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

learn more… | top users | synonyms (2)

1
vote
2answers
73 views

On the abstract bootstrap principle in the book “Nonlinear Dispersive Equations” by Terence Tao

In Terence Tao's book "Nonlinear Dispersive Equations", he gives the following "Abstract bootstrap priciple": "Let $I$ be a time interval, and for each $t \in I$ suppose we have two statements, a ...
0
votes
0answers
38 views

Perron solution and weak solution for a Dirichlet problem in a convex domain

Consider $\Omega \subset R^n$ an open, bounded and convex set. Then your boundary is Lipschtz. Then we can define the trace operator T. Consider $K \subset \partial \Omega$ a compact set with non ...
1
vote
0answers
15 views

about the Perron method for the Dirichlet problem

Consider $\Omega$ an open, convex and bounded set of $R^n$. Let $g: \partial \Omega \rightarrow R$ a function. Supose that $g$ is continuous except in one point. By the convexity of $\Omega $ we can ...
0
votes
0answers
5 views

characteristic structure of nonlinear system of hyperbolic conservation laws

For linear and nonlinear scalar conservation laws, i.e. : $$ q_{t} + f'(q)q_{x}=0 $$ we can solve this by method of characteristics, with $$ X(t) = f'(q(X(0),0)t + X(0). $$ Quite similarly, the ...
2
votes
0answers
49 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 ...
2
votes
3answers
132 views

Computing the spherical mean and showing it satisfies PDE

Compute the spherical mean of the function $h : \mathbb R^3 \to \mathbb R$ with $$ h(x,y,z) = x $$ and show that it satisfies the differential equation $$ u_{rr} + \frac{2}{r} u_r = u_{xx} + u_{yy} ...
0
votes
1answer
23 views

Solving a 2nd order PDE with boundary data

I have a feeling I may be making a trivial mistake here, but I would really appreciate it if someone could verify my method. I have a 2nd order PDE: $$u_{xx} - x^2 u_{yy} - \frac{1}{x} u_x$$ I can ...
2
votes
2answers
34 views

Solution of a PDE

Solve $\displaystyle \frac{\partial^2 z}{\partial x^2}+z=0$, given that when $x=0$, $z=e^y$ and $\displaystyle \frac{\partial z}{\partial x}=1$. My Attempt: Integrating w.r.t x twice (keeping y ...
3
votes
1answer
61 views

Solving the Hamilton-Jacobi equation

I can comprehend some but not all of the proofs. I do not understand how the limit definition of a derivative is derived in this context, and those are highlighted in $\color{#009900}{\text{green}}$. ...
1
vote
3answers
67 views

Linear PDE with Variable Coefficients Help

I would like to find the closed-form solution (for $x > 0$) to $$\frac{\partial u}{\partial t} = a \frac{\partial^2 u}{\partial x^2} +b \frac{\partial u}{\partial x} + (cx+d) u,$$ $$u(0,\,x) = ...
0
votes
0answers
13 views

Wave equation fundamental solution - Kevorkian.

I've been reviewing the Wave equation, and there are a few things that I don't understand in Kevorkian's book. It says the fundamental solution to the wave equation is defined by the solution to: ...
1
vote
1answer
34 views

Solving 2nd order hyperbolic PDE

I have a PDE: $$u_{xx} - u_{yy} = 0$$ And my boundary conditions are: $u = -\sin(2 \pi x)$ on $x+y = -1$ $u = \sin(2 \pi x)$ on $x-y=1$ Now I can find the characteristic variables $\phi = x+y$ ...
1
vote
1answer
42 views

How to solve $e^yu_x-u_y+u=xe^y$?

Solve $e^yu_x-u_y+u=xe^y$ $u_x-\frac{u_y}{e^y}=x-\frac{u}{e^y}$ $\frac{b}{a}=\frac{dy}{dx}=-\frac{1}{e^y}$ $e^y=-x+c$, $\eta=e^y+x,\xi=x$ $e^y=\eta-\xi$ ...
2
votes
0answers
20 views

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: $-X''=\lambda X$ $X'(0)-a_0X(0)=0$ ...
1
vote
0answers
11 views

seperation of variables $U_{tt}=c^2U_{xx}-rU_t$ when ${2\pi c\over l}<r<{4\pi c\over l}$

Consider waves in resistant medium that satisfy the problem $U_{tt}=c^2U_{xx}-rU_t$ for $0<x<l$ . $u=0$ at both ends. $u(x,0)=\phi(x)$ $u_t(x,0)=\theta(x)$, where r is a constant. Using ...
0
votes
1answer
29 views

Duhamels principle

I have a problem with the following exercise (I know how to proof for $u_{tt}$ but for $u_t$ I met a problem, for $u_{tt}$ proof works as we can assume that $v(x,t,t)=0$ and we add it artificially to ...
0
votes
0answers
22 views

Harmonic function in circle - exercise from Partial Differential Equations book by Y. Pinchover

