(\Omega)}$$ for some constant $f$. Let $T:H^1(\Omega) \to L^2(\partial\Omega)$ ...
1
vote
1answer
314 views

The uniform limit of a convergent sequence of harmonic functions is still harmonic

Is there someone who could please help me prove this statement? I know that the uniform limit of a convergent sequence of harmonic functions is continuous. Then my goal is to show this function ...
3
votes
1answer
1k views

Proving energy conservation for wave equation

Hi guys I have a midterm tommorow and I was doing this practice problem that I need help on. So any hints or solutions would be appreciated. Thank you for your time Problem The head of timpani is ...
1
vote
0answers
111 views

Proprieties of the Fractional Laplacian on unbounded domain

I'm interesting to the stochastic PDE $$\left\{\begin{array}{l}\dfrac{\partial u}{\partial t}(t,x)=\Delta_{\mathbf{\alpha }}u(t,x) + {\dot{W}}(t,x), \\u(0,x)=u_{0}(x),\,\,\,\, ...
8
votes
1answer
211 views

The constant in the Sobolev trace theorem inequality

The trace theorem for nice enough domains states that there is a operator $T:H^1(\Omega) \to L^2(\partial \Omega)$ such that $$|Tu|_{L^2(\partial \Omega)} \leq C|u|_{H^1}.$$ My question, is there an ...
0
votes
1answer
56 views

Second Order forward finite difference scheme

Show that $d^2u/dx^2(x_i)=[(-u_{i+3})+(4u_{i+2})-(5u_{i+1})+2u_i]/h^2 +O(h^2)$ provided all terms in the expression are well defined is a second order finite difference scheme for second order ...
5
votes
1answer
93 views

Getting the PDE using Laplace equations

Hey guys I need help on one of the past midterm question that I came across. I am pretty sure I got (a) right. But if it is wrong could you please let me know. But its (b) and (c) that I got in ...
4
votes
2answers
127 views

If $u$ and $v$ have weak derivatives,what about $uv$?

$\Omega$ is a domain in $R^n$, Let $u\in L^1_{\text{loc}}(\Omega)$. If there exists $g_i \in L^1_{\text{loc}}(\Omega)$ such that $$\int_\Omega g_i \phi \, dx=-\int_\Omega u \frac{\partial ...
2
votes
1answer
138 views

$ \frac{\partial^2 T}{\partial x\partial y} = 0 $, then $ T = ? $

Can we characterize all distributions $T \in \mathcal{D}'(\mathbb{R}^2) $ with the following property of distribution derivatives ? $$ \frac{\partial^2 T}{\partial x\partial y} = 0 $$ For functions it ...
1
vote
1answer
37 views

Klainerman's Null Forms (A Question of Dimension)

If $F$ has a particular form, then the wave equation $\square u = F(u,u')$ has a global solution for sufficiently small $C_0^\infty$ Cauchy data. Here $u'=(\partial_tu,\partial_1u,\dots,\partial_nu)$. ...
1
vote
2answers
78 views

Find the solutions of a differential equation

Let $u:\mathbb{R^+}\to\mathbb{R}$. Find the solutions of the following differential equation $$u''(x)+\displaystyle\frac{u'(x)}{x}=c\log x $$.
0
votes
1answer
187 views

Does a uniformly sequence of harmonic function converge to a harmonic function?

Let $f_n:\mathbb{R}^n \to \mathbb{R}$ is a sequence of harmonic function and $f_n$ uniformly converge to a function $f$. Prove that $f$ is harmonic function. Is this statement true?
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1answer
74 views

about powers of gradient operator

Let u be a smooth function and ∇ is the gradient operator in n dimensions such that $\nabla^2 u=\Delta u$ is obvious. However, if we set D=$(\partial_t,∇)$ as a PDE operator in $(n+1)$ dimensions, ...
3
votes
4answers
79 views

second order ODE.

I have this second order ODE: $$-au''+bu'=x^2$$ where $u=u(x)$, $a=\text{constant}>0$, $b=\text{constant}\ge0 $ and $$u(0)=1 ,u(1)=0$$ I tried to solve it and I get: $$u(x)=D+Ce^{bx/a}$$ ...
2
votes
2answers
308 views

Heat Equation identity with dirichlet boundary condition

Show an energy identity for the heat equation with convection and Dirichlet boundary condition. $$u_t -ku_{xx}+Vu_x=0 \qquad 0<x<1, t>0$$ $$u(0,t) = u(1,t)=0 \qquad t>0$$ $$u(x,0) = ...
4
votes
1answer
160 views

If $\{T(t)\}_{t\geq 0}$ is an uniformly continuous semigroup of bounded linear operators then $T(s)\to T(t)$.

