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

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Max Principle for Heat Equation with Neumann Boundary Condition

This is a homework problem. I am looking for an additional hint or reference to theorems/ideas that can help. Some of my thought process is presented below. Suppose \begin{align*} u_t - \Delta u ...
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186 views

Boundary Conditions for a Finite Difference Approximation of a Sixth Derivative

I am trying to use a finite difference scheme to numerically solve sixth order parabolic equations such as \begin{equation} u_t = u_{xxxxxx} \end{equation} with symmetry conditions \begin{equation} ...
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53 views

finding solution to a partial integro differential equation

I want to find a function (or a set of functions) such that $u(x,t)$ satisfies the following partial integro-differential equation with singular kernel \begin{eqnarray} &&u_x(0,t) = \int_0^t ...
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23 views

Existenence of the solution for a PDE-ODE system.

I have the PDE-ODE system below: $\frac{\partial c}{\partial t}= D \Delta c - \eta \nabla.(c\nabla v)+g(c,v)$ $\frac{dv}{dt}=-\alpha cv+\xi(c,v)$ with initial conditions and Neumann boundary ...
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38 views

mean curvature of a nonlinear parabolic PDE

I want to compute the mean curvature of a reaction diffusion equations of the form $$ u_t=u_{xx}-u^2 $$ Any suggestions is appreciated!
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52 views

First order partial differential equations in complex domain

Try to solve a first order linear partial differential equation $P(x,\partial)u(x)=f(x)$ in complex domain, while the operator is of the following form: $$ ...
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62 views

Differential Equation for brownian bridge?

For the brownian motion, we know that probability density of the particle's position at time $ t $, $ \rho(x,t) $ satisfies the diffusion equation pde: $ \partial_t \rho = d \; \partial_x^2 \rho $. Is ...
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38 views

Heat Kernel Inverse

Is it possible to construct an inverse heat kernel in the following restricted sense: given a nice function $ h: \mathbb{R} \rightarrow \mathbb{R} $, can one find a function $ g: \mathbb{R} ...
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43 views

existence and uniqueness of reaction diffusion equation

I am trying to prove uniqueness of equation of the form $$ u_t=u_{xx}+f(u) $$ Can anyone suggest the standard literature for existence and uniqueness for equation like above?
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163 views

Comparison and maximum principle for parabolic pde

I was told the comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the maximum principle. I also know comparison principle ...
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57 views

Taking partial derivatives over multiple summations

I have the following equation obtained from one of the models. $\mathcal{H} = \sum\limits_{D} \sum\limits_{W}n(d,w)\sum\limits_{Z} p(z|d,w)[\log{p(d)}+\log{p(z|d)}+\log{p(w|z)]}$ I need to take ...
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70 views

Sine-Gordon Equation application

Is it true that Sine-Gordon is satisfied for geodesics on the central Pseudosphere ( rotated surface of Tractrix)? If so, please cite text-book or article references. $$ \alpha''(s)=sin ( \alpha) ...
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62 views

Loss of derivatives

In many books on pdes the expression "loss of derivatives" is used when some estimates on solution are proved. Can someone clarify to me (maybe with an example) the meaning of this expression? For ...
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66 views

Derivative nonlinear Schrodinger equation

I'm dealing with the following DNLS $$iu_t+u_{xx}=i(|u|^2u)_x$$ Let's consider the following transformation $w=\exp(-i\int_{-\infty}^x|u|^2dy)u$. I'm interested in the equation satisfied by $w$. I ...
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45 views

Elliptic PDE on the Whole Space

Can anyone suggest a reference for elliptic PDE on the all of $\mathbb{R}^d$, as opposed to some bounded domain $\Omega$, covering the standard topics of existence, uniqueness, and regularity. I ...
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1answer
151 views

Deriving an estimate in regularity theory of the heat equation

I have another question from PDE Evans 2nd edition, this time from pages 380-381. It's about a step in the formal derivation of estimates. Given the initial-value problem for the heat equation ...
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3answers
46 views