Could I please ask about help with the following exercise: Let $u(x, y)$ be the harmonic function in $D = \{ (x, y) : x^2 + y^2 < 36\}$ which satisfes on $D$ the Dirichlet boundary condition: $$ ...
2
votes
1answer
27 views

$ U_{xx}+U_{yy}=0$ with rectangular boundary conditions

When solving $ U_{xx}+U_{yy}=0$ with $u(0,y)=u(a,y)=u(x,b)=0,u(x,0)=f(x)$. $0<=x<=a$ , $0<=y<=b$ by the method of separation of variables I have $-X''(x)-\lambda X(x)=0 $ ...
1
vote
0answers
13 views

Laplace equation on a rectangle

I have to solve$ U_{xx}+U_{yy}=0$ with $u(0,y)=u(a,y)=u(x,b)=0,u(x,0)=f(x)$. $0<=x<=a$,$0<=y<=b$ . I let $u(x,t)=X(x)Y(y)$. Then $X''(x)Y(y)+X(x)Y''(y)=0$. Then I took the minus sign to ...
1
vote
0answers
13 views

Question about extending a solution to Monge-Ampere solution

I am interested in solutions to the Monge-Ampere equation for a smooth function $h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is: ...
0
votes
1answer
38 views

intuition behind an identity related to fourier transforms

I saw the proof of this identity in a question about Fourier transforms : $F(f(−t))w=F(f(t))(−w)$ Can someone give the intuition behind it ? What I understand of Fourier transform of a function ...
2
votes
1answer
33 views

a question about Fourier transforms

I know it s simple but how to show that $\mathcal F(f(-t))w=\mathcal F(f(t))(-w)$ ? $\mathcal F(f(-t))w=\int^\infty_{-\infty}f(-t)e^{-iwt}dt$ $if -t=x\to -dt=dx$ ...
1
vote
0answers
20 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 ...
2
votes
0answers
25 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 ...
1
vote
0answers
29 views

Can I write the relation as a total derivative?

I am solving an hyperbolic system of PDE using the method of characteristics and I want to write the following expression as a total deivative. $$\left(\frac{\partial{u_1}}{\partial{t}}-\frac{x}{t} ...
0
votes
0answers
38 views

What happened to Otelbayev's proof on Navier-Stokes existence and smoothness? [duplicate]

I haven't heard any since the last comment on Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations? So what has gone on then? Sorry for asking a too soft question here. ...
0
votes
1answer
26 views

Example of non-local operator

Let $T: C_{0}^{\infty}(R^{n}) \to C^{\infty}(R^{n})$ be a linear operator. $T$ is local if $$\operatorname{supp} (Tu) \subset \operatorname{supp} (u),$$ for all $u \in C_{0}^{\infty}({R}^{n})$. We ...
2
votes
1answer
44 views

When is a real-analytic function harmonic?

I recently learnt that every harmonic function occurs as the real part of a complex analytic function. We also know that every harmonic function is real analytic. So, when is a real-analytic function ...
0
votes
1answer
30 views

Is there a (fundamental) solution of the laplace equation which is not radial?

In the approach given to solve the laplace equation ( With reference to PDE by L Evans ), we first observe that the laplace operator is rotation invariant .i.e., if we rotate the solution ,it still ...
0
votes
1answer
50 views

negative Sobolev space contains $L^1$ for a compact domain

I'd like to use something like Aubin-Lions lemma for the following spaces: $$ C^{0, \alpha}(B) \subset L^1(B) \subset W^{-1, q}(B),$$ with $B \subset \mathbb{R}^n$ being a compact, say a closed ball ...
0
votes
1answer
25 views

estimate to show bilinear form continouus

Consider the bilinear form $$a(u,v)=2\mu\int_\Omega \mathrm{trace}\left(\varepsilon(u)^T\varepsilon(v)\right)dx + \lambda\int_\Omega \mathrm{div}(u)\mathrm{div}(v)dx$$ with ...
1
vote
0answers
13 views

Continuity of the solution to a matrix PDE (mapping of a parameter to solution)

I'm considering the following PDE in $\Phi$: $\frac{\partial \Phi(t,s)}{\partial t}$ + $sR\frac{\partial \Phi(t,s)}{\partial s}$ + $\frac{1}{2} s^2 M \frac{\partial^2 \Phi(t,s)}{\partial s^2}$ + ...
2
votes
1answer
16 views

characteristic coordinate method to solve wave equation

In solving wave equation $u_{tt}=c^2u_{xx}$ by characteristic coordinate it is chosen that $\epsilon =x+ct$ and $n=x-ct$. But how was it decided that this was the transformation required.How do we ...
0
votes
1answer
46 views

The analytical solution for advection-diffusion equation with source term.

