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

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286 views

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
10
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232 views

Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
9
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157 views

Energy in the heat equation.

Before getting to question, some background. Let $u(x,t)$ be the temperature in a laterally insulated rod of length $L$, at position $x$ and time $t$. The temperature satisfies the heat equation ...
8
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266 views

How to solve a time-dependent Schrodinger equation in periodic Dirac delta potential

I'm trying to solve a 1D time-dependent Schrodinger equation: $$ i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)*x\right]\psi(x,t) $$ where $V(x)$ ...
8
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242 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: ...
8
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170 views

$W^{1,p} $ and $W^{2,p}$ Estimates.

In the beginning of section 4 in here the author says that one can easily adapt the methods in the preceding section to obtain $W^{1,p}$ estimate. I'm trying to do this. I think the following: the ...
7
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130 views

heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...
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160 views

Showing L infinity norm bounded by L2 norm on a manifold

I have the following problem that I'm working on: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality ...
7
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112 views

Two definitions of $H^1(\partial\Omega)$, one using charts and one use tangential gradients

Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$. There are two ways to define a space $H^1(\partial\Omega)$: By using charts, we can define $H^1(\partial\Omega)$ to ...
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280 views

Connection between distributional and renormalized solutions for Boltzmann equation

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read through some papers by DiPerna and Lions concerning the Cauchy Problem ...
6
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130 views

Uniqueness of solutions to $u_{tt} - c^{2}u_{xxxx} + au_{t} = 0$

The problem I am working on is to show that there is a unique compactly supported solution to the PDE $u_{tt} - c^{2}u_{xxxx} + au_{t} = 0$, $(x, t) \in \mathbb{R} \times [0, \infty)$ with $u(x, 0)= ...
6
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71 views

Existence of classical solution for equation $\Delta u + v(x) u = 0$

It is known that if $D$ is a bounded domain in $\mathbb R^n$, $n \geq 2$, with $\partial D \in C^2$ then the Dirichlet problem $$ \begin{array}{rl} \Delta u & = 0 \quad \text{in $D$}, ...
6
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118 views

When is separation of variables possible?

In classical PDE courses it is common to learn to perform a change of variables without really learning how to find the adequate equations of the change (polar, cylindrical or spherical coordinates ...
6
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134 views

How to solve a particular initial-boundary value problem

I have the following initial-boundary value problem $$\begin{cases}\dfrac{\partial^2 u_1}{\partial x^2}=A_{11}\dfrac{\partial u_1}{\partial t}+A_{12}\dfrac{\partial u_2}{\partial ...
6
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162 views

Separation of Variables for the Wave Equation

I'm trying to understand the method of separation of variables. I'm probability overlooking something simple, regarding the justification for the term-by-term differentiation that comes up when an ...
6
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178 views

References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried with Hamilton (1982) The Inverse Function Theorem of Nash and Moser. But, the article is very encyclopedic. I have a background in ...
6
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81 views

How are boundary conditions formally captured by the jet bundle approach to differential equations?

In the jet bundle approach to differential equations https://en.wikipedia.org/wiki/Jet_bundle#Partial_differential_equations one identifies the equation with the set of a solution of the ...
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93 views

Understanding orthogonality in two-scale asymptotic expansion (cf. G. Allaire)

This question is about equation (2.16) of Lecture 2 on Homogenization in Porous Media_ by Allaire page 28. There are two spacial scales: $x$ being macroscopic and $y=\dfrac{x}{\varepsilon}$ being ...
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1k views

First Order PDE Solution Method Issues

I'd really appreciate help with two little questions relating to first order partial differential equations. Just to quickly let you know what I'm asking, the first is about solution methods to first ...
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184 views

Show that, given spherically symmetric initial data, a solution to the heat equation is spherically symmetric

Let $\phi \colon \mathbb{R}^n \to \mathbb{R}$ be continuous with compact support. Furthermore, suppose that $\phi$ is spherically symmetric. That is, suppose that $\phi(Tx) = \phi(x)$ for every ...
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51 views

Elliptic and Fredholm partial differential operators

As I learn from the comments to this question a non elliptic operator on a compact manifold can not be a fredholm operator. On the other hand it seems that the following operator is fredholm(At ...
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44 views

Solving a partial differential equation by transformation of variables?

I found an exercise in a book, where one was supposed to transform the differential equation $$y\frac{\partial f}{\partial x}-x\frac{\partial f}{\partial y}=xyf(x,y)$$ by using the substitutions ...
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42 views

Vector Laplace equation with constraint

I want to solve Laplace equation for a vector $\boldsymbol v=(v_x,v_y)$: $$\nabla^2 \boldsymbol{v}=0$$ but under the constraint that $$(1+v_x)^2+v_y^2=1$$ which becomes $v_y = -(2v_x+v_x^2)^{1/2}$. ...
5
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70 views

General form of a connection with zero curvature

I am looking for proofs of the following two theorems: Theorem 1. On a connected and simply-connected open set $\Omega\subset\mathbb{R}^3$, functions $L^p_{ij}\in C^1(\Omega)$ are given that satisfy ...
5
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59 views

Weakening Sobolev coefficient in an elliptic estimate.

