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

learn more… | top users | synonyms (2)

4
votes
0answers
128 views

Curvatures of contours of solutions of 3d Poisson's equation

Let $f(x,y,z)$ be a complex function in a 3d euclidian space that fulfill the Poisson's equation $$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f + \frac{\partial^2}{\partial ...
4
votes
0answers
305 views

The canonical form of a nonlinear second order PDE

Can anyone help me find the canonical form of $$x^2u_{xy} - yu_{yy} + u_x - 4u = 0?$$ I don't know how to solve it because $a = 0$. I just got that it's hyperbolic since $a=0$ , $b =(x^2)/2$, $c= ...
4
votes
0answers
185 views

Asymptotic behaviour of a two-dimensional recurrence relation

This problem comes out of a research in models of firm growth. The model is simple: A firm has two parameters which are its size (number of employees) and job vacancies. A firm of size $n$ will ...
3
votes
0answers
9 views

Imposing boundary conditions AND self-similarity on a PDE

I have a PDE in the form $$u_t=F(u,u_x)$$ where the unknown is $u(x,t)$ on say $\mathbb R\times[0,\infty)$. $F$ is very nonlinear so I was told to assume self-similarity in the form $$u(x,t)=t\tilde ...
3
votes
0answers
27 views

Dissipation term in wave equation

If we're given a string with mass density $\rho$ in units $\frac{M}{L^3}$ with constant cross-section $A$, tension $T$ in units $\frac{F}{L^2}$, and whose length is $L$; and then we assume that the ...
3
votes
0answers
23 views

Vector Calculus: solution to Poisson equation

This is problem 8.4.17. from Marsden Vector Calculus book. Let $\rho$ be a continuous function which vanishes outside a 3D region $W$. Define ...
3
votes
0answers
135 views

Complete integral of pde without independent variables

Show that the complete integral of pde $F(u,p,q)=0$ ($p=u_{x}$ and $q=u_{y}$) is $$ f(x,y,u,a,b) = x + ay + b - \int\frac{du}{g(u,a)}, $$ where the function $p=g(u,a)$ is computed from the ...
3
votes
0answers
22 views

solving linear gradient PDE

Let $f:\mathbb R^n \to \mathbb R$ be a function that satisfies the equation: $$\nabla^T f = - (\nabla^T \phi ) f $$ where $\phi:\mathbb R^n \to \mathbb R$ is some given function and $\nabla^T$ is the ...
3
votes
0answers
24 views

Exact solution of 2d poisson equation on an annulus

I have to find the exact solution of the following: $$-\nabla^{2}{u}=1$$ on the two dimensional domain $$\Omega=\{z=(x,y):|z|\ge{1} \wedge |z|\le{R}\} $$ On the boundaries of $\Omega$ we impose ...
3
votes
0answers
42 views

Equation of the form $\overline{\partial}_{J}f=g$ made holomorphic

In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried ...
3
votes
0answers
40 views

Maple Code for Ricci Flow

I am trying to generate a Maple Simulation for Ricci Flow assuming general Solids of Revolution and such. I assume that there is a surface with the following parametrization: ...
3
votes
0answers
23 views

Solve 1st order nonlinear pde $u_xu_y-xy+u=0$

Given a first-order pde $$u_xu_y-xy+u=0$$ with boundary value $u(x,y)=0$ at unit circle. I try to use the method of characteristic lines and obtain the following $$\begin{cases}\dot x=q\\\dot y =p\\ ...
3
votes
0answers
53 views

Example of an oscillation Young measure

I'm taking a course in which Young measures are introduced for oscillation and concentration. I have understood the examples the lecturer has given us for concentration Young measures, but cannot get ...
3
votes
0answers
46 views

Is there a version of Friedrich's lemma for Bochner spaces?

Let $u\in H^{1}([0,T], H^{1}_{0})$ that satisfies $u_{tt}+L u=f\in L^{2}(0,T; L^{2})$. where $L$ is a uniformly elliptic operator in divergence form given by $$L=(a^{ij}(t,x)u_{x^{i}})_{x^{j}}$$ ...
3
votes
0answers
32 views

Steps in alternative proof that if $u \in H^1(\Omega)$, then $Du = 0$ a.e. on set $\{u = 0\}$

Let $\Omega$ be an open subset of $\mathbb{R}^n$, and let $\phi$ be a smooth, bounded and nondecreasing function, such that $\phi'$ is bounded and $\phi(z) = z$ if $|z| \le 1$. Set$$u^\epsilon(x) := ...
3
votes
0answers
32 views

$L^p$-bounding inequality

Do we have that$$\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over2}$$for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$? Here, $U$ denotes an open subset of $\mathbb{R}^n$. ...
3
votes
0answers
32 views

Continuous inclusions Sobolev theorem, inequality

How do I see that if $f$, $g \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $fg \in H^s(\mathbb{R}^n)$ and$$\|fg\|_{H^s(\mathbb{R}^n)} \le C\|f\|_{H^s(\mathbb{R}^n)}\|g\|_{H^s(\mathbb{R}^n)},$$the ...
3
votes
0answers
55 views