Definition: Let $X$ be a Banach space and $I$ the identity operator on $X$. A family $\{T(t)\}_{t\geq 0}$ of bounded linear operators from $X$ into $X$ is a semigroup of bounded linear operator on ...
0
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1answer
122 views

Nonlinear initial-boundary value problems using Taylor expansion of parameter

Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value problem $$ u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1 $$ $$ u(0,t) = 0 \\ u(1,t) = 0 \\ u(x,0) = \epsilon f(x) ...
3
votes
1answer
315 views

Determining what technique to use to solve a nonhomogeneous second-order PDE

$$ u_{tt}-u_{xx}=1-x, \space\space\space x \in (0,1) $$ with the following boundary conditions: $$ u(x,0) = x^2(1-x), $$ $$ u_{t}(x,0) = 0, $$ $$ u_{x}(0,t) = 0, $$ $$ u(1,t) = 0 $$ $u_{tt}$ and ...
1
vote
0answers
95 views

A system of nonlinear partial differential equations

Here are non-linear partial differential equations, where $f$ and $g$ are functions of $x,t$ : $g^2 (\partial_{x}f)(\partial_{t}f) - (\partial_{x}g) (\partial_{t}g) = 0, \quad ...
3
votes
1answer
399 views

Wave equation 1D inhomogeneous Laplace/Fourier Transforms vs Green's Function

I am trying to solve the following 1D inhomogeneous wave equation. Forgive me if I a miss any rigorous mathematical concept. $$ \frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 ...
2
votes
1answer
193 views

Why is this space dense in this Sobolev space? (Bochner spaces)

Let $V$ be a separable Banach space, and let $H$ be a Hilbert space. We have the Hilbert triple $$V \subset H \subset V^*.$$ By separability, there exists subspaces $V_k$ with $V_k \subset V_{k+1}$ ...
0
votes
1answer
39 views

How to show this $H^1$ space is separable?

Let $H = \{u \in L^2(0,T) : u' \in L^2(0,T), u(T) = u(0)\}$ be the space of $H^1$ functions $u$ with $u(T) = u(0)$. How to show that this space is separable with the usual $H^1$ norm?
0
votes
1answer
154 views

If $f(x,y,t):= u(r) \cos ( \omega t)$, use the multivariable chain rule to obtain an ODE for $u$ from the PDE for $f$.

Let $f(x,y,t) :=u(r)\cos \omega t$, where $r= \sqrt{x^2 +y^2}$. Physics tells us the following: For $f(x,y,t)$ to describe a vibrating membrane, with $f(x,y,t)$ telling how high the mem- brane is ...
1
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0answers
84 views

a version of the comparison lemma (harmonic functions)

I am trying to solve this exercise : Consider $U$ open , bounded set in $R^n$ with smooth boundary. Let $u \in C^2(U)$ a harmonic function. Suppose that for each $x \in \partial U $ there exists a ...
2
votes
0answers
59 views

Inequality in H-curl function space

Define a function space V , $$ V:=\{\mathbf{v} \in \mathbf{L}^{1+\alpha}(\Omega), \mathbf{curl}~\mathbf{v} \in \mathbf{L}^2(\Omega)\}, $$ equipped with graph norm $$ \|\mathbf{v}\|_{V} := ...
3
votes
0answers
131 views

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} ...
0
votes
0answers
152 views

Trouble learning PDEs

My college has started teaching PDEs, and I must say, it's the worst math course I've taken so far. I believe this is because of the sluggish curriculum of my college rather than the difficulty of the ...
1
vote
1answer
169 views

Support of Convolution and Smoothing

I just want to know how it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in C^{\infty}(V)$ by using the translations, but I'm having difficulty seeing how it ...
2
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0answers
40 views

subharmonic function and support functions

$M$ is a Riemannian manifold and $f$ is a continuous function on $M$. $f$ has the property that for any $p \in M,\epsilon>0$, we can find a smooth function $f_{\epsilon}$ such that ${f_\varepsilon ...
2
votes
1answer
71 views

Find the first four terms in the Fourier series for a solution of the wave equation

The question is to find the first four terms in the Fourier series for $u(x,t), t>0$. It is for a plucked string of length L, has zero initial displacement (I.e. $u(x,0)=0, 0<x<L$) and ...
1
vote
0answers
51 views

Definition of global weak solution to PDE

What is the definition of a global weak solution to a parabolic PDE? Is it a solution $u \in L^2_{loc}(0,\infty;V)$ with $u' \in L^2_{loc}(0,T;V')$ or is it a solution $u \in L^2(0,\infty;V)$ with ...
2
votes
1answer
251 views

Solving Hamilton-Jacobi-Bellman equations numerically?