One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ ...
5
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141 views

Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
5
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135 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
5
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51 views

“Advective”, “diffusive”, “dispersive”, and related terms in the realm of PDEs

Whenever I read a paper involving PDEs, the discussion inevitably refers to “the dispersive term” or “the advective term” or similar. From context it is usually possible to figure out the antecedent, ...
5
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154 views

What differential equation might model this almost-harmonic oscillator?

I need to precisely control the motion of a damped, driven (nearly) harmonic oscillator: $$ \ddot x(t) + \alpha\dot x(t) + \omega_0^2 x(t) \approx V(t) $$ I use the $\approx$ symbol because this is ...
5
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2k views

How prove this hypersingular operator $(T\psi)((z(t))$

Let $\Omega\subset R^2$ be a simply connected bounded domain with infinitely differentiable boundary $\partial\Omega$and unit normal vector $v$ directed into the exterior of $\Omega$ ...
5
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155 views

An imcomprehensible proof on Arnold's Lectures on PDE, contact diffeomorphism

In Lectures on Partial Differential Equations, Arnold gives a theorem in the chapter Huygens' Principle in the Theory of Wave Propagation: Theorem 1 (the theory of support functions). The manifold ...
5
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64 views

Inhomogeneous Wave Equation in 3 dimensions

From section 7.5 in this source, I see that, for $\vec{x} \in \mathbb{R}^3$, if $$ \frac{\partial^2 u\left(\vec{x},t\right)}{\partial t^2} - \nabla^2u\left(\vec{x},t\right) ...
5
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137 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 ...
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700 views

3d-diffusion equation in spherical coordinates (numerical), boundary problem

There is one boundary problem $$\frac{\partial u}{\partial t}= \operatorname{div}\left(a^2 E \nabla u\left(r,\varphi,\psi \right) \right) $$ in a ball $$ B_{1}(0)=\left\{x \in \mathbb{R^3}: \left\| ...
5
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128 views

Generating Functions, Recursive Polynomials

At the CMFT international conference in Turkey (2009), the following open problem was given: Show that $$p_n(x):=\sum_{k=0}^n \frac{(n-k)^k}{k!}x^{n-k}$$ has only real simple zeros for every $n$. ...
5
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128 views

What does “smooth solution” of Ricci flow mean?

A way to formalize smoothness of a flow is to think of the spacetime, see e.g. here. Let's say we are flowing a compact manifold $M$. Which of the following is true? $g_{ij}$ is a $C^1$ in time and ...
5
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100 views

Scattering of waves by an obstacle

I wish to study the paper by Melroe and Taylor: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv in Math, Vol 55 (3), 1985, 242–315. ...
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149 views

Operator completly continuous

For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$ and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
5
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93 views

Solution to $\Delta_g u = \delta-1$ on a 2-sphere.

Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the ...
5
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114 views

What is visualization of gradient flow of a functional?

I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a ...
5
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352 views

How can I solve the Poisson PDE efficiently and fast in cylindrical coordinates?

I am trying to numerically solve the Possion PDE in cylindrical coordinate system. $$\Delta f = {1 \over \rho} {\partial \over \partial \rho} \left(\rho {\partial f \over \partial \rho} \right) + {1 ...
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163 views

Existence of the degenerate elliptic PDE coefficient condition

This is a question related to the theory presented in a book on degenerate elliptic PDEs. The book builds a theory for equations of the form: $$\sum a_{ij}u_{x_ix_j}+\sum b_{i}u_{x_i}+cu=0$$ with ...
5
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109 views

Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
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439 views

Does the infinite propagation speed of the heat equation violate special relativity?

I realize that Special Relativity is more of a physics concept than a math one, but I figured that since I learned the heat equation, $$\frac{\partial u}{\partial t} = \kappa\frac{\partial^2 ...
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98 views

Can we do some scaling argument in the presence of inhomogeneous norms?

Notation: $B^n_R$ stands for the ball of radius $R$ in $\mathbb{R}^n$. $\hat{f}$ stands for the Fourier transform of $f$. Question. The following inequality holds true for all $f\in ...
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192 views

Recommendation textbooks on D-module

I am going to take part in a seminar on D-module and applications, the textbooks that will be used are : D-modules, Perverse Sheaves, and Representation Theory, and A Primer of Algebraic D-Modules ...
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203 views

How should I understand $u_{\infty}$ in this theorem?

I learned the extension of Green's formula to unbounded domains in Kress's Linear Integral Equations (p.71): Theorem 6.10 Assume that $D$ is a bounded domain of class $C^1$ with a connected boundary ...
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292 views

How to use Galerkin method to prove uniqueness of solutions of hyperbolic equations?

Galerkin method is used heavily in finite element method, which can conveniently convert continuous problems to discrete ones. Particularly, Galerkin method can be used to prove uniqueness existence ...
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42 views

What does a well-posed problem imply?

A well-posed problem in the sense of Hadamard states that: A solution exists The solution is unique The solution's behavior changes continuously with the initial conditions. Now in order to prove ...
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107 views

Question 13 in Taylor's PDE vol III section 16.1.

my question comes from Taylor's PDE textbook, volume III. Consider a semilinear hyperbolic system, $u_t=Lu+g(u)$, $u(0)=f$, where $Lu=\sum_j A_j \partial_{x_j}u$, $g(0)=0, \ |g'(u)| \le C$, take ...