We have: $$\frac{\partial w}{\partial t} + a(x) \frac{\partial w}{\partial x} - v \frac{\partial^2 w}{\partial x^2} = f(t)$$ within a domain $x \in [0,1]$ Simplest Sample is $a(x) = 1$ (constant) and ...
0
votes
0answers
8 views

Category of a PDE and its properties

Now I am working on numerical method for a PDE. I am considering the following PDE: $$ u_t+a^2u_{xx}=f\\ u(x,0)=u_0\\ u(x,t)|_\Gamma=u_g $$ That equation seems very like heat equation which only ...
1
vote
2answers
17 views

Solve this PDE using a change of variable.

Solve the following PDE using a change of variable: $$ \alpha^2 \dfrac{\partial^2 z}{\partial x^2} - \beta^2 \dfrac{\partial^2 z}{\partial y^2} = 0$$ This is my attemp: Let the following change of ...
1
vote
1answer
22 views

PDE with variable coefficient equation

Solving $U_x+yU_y=0$.The curves in the x,y plane with (1,y) as tangent vectors have slopes y. Their equations are $dy/dx=y$.This Ode has the solution $y=Ce^x$. Hence $u(x,y)=u(x,Ce^x)=U_x+Ce^xU_y=0$. ...
1
vote
1answer
26 views

derivation of transport equation

The amount of pollutant in the interval at time $t$ is $M=\int_0^bu(x,t)dx$ .At later time t+h,the same molecules of pollutant have moved to the right by $ch$ centimetres. ...
8
votes
1answer
127 views

Is every scalar differential operator on $(M,g)$ that commutes with isometries a polynomial of the Laplacian?

On $(\mathbb{R}^n, g_{\text{std}})$ with $\Delta$ the Laplacian, the following holds: Fact: Every scalar differential operator $D$ that satisfies $D \circ F^* = F^* \circ D$ for all isometries $F ...
2
votes
0answers
48 views

Why has no body retypeset Ladyzhenskaya et al's “Linear and quasi-linear equations of parabolic type”? [closed]

The book "Linear and quasi-linear equations of parabolic type" is one of the ugliest books I have ever seen in my life. The fonts are awful, the notation is difficult to understand and recall and the ...
1
vote
1answer
47 views

Question about the solution of the wave equation

The solution of the wave equation $$u_{tt}-c^2u_{xx}=0$$ is given as $$u(x,t)=f(x+ct)+g(x-ct)$$ where $f,g$ are arbitrary functions of one variable. One way to prove this is the following: ...
1
vote
0answers
13 views

Causality and viscous wave equation

I've seen several papers related to the causality condition concerning the viscous wave equation resolution but never understood how causality and stability are linked ? Conceptually, it seems hard to ...
0
votes
1answer
32 views

Method of characteristics-Hypebolic PDE

I am looking at the following example of solving an hyperbolic system of equations using the method of characteristics: $$\left.\begin{matrix} \frac{\partial{u_1}}{\partial{t}}+a ...
0
votes
0answers
11 views

Klein Gordan equation with harmonic source function

I have come across the following differential equation: $$u_{tt}-u_{xx}-u_{yy}-u_{zz}-\mu^2\,u=f(x,y,z)\,,$$ where $f(x,y,z)$ is a harmonic function satisfying Laplace's equation $\Delta f=0$. Does ...
0
votes
0answers
24 views

Maximum principle in PDE [duplicate]

I was told Maximum principle is a common method in proving uniqueness of the solution to certain PDE. Could anyone explain 1) How does Maximum principle work in the context of PDE theory? 2) How ...
0
votes
1answer
42 views

Show that $(\log |x|)^2\notin \text{BMO}([-1,1])$.

I am trying to show that $u(x)\equiv (\log |x|)^2\notin \text{BMO}([-1, 1])$ by showing that it doesn't satisfy the John-Nirenberg inequality. If $u\in\text{BMO}[-1, 1])$ then this inequality says ...
1
vote
1answer
50 views

When is a given matrix-valued function the Jacobian of something?

Let $F$ be an $n\times m$ matrix of real-valued functions which are defined and smooth on a neighborhood of a point $p\in \mathbb{R}^m$. Under what conditions is it possible to find a smooth function ...
0
votes
0answers
86 views

Prove a differential equation:

The partial differential equation $$\frac{d^2u}{dt^2}=c^2\left(\frac{d^2u}{dx^2}+\frac{d^2u}{dy^2}+\frac{d^2u}{dz^2}\right) \tag{$*$}$$ is the three dimensional wave equation. In the case of ...
0
votes
1answer
21 views

How to get comparison principle from contraction principle for PDE

let $u$ and $v$ be two solutions to some PDE with initial data $u_0$ and $v_0$. Is $$|u(t)-v(t)|_{L^1} \leq |u_0-v_0|_{L^1}$$ a contraction principle? I read that "contraction principle gives ...
1
vote
1answer
28 views

Dimensional analysis of a PDE (so called “transport equation”)

I need to know what the dimensions are of things! It seems horrible! Okay the equation models a thin channel with some pollution or something in it. The notes say: "$u(x,t)$ denotes the pollution per ...