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

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2
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1answer
55 views

Symmetry method for 2d heat equation

Suppose we have the pde $$ \frac{\partial p}{\partial t} = \frac{1}{4}\left( \frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial y^2} \right) $$ Assuming a solution of the form, $$ ...
3
votes
1answer
29 views

Defining a distribution

We fix the space $\mathcal{D}=\mathcal{C}^\infty_0(\mathbb{R}^n)$ as space of testfunctions. Let $(f_n)$ be a sequence of distributions with $\lim_{n\to\infty} f_n(\varphi)$ existing for all ...
1
vote
0answers
34 views

Boundedness of Helmholtz projection

Let's cosider the Helmoltz projection $P$ into solenoidal subspace defined as $$P=I-\nabla div\Delta^{-1}$$ What can I say about its boundedness as an operator in $H^s(\mathbb{R^3})$?
2
votes
2answers
47 views

Strong solutions to an elliptic PDE

I would like a reference for the following result (you can assume more regularity and replace $C^2(\bar\Omega)$ with $C^2(\mathbb R^n)$ if needed): Let $\Omega\subset\mathbb R^n$ be a bounded ...
1
vote
0answers
31 views

Regularity for a vector transport equation

Let $\Omega\subset\mathbb{R}^N, N=2$ or $N=3$, be a smooth bounded domain and $T>0$. Given a vector function $h: \Omega\times (0,T)\to \mathbb{R}^N$ such that $\nabla\cdot h=0$. For any function ...
2
votes
1answer
32 views

Heat Equation, possible solutions

NOTE: This is a homework problem. Please do not solve. I was given a problem that asked me to find a function of the form $u_n(x,t)=\chi_n(x) \cdot T_n(t) $ that solves the heat equation with the ...
2
votes
1answer
38 views

Intuition to solving partial differential equations

I do not understand how to solve PDEs using the geometric method. I just do not understand the logic behind the solution. For example, the constant coefficient equation $$au_x + bu_y = 0,$$ where a ...
2
votes
1answer
50 views

Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
2
votes
1answer
56 views

Partial Differential Equation $\frac{\partial}{\partial t} p(x,t) = \frac{\partial^2}{\partial x^2} \left[ x^2 p(x,t) \right]$

In my research I have come across the partial differential equation \begin{equation} \frac{\partial}{\partial t} p(x,t) = \frac{\partial^2}{\partial x^2} \left[ x^2 p(x,t) \right]. \end{equation} ...
1
vote
1answer
23 views

Uniqueness of solutions to the wave equation

we are given the problem $u_{tt}-c^2\Delta u=0$ with conditions $u(x,0)=u_0(x)$ and $u_t(x,0)=u_1(x)$ where $x\in\mathbb{R}^n$ and $u_0,u_1\in\mathcal{C}^1$ having compact supports. If a solution ...
1
vote
0answers
20 views

How to improve stability of numerical solutions to partial differential equations

This is a quite general question, but I am working with a system of partial differential equations in two variables. There is one time direction $t$ and one spatial direction $z$ and the numerical ...
1
vote
0answers
31 views

Show $A:= -\Delta^2 - \beta\Delta$ is sectorial

Let $U$ be a open, bounded with boudary sufficiently regular domain. I want to show that the operator $$ A:= -\Delta^2 - \beta\Delta $$ defined on $U$, for some $\beta\in\mathbb{R}$ and ...
1
vote
0answers
47 views

Leibnitz rule for fractional derivatives

I need to estimate the following norm $$\Vert fg\Vert_{H^{\frac{1}{2}}(\mathbb{R}^3)}$$ Is there some product rule for the fractional derivative?
2
votes
1answer
45 views

Laplacian and Hodge Laplacian

I am new to the theory of differential forms, but there is one thing that I don't get at all. Imagine that you are on the sphere $\mathbb{S}^2$, then the Laplacian $- \Delta$ is known to be a ...
3
votes
0answers
16 views

How to derive the Sperically symmetric Wave Equation (in spherical coordinates) using first principles?

I want to use first principles (of mass and momentum conservation) on a spherical shell and derive the wave equation given below, where $p'$ is the pressure perturbation : $\frac{\partial^2 ...
2
votes
1answer
36 views

Poisson's Equation $-\Delta u= f$ where $f\in C_c^1(U)$

Evan's PDE book discusses Poisson's Equation $-\Delta u= f$ where $f\in C_c^2(U)$ as Theorem 1.1. With such a condition on $f$, we can basically pass all differentiations to it in order to show that ...
0
votes
0answers
25 views

Prove asolution to heat equation in 1D

I want to prove that \begin{equation} u(x,t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \phi(y)e^{-\nu k^2t+ik(x-y)} dkdy \end{equation} is the solution to the heat equation $u_t ...
1
vote
3answers
86 views

Video lectures on Partial Differential Equations

Would anyone happen to know any introductory video lectures / courses on partial differential equations? I have tried to find it without success (I found, however, on ODEs). It does not have to be ...
1
vote
1answer
43 views

Stuck with this function

im trying to find the values of $\alpha \in \mathbb{R}$ for which the function $f: B_2(0,\frac{1}{2}) \rightarrow \mathbb{R} $, $ x \mapsto |\log(\|x\|_2)|^\alpha$ is in $L^2(B_2(0,\frac{1}{2}))$ and ...
0
votes
0answers
33 views

How to compute an integral involving the error function

Solving a problem with one dimensional diffusion the following identity naturally arises $$\int _{-\infty }^{\infty }\frac{q}{2}{{\rm e}^{\eta\,t{\alpha}^{2}-\alpha \,x}} \left( {{\rm ...
0
votes
0answers
36 views

$\sup$ of a $C^s$ smooth function.

