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

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existence of entropy solution for a nonlinear degenerate PDE

I have come across this paper: UNIQUENESS OF THE ENTROPY SOLUTION OF A STRONGLY DEGENERATE PARABOLIC EQUATION, by Roberta Dal Passo. He tries to solve the equation $u_t=(\psi(u)\varphi(u))_x$,in R^...
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Proving Navier-Stokes bilinear operator is bounded on $H^s$

I'm trying to understand the proof that the bilinear operator arising in the Navier-Stokes equation, $$B(u,v)=\int_0^te^{(t-s)\Delta}\mathbb P\nabla\cdot(u\otimes v)ds$$ where $\mathbb P$ is the Leray ...
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0answers
14 views

Show that: $u_{tt}-u_{xx}=0$, $u(x,0)=x^2+x$, $u_t(x,0)=3$ with $x,t \in \mathbb{R}$.

I'm trying to solve the PDE $$u_{tt}-u_{xx}=0$$ $$u(x,0)=x^2+x$$ $$u_t(x,0)=3$$ with $x,t \in \mathbb{R}$. By applying d'Alembert's formula I get $$u(x,t)=\frac{1}{2}\bigg((x+t)^2+x+t+(x-t)^2+x-t +...
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1answer
15 views

Laplace Equation on Prolate Spheroidal Coordinates

I'm currently trying to solve Laplace's equation outside of a prolate spheroid. The scalar field has to vanish far from the spheroid. Because of the geometry I thought it might be convenient to use ...
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1answer
17 views

specific solution the the Burger's equation $u_t + u u_x =0$ with boundary condition $u(x,0)=e^{-x^2}$

I have difficulty with finding specific solution to the below PDE $$\left\{\begin{matrix} u_{t}+uu_{x}=0\\ u(x,0)=e^{-x^2} \end{matrix}\right.$$ My attempt: It is stright forward to get the ...
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1answer
21 views

Kolmogorov backwards equation / stationary distribution

One can in the case of the Fokker-Planck / forward Kolmogorov equation, set the time derivative term to zero, and solve the remaining ODE to obtain the "forward-time" stationary distribution. Does ...
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19 views
+50

Crandall-Ishii lemma on unbounded domains

In the User's Guide to Viscosity Solutions, there is an example (Section 5.D; page 30) using the Crandall-Ishii lemma on unbounded domains. Equation (5.17) claims $$ -3\alpha\left(\begin{array}{cc} I\...
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18 views

PDEs - traffic light, allocation of green to red light time

first time poster, so forgive me if I haven't done well with mathjax. doing some undergrad PDE traffic flow problems and we have a nasty. The set up is as follows. If you are looking at the correct ...
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1answer
20 views

Sobolev Extension on boundary of Lipschitz domain

If $\Omega$ is Lipschitz and $f\in C^{0,\alpha}(\partial\Omega)$, then there exists $g\in C^{0,\alpha}(\Omega')$ such that $f=g$ on $\partial\Omega$, where $\Omega\subset\subset\Omega'$. Is there a ...
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17 views

Show that $\lim_{|y|\to\infty}|y|^{n-2}u(y)>0$

For $n\ge 3$,Let $u\in C^2(R^n) $, $\Delta u\le 0, u>0$ in $R^n$ ,show that $\lim_{|y|\to\infty}|y|^{n-2}u(y)>0$. I consider the maximum principle,but I don't know how to deal with.
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9 views

Can we write $\text{tr}[Q(x)^*\nabla^2u(x)Q(x)]$ for $u∈C^2(ℝ^d)$ and $Q:ℝ^d→\text{HS}(H,ℝ^d)$ in terms of a differential operator?

Let $H$ be a separable $\mathbb R$-Hilbert space $u\in C^2(\mathbb R^d)$ and $\nabla^2u(x)$ denote the Hessian of $u$ at $x\in\mathbb R^d$ $\operatorname{HS}(H,\mathbb R^d)$ denote the space of ...
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2answers
53 views

How to tell if you have specified sufficient initial data for a differential equation?

I recently learnt that the following 'wave equation' is not well-posed $$ \begin{cases} \partial_{tt}u=\partial_{xx} u, & (0,1)\times\mathbb R\\ u(t,0)=u(t,1)=0,&t\in [0,1] \end{cases} $$ ...
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0answers
41 views

PDE basic traffic flow problem

I am analyzing a basic example of traffic flow presented here http://people.uncw.edu/hermanr/pde1/PDEbook/PDE_Main.pdf and have a question to the last transition in the traffic flow equation ...
2
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1answer
510 views

PDE Evans Chapter 7 problem 16

Problem 16 of chapter 7 states Use problem 15 to prove that if $u$ is the semigroup solution in $X=L^2(U)$ of $$ \left\{ \begin{array}{rl} u_t - \Delta u =0 & \text{in } U_T \\ u=0 & \...
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0answers
16 views