I've been told that HJB equations can be solved numerically. I know very little about the subject, could someone provide a couple of comments or a reference (ideally, one that is accessible for a ...
4
votes
1answer
95 views

How do I derive p-Laplacian?

How do I obtain the p-Laplacian equation $$\Delta_p u = \nabla \cdot (|\nabla u|^{p-2} \nabla u) = 0$$ as the minimiser of the integral $$\int_{\Omega}|\nabla u|^p$$ ? I can't expand $|\nabla u + ...
5
votes
1answer
116 views

Need source for elliptic regularity on unbounded domains

I need a source that provides a $W^{2,2}$-regularity result for linear elliptic systems on unbounded domains. To be more specific, I study the equation $$ -\Delta w - ic \partial_1 w + \left( ...
2
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0answers
93 views

How to prove and what are the necessary hypothesis to prove that $\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial x_i}(x)$ uniformly?

Let $U\subset\mathbb{R}^n$ be a open set and $f:U\to\mathbb{R}$ a function in $C^\infty_c(U)$. Evans PDE book uses the following result $$\frac{f(x+te_i)-f(x)}{t}\to\frac{\partial f}{\partial ...
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0answers
79 views

question about the notation $C^2(\Omega)\cap C(\bar{\Omega})$

The notation $C^2(\Omega)\cap C(\bar{\Omega})$ seems to be prevalent in PDE. Suppose that $u$ is a solution of the PDE and that $u$ is $C^2(\Omega)\cap C(\bar{\Omega})$. Does it make sense to adopt ...
2
votes
1answer
1k views

Find the steady state temperature of the rod

A rod occupying the interval $0 \leq x \leq l$ is subject to the heat source $f(x) =0, $ for $ 0 < x < L/2$, $f(x) =H $ for $ L/2 <x <L ,H>0$ (1)The rod satisfies the heat equation ...
2
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0answers
111 views

Don't understand proof of global solution to parabolic PDE

Let $u \in L^p(0,T;V)$ denote the solution to the parabolic PDE $$u_t + \Delta u = f\qquad\text{a.e $t \in [0,T]$}$$ where $u_t \in L^q(0,T;V^*).$ We have the usual assumptions on $V \subset H \subset ...
0
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1answer
68 views

$(u_n)$ is bounded in $H^1(\mathbb{R}^N)$, some results about the convergence.

If $(u_n)$ is bounded in the Hilbert space $H^1(\mathbb{R}^N)$, we have that, up to a subsequence, \begin{eqnarray} &&u_n \rightharpoonup u\ \mbox{ weakly in }H^1(\mathbb{R}^N),\\ ...
3
votes
1answer
157 views

The form of the Poisson Integral Formula for a Temperature Profile

The question: consider a hole of radius a in a two-dimensional plane. We let the temperature on the hole's boundary be given by $f(\theta)$, where $\theta$ is a polar angle. From the temperature ...
2
votes
0answers
57 views

Integral with $\cosh$ and $\log$ in the integrand

I am trying to find a good way to simplify (or even solve) the following integral: $$ \int_0^{-\log \varepsilon} \cosh^3r \frac{\partial^2}{\partial r^2} \log (\det E) dr $$ where $r > 0$ is a ...
0
votes
1answer
394 views

Heat equation on a three dimensional box

Consider the heat equation on a three dimensional box with $0 < x < L$, $0 < y < H$ and $0 < z < W$. The heat equation is given by $$u_t=k(u_{xx}+u_{yy}+u_{zz}).$$ and the ...
4
votes
1answer
326 views

Evans PDE example 4, p.260 clarification

I am having trouble understanding the following example from Evans' PDE book. It is as follows (Example 4, p.260, Evans, Partial Differential Equations, 2nd ed.,) Let $\{r_k\}_1^\infty$ be a ...
1
vote
1answer
332 views

question in Evans PDE book

This question might sound naive but on p.34 of his book, he is considering the Poisson-dirichlet problem with $\Delta u=-f$ on $U$ with $u=g$ on $\partial U$. He then derives a formula for the general ...
5
votes
0answers
154 views

Importance of Schwartz kernel theorem

I am currently reading the proof of the Schwartz Kernel Theorem from Hormander Vol I. At the risk of sounding naive, what is the importance of Schwartz kernel theorem? What are certain insights that ...
1
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1answer
474 views

steady state solution to differential equation - checking my work

EDIT: fixed a stray negative sign. The problem as given: $y'' + 2y' + 5y = 10\cos t$ We want to find the general solution and the steady-state solution. We're using $\mu y'' + c y' + k y = F(t)$ ...