I want to prove that for a function $F\in C^k(\mathbb{R}^n)$ which vanishes at zero, and a function $u\in H^k(\mathbb{R}^n)$we get: $$\left\| \int_{r=0}^1 F'(ru)(\cdot)dr \right\|_{L^{\infty}} \leq ...
0
votes
0answers
33 views

Existence of a solution of a PDE

Let $a,h\in\mathcal{C}^1(\mathbb{R})$. I found out that a solution of the problem $u_t+a(u)u_x=0$, $u(0,x)=h(x)$ for $x\in\mathbb{R}$ can be represented (implicitly) by $$u(t,x)=h(x-ta(u(t,x))).$$ ...
1
vote
1answer
30 views

Need a estimate for the norm in $H^{-1}(\Omega)-$dual space of $H_0^1(\Omega)$!

Let $\Omega$ be a bounded domain in $\mathbb{R}^N, N\geq 2$. Let $v\in H^{-1}(\Omega)$-dual space of $H_0^1(\Omega)$ and I want to find the assumptions on $u\in X, (X=?)$ such that the following ...
1
vote
1answer
31 views

Evans' proof of parabolic strong maximum principle

I'm reading his PDE in chapter 7.1 the last theorem is about the strong maximum principle for parabolic equation when $c\geq 0$ at page 398. I have some problem at the second step: Since $u_t ...
1
vote
0answers
28 views

General solution to 3D linear 2nd order PDE using Wronskian and Integrals?

Using the Wronskian and Indefinite Integrals, I can write the solution to the general one dimensional second order linear non-homogeneous differential equation $$ y''+p(x)y'+q(x)y=g(x) \\ y^*(x)= ...
4
votes
0answers
58 views

why $ \nabla v_n \to \nabla v \ \ (a.e.)$ and $ v_n \to v $

Can someone see the 10th line of page 9 in this article and give a hint that why $$ \nabla v_n \to \nabla v \ \ (a.e.)$$ and $$ v_n \to v $$ and how with theorem 2.1 we could conclude there exists $ ...
0
votes
0answers
19 views

Show system has a solution using Frobenius method

System: $$\frac{\partial u}{\partial x}=v,\frac{\partial u}{\partial t}=w$$$$\frac{\partial v}{\partial x}=G(x,t),\frac{\partial v}{\partial t}=-\dot{a}(t)G(x,t)$$$$\frac{\partial w}{\partial ...
0
votes
1answer
34 views

analysis about elliptic PDEs

I want to study elliptic PDEs,but i have no knowlegde the analysis behind it, such as Arzelà–Ascoli theorem,sobolev embedding,campanato space,Rellich theorem,Poincare inequality... Do you have some ...
2
votes
0answers
35 views

if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
1
vote
0answers
39 views

Coefficients of spherical solution to Laplace's equation with difficult Robin boundary conditions

I'm trying to solve Laplace's equation in an (axisymmetric) external spherical domain. The controlling equation is: $$\nabla f = 0$$ $f$ must dissappear at infinity, and at the surface of the ...
1
vote
1answer
39 views

help to prove $||u||_{W^{2,2}(\Omega) }\le C ||\Delta u ||_{L^2(\Omega )} $

Can some one give a reference or hint for proving $$||u||_{W^{2,2}(\Omega)} \le C ||\Delta u ||_{L^2(\Omega )} $$
1
vote
0answers
23 views

Distribution annihilated by a vector field

Let $u$ be a distribution in $\mathcal{D}'(M)$ (the continuous dual of $\mathcal{D}(M) = C_0^\infty(M ; \mathbb{C})$), where $M$ is a smooth manifold. Let also $X$ be a smooth vector field on $M$, ...
1
vote
1answer
50 views

PDE solving with seperating of variables and SL-problem

I am stuck at the beginning of an excercise of PDE. The question is the following: A thin bar with length L, so that x=0 --> x=L. The bar is totally insulated and has a temperature of $100°C$. The ...
3
votes
1answer
81 views

$-\Delta u - \alpha u^{1/3} = 0$ implies $u \equiv 0$ if $\alpha$ is small

Let $\Omega$ be a domain in $\mathbb{R}^{d}$ with smooth boundary. Let $u(x)$ be a $H^{1}(\Omega)$ solution of the equation $-\Delta u - \alpha u^{1/3} = 0$, $u|_{\partial \Omega} = 0$. The problem I ...
0
votes
1answer
32 views

Heat equation in 1D with collocation method

I want to use the collocation method to solve $u_t=u_{xx}$. I impose the PDE pointwise and expand the solution in Fourier Series: $$ \partial_{t}\sum_{k=-K}^{K}\hat{u}_{k}(t)\ ...
1
vote
2answers
32 views

How should terms be scaled by finite dx and dt in numerical integration of 1D diffusion?