Constant Laplacian with some boundary data

Let $B$ be the open unit ball in $\mathbb{R}^2$ and $r=\sqrt{x^2+y^2}$, consider $$ \begin{cases} \Delta u = c \qquad \text{in} \ B \\ \frac{\partial u}{\partial r} =0 \qquad \text{on} \ \partial B \...
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11 views

Fisher-KPP equation with a compactly supported initial condition

Let us consider the classical Fisher-KPP equation on the real line : $$ \partial_t u - \Delta u = u(1-u) $$ with an initial condition $u_0$ that satisfies : $u_0$ is smooth $u_0(x) = 1$ if $|x|\leq ...
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0answers
30 views

A Simpler way of Thinking About Integrability

I have recently been looking into the concept of integrability, and the more I look into it, the less sense it seems to make. There seems to be several kinds of integrability and their definitions ...
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11 views

Equivalence of different definitions of (Laplace) Green function

Fix an open set $D \subset \mathbb{R}^d$. Usually the (Laplace) Green function is defined as the solution to the boundary value problem $$ \begin{cases} \Delta u(x) = \delta_y(x) \quad x \in D\,, \\u(...
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0answers
29 views

Fixed Point method and existence of entropy solution of a Nonlinear Parabolic PDE [on hold]

Could someone point me to literature that addresses this kind of problem? Actually I want to prove the existence of entropy solution for a nonlinear degenerate PDE for an equation that cannot solve ...
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0answers
12 views

Solution for the inhomogeneous 3D heat equation with initial temperature distribution

Can anyone describe the general solution for the inhomogeneous 3-dimensional heat equation: $u_t = K\nabla^2u + \frac{1}{c\rho}f$, with initial condition $u(x, 0) = g(x)$, no boundary conditions. ...
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0answers
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solve the initial value problem on the half line for the diffusion equation $U_x(t,0)=\sin t$ [on hold]

solve $U_t-U_{xx}=0$ for the half line with initial conditions: $$\quad Ux(t,0)=\sin t\\ U(0,x)=x$$
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0answers
9 views

Do conformal maps preserve subsolutions of elliptic PDE?

The fact is well-known for the Laplace equation for regions in $\mathbb R^n$ but I'm wondering if it extends to general elliptic PDE.
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15 views

Is the subspace $\{f \in C^{\infty}(\Omega) \;:\; \Delta^nf=0 \;, f=0 \mbox{ on } \Omega \}$ dense? [on hold]

As the title asks, is the subspace $S = \{f \in C^{\infty}(\Omega) \;:\; \Delta^nf=0 \mbox{ for some n, } f=0 \mbox{ on } \partial\Omega \}$ dense in $C^\infty(\Omega)$ (for nice enough $\Omega$)? ...
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1answer
34 views

Best Differential Equation, Partial Differential Equation and Calculus of Variations books?

Electrical Engineer here thinking of switching to physics. What are the best Differential Equation, Partial Differential Equation and Calculus of Variations books? Ideally they explain the topic ...
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1answer
51 views

Show that this function is weakly differentiable

I need to show that the function \begin{equation} u(x_1,x_2) = 1-x_1^2 \quad x_1>0 \\ u(x_1,x_2)=1+x_1^2 \quad x_1 \leq 0 \end{equation} is weakly differentiable on the unit ball. It is clear ...
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1answer
379 views

Method of characteristics - Burger's equation

Using the method of characteristics, find a solution to Burger's equation \begin{cases} u_t+\left(\frac{u^2}{2} \right)_x =0 & \text{in }\mathbb{R}\times(0,\infty) \\ \qquad \qquad \, \, u=g &...
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2answers
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Bootstrap argument to prove regularity of a special solution

Rabinowitz proves (using the Mountain Pass Theorem) that for a bounded smooth domain $\Omega \in \mathbb{R}^n$, and $f(x,\xi)\in C(\bar{\Omega}\times \mathbb{R},\mathbb{R})$ satisyfing the growth ...
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Solving heat equation

Given $k>0$ and $u_0\in\mathcal{C}^1[0,L]$, $u_0\geq 0$ we have the problem $$\left\{\begin{matrix} \frac{\partial u}{\partial t}(x,t)=k\frac{\partial^2u}{\partial x^2},&0<x<L,&t>0\...
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24 views

PDE IVP: Dirac delta function

I'm preparing for the candidacy exam on PDE and solving some past questions. The theory of 'delta' functions wasn't covered when I took the course. Any suggestions on how to solve this problem or ...
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Equivalence of definitions of harmonic (or wave) coordinates

In GR, one often uses harmonic (or wave) coordinates to simplify things. Now, one definition involves the coordinates themselves: $$ \Box_g x^{\alpha} = 0 $$ where $ \Box_g = g_{\mu \nu}\nabla^{\mu}...
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30 views

Singularities in a PDE

This is more of a general question rather than anything specific but I was just wondering if someone could point me toward resources which discuss singularities in a PDE rather than in an ODE (by ...
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Why is there no classification scheme for linear pde of third order or above

I am well aware of the classification of 2nd Order PDE into Elliptic, Parabolic or Hyperbolic type depending on the analogy with conic sections from analytic geometry. I have seen third order PDE ...
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426 views

Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output. [on hold]

I've been trying to solve the following Schrödinger equation numerically, \begin{equation} -(\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2})\Psi + \frac{\sinh^2(y) + \sinh^2(z)}{(\...
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27 views

PDE: Classical Solution?