Fokker Planck and SDE

I have the following Fokker-Planck equation in spherical coordinates $(\theta,\phi)$: $$ \partial f/ \partial t= D \cot\theta \quad \partial f/\partial \theta + \quad 1/\sin^2\theta \quad ...
3
votes
0answers
28 views

Heat equation with mixed boundary conditions

I am trying to solve the following problem $$\left\{\begin{matrix}\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},& 0<x<1,t>0,\\ u(0,t)=\frac{\partial u}{\partial ...
3
votes
0answers
18 views

Find explicit value of ${{u}_{x}}$ and ${{u}_{y}}$ of Burgers equation

Consider the first order quasi-linear equation with initail condition for a function $u(x,y)$ of two variables $x, y$ : $$\left\{ \begin{align} & {{u}_{y}}+u{{u}_{x}}=0 \\ & u\left( x,0 ...
3
votes
0answers
43 views

Dirichlet energy of solution to Laplace equation

Suppose $V\subseteq\mathbb{R}^3$ is compact with a smooth boundary. I'm interested in the Dirichlet problem $\Delta u=0$ subject to boundary conditions $u|_{\partial V}=f$ for a given function ...
3
votes
0answers
44 views

Solving the wave equation with Neumann conditions(?) using Green's functions.

I either have to solve the wave equation using Green's functions or Fourier transforms. I am given an infinite string which is fixed at $x_0=0$ and at rest for $t<0$. At $t=0$, it receives an ...
3
votes
0answers
40 views

What is a “standard symbol” in analysis?

I was reading some Wikipedia pages about analysis when I came across this strange "standard symbol" terminology in the "Fourier integral operator" page. It seems to be a function (or a distribution) ...
3
votes
0answers
71 views

Energy method for one dimensional wave equation with Robin boundary condition

Show that the initial-boundary value problem \begin{align} & {{u}_{tt}}={{u}_{xx}}\text{ }(x,t)\in \left( 0,l \right)\times \left( 0,T \right),\text{ }T,l>0 \\ & u\left( x,0 ...
3
votes
0answers
21 views

Using characteristics and picard iteration to solve nonlinear wave equation

My question: I want to solve $u_{tt}-u_{xx}+\lambda u=0$ in $0 \leq |t|\leq x \leq 1$ with the conditions that $u(x,x)=u(x,-x)=1$. My attempt: I let $\xi=x+t$ and $\nu=x-t$. This transformed the ...
3
votes
0answers
37 views

The behavior of the 3D wave equation close to the origin

The general solution to the three dimensional wave equation is \begin{equation}u(r,t) = \frac{F(x+ct)}{r} + \frac{G(x-ct)}{r} \end{equation} where $F$ and $G$ are arbitrary functions. I want to ...
3
votes
0answers
102 views

Solve Burger's equation after shock forms

Solve the Burger's equation: $$u_t+(\frac{u^2}{2})_x=0,\quad 0<x<2,\quad 0<t<\infty,$$ with periodic boundary conditions and the initial condition $$u(x,0)=\alpha+\beta\sin(\pi ...
3
votes
0answers
100 views

Diagonalizable operators on compact Riemannian manifolds

Let $M$ be a compact Riemanniam manifold and $\Delta$ the Laplace-Beltrami operator. Let $\lambda_0 \leq \ldots \leq \lambda_j \leq \ldots$ be the eigenvalues and $\phi_0, \ldots, \phi_j, \ldots$ be ...
3
votes
0answers
31 views

Second partial derivatives and integrals

Problem: Verify that $u(x,t)=\frac{1}{2c}\int_0^t\int_{x-ct+cs}^{x+ct-cs}f(y,s)\,dy\,ds$ is a solution to $u_{tt}-c^2u_{xx}=f$ with BC $u(x,0)=u_t(x,0)=0$. Attempt: First find $u_{tt}$ ...
3
votes
0answers
38 views

Have I gone wrong here? (partial differentiation)

Starting with $T = \frac{1}{2}M_{w}\dot{x}^{2} + \frac{1}{2}I_{w}\frac{\dot{x}^2}{r^2} + \frac{1}{2}M_{b}((\dot{x} + L\dot{\theta}cos(\theta))^2 + (L\dot{\theta}sin(\theta))^2) + ...
3
votes
0answers
29 views

A specific partial differential equation using Fourier Transform

I have the following PDE problem which I think sounds like a job for the Fourier transform: $ u_t + 2u_x = u_{xx} \space \space \space -\infty < x < \infty \space \space \space t>0 $ ...
3
votes
0answers
128 views

Heat equation in 2d Circle polar coordinates

I was presented this problem in PDE class involving heat equation on unit circle in polar coordinates using separation of variables, giving the following heat equation problem: $ u_t = 9\Delta u ...
3
votes
0answers
54 views