Finding the change of variables to transform $u_{tt} - u_{xx} = 0$ into $u_{rs} = 0$

I'm just beginning to introduce myself to partial differential equations and one of the first problems presented in the textbook I have literally no idea how to do. I think the author intended the ...
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2answers
122 views

Confused over the solution of partial differential equation $xu_x+u_t=0$

Consider, $$ \displaystyle x\frac{\partial u}{\partial x}+\frac{\partial u}{\partial t} = 0 $$ with initial values $ t = 0 : \ u(x, 0) = f(x) $ and calculate the solution $ u(x,t) $ of the above ...
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53 views

Proof of Second Partials Test

How does one rigorously prove the second partials test without firstly assuming that $D(a,b)=AC-B^2$ that states the following: $ A=\frac {\partial^{2}f(a,b)}{\partial x^{2}},B=$$\frac ...
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35 views

harmonic functions: comparison of gradients

Consider $\Omega$ a open, bounded, convex domain in $R^n.$ I am trying to justify this: Let $u, v$ non negative harmonic functions in $\Omega$ (in the Sobolev sense). Suppose that $ u = v =0$ in ...
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51 views

Finite difference scheme for hyperbolic system

I'm having a bit of trouble understanding the following, so it'd be great if anyone has any nice explanations! Thanks in advance! Consider the hyperbolic system $$u_t = Au_x + Bu$$ where $A$ and $B$ ...
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61 views

Studying Hamiltonian PDEs - Where to start?

I first got in contact with Hamiltonian PDEs when I wrote my first thesis about the Nash-Moser-Theorem as some books mentioned them as a field of application of this theorem. As those examples were ...
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52 views

Growth condition in differential equation and vanishing solution at boundaries

In a discussion on solving a partial differential equation I lately read: "Under a standard growth condition on the solution at infinity, the resulting PDE is fully specified without boundary ...
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0answers
44 views

Finding gradient of an objective as a PDE

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
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0answers
63 views

Compute average and maximum value of a field over a streamline

I'm working on a code solving a set of PDEs. I have a vector field, $\vec{v}(x,\theta,z,t)$ (it's a velocity) and a scalar field, $c(x,\theta,z,t)$. I have a $2\pi$-periodicity in $\theta$. The ...
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1answer
169 views

Green's function of a Markov Chain, and maybe of a Feller Process?

How are the Green's functions of a Markov chain related to the notion from PDE theory? For instance, if the Markov chain (i.e. discrete state space) is continuous time, then the Green's function I'm ...
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1answer
33 views

Why orthgonality matters for numerical treatment of PDEs?

It seems to me that general orthogonal coordinates are quite popular in numerical treatments of PDEs. Sometimes people even use conformal maps to generate the locally orthogonal grids. But the actual ...
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1answer
72 views

Equivalence of Localized Fourier Restriction Estimates

I'm reading Tao's Park City notes on the restriction conjecture http://arxiv.org/abs/math/0311181. He says at some point that the estimate: there is a constant $C > 0$ such that, for any $R \ge ...
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67 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
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0answers
57 views

Mollification of functions in $L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$

Let $u \in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain. I know that (eg. from Wloka or Hunter's PDE notes) that there is a mollification ...
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162 views

Eigenvalues problem for generalized Kuramoto-Sivashinsky equation

I been working on Kuramoto-Sivashinsky Equation. In the process of analysis, I need to solve the following eigenvalues problem $$ -u_{xxxx}-\lambda u_{xx}=\beta(\lambda)u $$ where $\lambda$ is a ...
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81 views

Meaning of “Canonical System of the First Order”

I am learning about PDEs and came across the following. "Convert a partial differential equation of higher order into a canonical system of the first order" What does the above statement mean/imply? ...
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67 views

Solving PDE via Fourier Transform & Uniqueness

When a PDE is solved via Fourier transform, is there already a uniqueness assertion that comes for free? For example, if we Fourier the heat equation \begin{align} \partial_t u(x,t) &= \Delta ...
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47 views