Part (a) is a standard problem which can be solved either using separation of variables or Fourier transform.I haven't seen a proper definition for 'classical solution' to answer (b). Any help is much ...
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0answers
32 views

Heat Equation : Commutation of partial derivatives and summation

I'am having a problem when checking the validity of the solution i found for the heat equation: \begin{cases} U_{t}(x,t)=U_{xx}(x,t),\ {(x,t)\in (0,1)\times(0,+\infty)} \\ U(x,0) = x^2 - x\\U(0,t)=0\...
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$-\Delta u = u^p$ in bounded domain

In my PDE lecture we had the following theorem and I am wondering how strong it is: Theorem Let $\Omega \subset \mathbb{R}^n$ be a bounded domain (with $\partial \Omega$ sufficiently smooth, lets ...
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8answers
2k views

Why is uniqueness important for PDEs?

Every text on PDEs I come across will spend alot of time on showing the existence and uniqueness of solutions to a particular PDE. The importance of the existence of a solution to a PDE is obvious, ...
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1answer
476 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: $$\frac{d}{dt}\...
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1answer
394 views

Simple heat equation, solution regularity

I have a small problem with a regularity result for a simple parabolic heat equation: Given a $C^2$ open subset of $\mathbb R^n$ called $\Omega$, and a time $T > 0$, i have the following heat ...
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1answer
31 views

Unable to reach the desired result by substituting a given solution into the Schrodinger equation

A Textbook question asks me to: From the time dependent Schrodinger equation: $$-\frac{\hbar^2}{2m}\nabla^2\Psi+V\Psi=i\hbar\frac{\partial\Psi}{\partial t}\tag{...
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1answer
386 views

application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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1answer
17 views

How to derive a logarithmic potential from Newtonian?

Suppose we believe that the formula for Newtonian potential in $R^3$ is correct: $\varphi(\bar{x}) = \frac{1}{|x|} = \frac{1}{r}$, disregarding the constant. What is the justification of the fact ...
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31 views

The set of all maximal ideals (Wiener Algebra)

I am trying to prove a proposition and in my proof I somehow need to find the set of all maximal ideals of a Banach Algebra. This is my working environment: Let $A(\mathbb{R}^2)$ be the (Wiener ...
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1answer
33 views

How can I use this initial condition for the heat equation

How can I use the following initial condition for a partial differential equation describing heat diffusion? $$f(x) = \begin{cases} 0, & 0<x<0.45 \\ 1, & 0.45<x<0.55 \\ 0, & 0....
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Singularities in an Equation

these might sound like extremely trivial questions but since my background is more in probability and statistics I'm not too sure what to do, or even what to read up on to understand what to do. I ...
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Problem 16 Chapter 2. Evans PDE 2nd edition [closed]

This is a Problem from Evans PDE 2nd Edition,Chapter 2 Problem 16. Give a direct proof that if $U$ is bounded and $u\in C_1^2(U_T)\cap C(\overline U_T) $ solved the heat equation, then $$\begin{...
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2answers
94 views

Why this abuse of notation correctly solves the heat equation

Here's a stupid method I observed to solve the heat equation in $\mathbb R^d$, \begin{align*} \partial_tu=\Delta u,\quad u|_{t=0}=u_0. \end{align*} Pretend that $\Delta$ is a constant so this just ...
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1answer
485 views

How to decide whether PDE is Homogeneous or non-homogeneous.

I am studying second order PDE. And I have seen homogeneous and non-homogeneous PDE. But I cannot decide which one is homogeneous or non-homogeneous. For examples; (1) $(D^3-3D^2D'+4D'^3)u=0$ ...
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3answers
59 views

Separation of Variables and Linear PDEs

Separation of variables is a powerful method which comes to our help for finding a closed form solution for a linear partial differential equation (PDE). For example, we all know that how the method ...
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1answer
17 views

PDE argument about passage to limit in Bochner space and weak derivative

I have a function $u \in L^2(0,T;H^1_0)$ with $u' \in L^2(0,T;H^{-1})$, all on some smooth bounded domain $\Omega$. In addition I have a sequence $u_n$ such that $u_n \to u$ in $L^2(0,T;H^1_0)$ and ...