Wave equation with fixed end

A wave $f(x + ct)$ travels along a semi-infinite string $(0 < x < \infty)$ for $t < 0.$ Find the vibrations $u(x, t)$ of the string for $t > 0$ if the end $x = 0$ is fixed Answer: $f (x ...
3
votes
0answers
57 views

Learning Roadmap to Function Spaces

We are a small group of people that would like to start a reading course with the topic 'Function Spaces'. So far, we have all attended some graduate courses in Functional Analysis, Measure Theory, ...
3
votes
0answers
40 views

Eigenfunctions of the Laplacian on singular spaces

Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian ...
3
votes
0answers
84 views

Is the space $\mathcal{H}=\{v\in H^1(\Omega):\Delta v\in H^1(\Omega)^*\}$ a Banach space?

Let $\Omega$ be a Lipschitz domain in $\Bbb R^n$, is the space $$\mathcal{H}=\{v\in H^1(\Omega):\Delta v\in H^1(\Omega)^*\}$$ a Hilbert space when endowed with the norm $\|\cdot\|_\mathcal{H} = ...
3
votes
0answers
50 views

When are $\frac{1}{|x|^s}$ and $\log|x|$ integrable near the origin?

When are $\frac{1}{|x|^s}$ for $s>0$ and $\log|x|$ integrable near the origin? I'm reading Evans PDE and in the construction of the fundamental solution of Poisson's equation, he defines $$ \Phi(x) ...
3
votes
0answers
41 views

Coupled partial differential equation, with boundaries specification

Please, help me to find a books or samples to learn how to solve such coupled equations $$\begin{eqnarray} \frac{\partial T_1(x,t)}{\partial t}&=& \alpha_1 \frac{\partial^2 T_1(x,t)}{ ...
3
votes
0answers
31 views

Analytic version of Hilbert's XIX problem

The famous Hilbert's nineteenth problem, initially stated in the $C^\omega$ category, was reduced by Bernstein and Petrowsky to the analogous statement in the $C^\infty$ category (and, after ...
3
votes
0answers
45 views

looking for some exercise to test understanding of covector

I am trying to understand the concept of "covector" so that I can work with it for problems I come across in PDE. My knowledge on differential geometry is very little. And my topological background ...
3
votes
0answers
76 views

Reference request: The compactness and compact embedding in Besov Space?

This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions. Thank you guys! Let $\Omega\subset \mathbb R^N$ be open bounded with ...
3
votes
0answers
45 views

The best constant in Poincare-liked inequality in BV space

Let $\Omega\subset \mathbb R^N$ open bounded smooth boundary. Then we could prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E $, where $\mathcal E$ denotes the ...
3
votes
0answers
68 views

How can we show that $u$ as a weak solution has properties $u \in L^{\infty}(\Omega)$ , $ u>0 $

Let $\Omega$ be an open domain in $\mathbb {R^n}$ and $f \in C^{\infty}(\Omega)$ then how can we prove there is a weak solution $u \in ‎‎ W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \cap ...
3
votes
0answers
38 views

Minimum Knowledges to precisely calculate PDEs (integral equations)

Basically all I want to do is to calculate (or prove) precisely equations such as $$ \frac{1}{n\alpha(n)r^{n-1}}\int_{\partial B(x,r)} u(y) dS(y) = \frac{1}{n\alpha(n)} \int_{\partial B(x,r)} ...
3
votes
0answers
25 views

Proving parabolic PDE has maximum on boundary

Given $$u_t=u_{xx} - \frac{x}{2t}u_x - u^3$$ for $x,t>0$, with $u \rightarrow \frac{1}{2\sqrt t}$ as $x \rightarrow \infty$ for fixed any $t>0$, then I want to show that a maximum of $u(x,t)$ ...
3
votes
0answers
40 views

Resolvent estimate of hyperbolic Laplacian

Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form $$\Vert (-\Delta - \lambda ...
3
votes
0answers
40 views

$\Box u= | u |^2 u$ global solution in $C^\infty$

Let $u_0, u_1 \in C^\infty( \mathbb{R},\mathbb{R}^3)$. Consider the cubic defocusing NLW $$(\ast)\begin{cases} \Box u= |u|^2 u \\ (u,\partial_t u) \restriction_{t=0} = (u_0,u_1) ,\end{cases}$$ where ...
3
votes
0answers
83 views

Is this derivative well defined?

Let $\Omega$ be an open bounded domain in $\mathbb{R}^n$. Let $z_1,z_2,\dots,z_k\in \bar \Omega $ be $k$ distinct points. Let $\bf Z$ denote the k-tuple $Z = (z_1,z_2, \dots, z_k)$ Define $R_i = ...
3
votes
0answers
79 views

What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
3
votes
0answers
51 views

Estimating/approximating a very high dimensional unbounded poisson's equation

Consider the poisson equation on an unbounded domain. Suppose that the solution is known to exist. $$ \Delta u=f $$ I would like to estimate the solution of the this equation at a given point $x_0$. ...