Local Mollification of a $L^1$ function

Let $f$ be a $L^1(\Omega)$ function, where $\Omega\subseteq \mathbb{R}^2$ is a smooth and bounded domain. Then can we find a fuction $\phi\in W_0^{2,2}$ suct that $\phi>0$ in $\Omega$ and ...
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1answer
174 views

Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...
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174 views

Compact support

From PDE Evans, 2nd edition, page 204 Example 9 (Wave equation from the heat equation). Next we employ some Laplace transform ideas to provide a new derivation of the solution for the wave ...
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2answers
93 views

What is the right domain for this Hamiltonian

I want to define a proper domain $D(H) \subset L^2$ for this Hamiltonian ( $\theta$, $\phi$ are the standard angles in spherical coordinates). Furthermore, the wave function is supposed to satisfy ...
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36 views

Hamiltonian form of PDE

I am quite new in this field and I was wondering what does exactly mean to write a PDE (in 2 or 3 dimensions) in an Hamiltonian form. More in detail, is there any standard procedure to write a given ...
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2answers
85 views

Analytical Solution of a PDE

I need to solve a PDE which seems to be quite simple and to have an analytical solution. I tried the method of separation of variables, but could not complete the solution. Could you please let me ...
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0answers
76 views

Why is this bilinear form elliptic on $H^1_0$?

I have the following bilinear form: $$ a(v,w) = \int_\Omega \nabla v(x) \cdot \nabla w(x)dx + \int_\Omega \left( \sum_{i=1}^n \beta_i v_{x_i}(x) \right) w(x) dx $$ where $\Omega \subset ...
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42 views

Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
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60 views

Imposing boundary conditions in a finite element sense

Say we have a PDE which is well posed with the boundary condition $|\nabla u|=r$ ($r$ constant) on $\partial\Omega$, where $\Omega\subset \Bbb R^n$ is uniformly convex. How would one impose this ...
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61 views

Explicit computation of spectrum of Hodge-Laplacian on forms

While I know of some explicit examples (sphere, flat torus) for the spectrum of the Hodge-Laplacian on 0-forms (i.e. the Laplace-Beltrami operator on functions), I haven't found anything for "actual ...
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51 views

When does weakly elliptic $\Rightarrow$ strongly elliptic?

While learning more about the analytic background for the Atiyah-Singer Index Theorem, I was curious about the following question (although not needed for the ASID): what are some general conditions ...
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38 views

Missing Step in a Paper of Struwe.

In this paper on Page 4, in the last line of the proof, the author asserts that if a radial function $u:\mathbb{R_t}\times \mathbb{R}^2\to \mathbb{R}$, smooth outside the origin $(0_t,0_x)$, admits ...
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38 views

intuition for entropy solutions

For a hyperbolic PDE of the form $$u_t + f(u)_x = 0$$ it turns out that the right notion of solution is entropy solution. Now, the notion of classical solutions are obviously very natural, and also ...
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107 views

Relation between fractional integral operator and solution of poisson equation

For $0<\alpha<d$, fractional integral operator $I_{\alpha}$ is defined by $$I_{\alpha}f(x)=\int_{\mathbb{R}^d} \frac{|f(y)|}{|x-y|^{d-\alpha}} dy$$ for any suitable function on $\mathbb{R}^d$. ...
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124 views

Solving a system of integral-partial differential equations

Hi I am a student in electrical engineering. Currently I am facing a difficult problem solving a coupled integral-differential partial equations arising from mean field game. The problem is similar as ...
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1answer
45 views

the differentiability from an oscillation estimate

If we have an oscillation estimate of $u\in W^{2,n} (B^{+}) \cap C^0(\bar B^{+})$, and $u=0$ on $T = B \cap \partial R^{n}_{+}$, that $$osc_{B^{+}} \frac{u}{x_{n}} \leq C.$$ Then how comes that $u$ is ...
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38 views

Greens function of a uniformly charged sphere

The potential $\phi(\boldsymbol{x})$ satisfies $\nabla^2\phi=f$ It may be shown that by defining an appropriate Green's function $g(\boldsymbol{x},\boldsymbol{\xi})$ that ...