I am familiar with numerically integrating systems of ordinary-differential equations, but I feel that I am missing something important in terms of how numerically integrating ODEs differs from ...
1
vote
0answers
50 views

Difficult question on integral

we denote by $\overline{u}$ a positif fuction "radially symmetric about the origin" that realize $$\inf\{\int_{\mathbb{R}^N} (|\nabla u|^2+\lambda u^2) dx, u\in H^1_0({\mathbb{R}^N}), ...
1
vote
0answers
20 views

Compactness of Pseudo-differential Operators on $H^s(\mathbb R^n)$?

The Sobolev space of order $s\in\mathbb R$ in $\mathbb R^n$, denoted by $H^s(\mathbb R^n)$, is defined as follows: $$H^{s}(\mathbb R^n):=\{u\in\mathscr{S}^{'}(\mathbb R^n): \exists f\in ...
1
vote
1answer
30 views

Limit of an average integral?

Lebesgue's differentiation theorem states that if $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is locally summable then \begin{align*} \frac{1}{|B(x_0,r)|}\int_{B(x_0,r)}f(x)\,dx\rightarrow f(x_0), ...
0
votes
0answers
28 views

Why $\mathcal{D}(\Omega)$ be a topological vector space is important?

Let $\Omega\subset\mathbb{R}^N$ be an open set and $\mathcal{D}(\Omega)$ the set of all infinitely differentiable functions with compact support on $\Omega$. In the study of PDEs, we use the ...
2
votes
1answer
73 views

Is every hypersurface in $\mathbb{R}^n$ the boundary of an open domain?

We know if $\Omega \subset \mathbb{R}^{n}$ is a bounded $C^k$ domain, then its boundary $\partial\Omega$ is a $C^k$ compact hypersurface of dimension $n-1$. Is it true that every $m-$dimensional ...
1
vote
1answer
29 views

Show that a distributional solution of $\Delta u = f u $ is smooth for smooth $f$

As in the title - I would like to show that if $f$ is a smooth ($C^{\infty}$) function then for any distribution $u$ satisfying $$ \Delta u = fu$$ in the distributional sense we have, in fact, $u ...
2
votes
0answers
18 views

Solution operator for Laplacian is a continuous operator $C^{1,\alpha}(\partial D) \to C^1(\overline D)$

I consider the Dirichlet problem $$ \begin{cases} \Delta u(x) = 0, \quad x \in D,\\ u|_{\partial D} = \varphi, \end{cases} $$ where $\varphi \in C^{1,\alpha}(\partial D)$ and $D$ is a ...
0
votes
0answers
65 views

How could I solve this PDE?

Can anybody please let me know an idea about solving the following PDE for a given initial condition $T_0=0$ and boundary conditions $T(0,w)=u_0$? \begin{equation*} \nabla_w T_t(t,w) + T(t,w) . ...
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vote
0answers
30 views

Evans PDE derivation of solution for Poisson's equation - integration by parts

In Evans' PDE text, when proving the solution to Poisson's equation, when he integrates by parts he takes the inward pointing normal, whereas in the integration by parts formula in the appendix, it ...
1
vote
1answer
31 views

Unique solution for nonlinear heat equation

I have the following initial value problem: $$ \dfrac{\partial}{\partial t} u(x,t) - \frac{1}{2} \dfrac{\partial^2}{\partial x^2}u(x,t) = (u(x,t))^2$$ $$ u(x,0) = u_0(x)\in C^2(S^1)$$ and want to show ...
2
votes
1answer
27 views

inviscid Burgers' Shock Solution using Vanishing Viscosity Method

Ok, lets say I am going to solve the following equation: $u_t + (\frac{u^2}{2})_x = \epsilon u_{xx}$ which connects the end conditions $u_-=1$ and $u_+ = 0$. According to my understanding of the ...
1
vote
0answers
19 views

Lie point symmetry of KdV.

I'm asked to consider the 1-param. group of transformations generated by $V = \dfrac{\partial}{\partial u} + \alpha t \dfrac{\partial}{\partial x}$, which easily enough yields $g^{\epsilon}(x,t,u) = ...
0
votes
1answer
30 views

string displacement function by d'Alembert's formula

Consider an infinite string stretched taut on $x$ axis from $-\infty$ to $\infty$ . Let the string be drawn aside into a curve $y=f(x)$ and released, and assume that its subsequent motion is described ...
0
votes
0answers
23 views

numerical simulation of 4 coupled nonlinear PDEs

i have 4 nonlinear coupled PDEs with 8 BC eqs and i'm looking for a numerical method to simulate those in matlab. i can solve any ODE in M file by discretizing equations by writing the basic ...