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

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Solving the PDE's naturally arising from requiring vector fields to commute

If one has two vector fields $X$ and $Y$ in $\mathbb{R}^4$ whose coefficients are indeterminate functions (some coefficients are linearly related to others, however), then the condition that the ...
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diffusion equation [on hold]

I'm kinda lost with this problem. I don't know how to solve it. If somebody can help me I will be so thankfully. I'm so confuse.If somebody know a reference problem that would help a lot
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22 views

heat equation, total heat energy [duplicate]

I'm having a hard time with this problem. I get the situation, but I just don't know how to model it and show part b and part c. I will be so thankfully.
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1answer
22 views

Boundary conditions

I am kinda confuse with the second part of my homework. I did the first part (3/a and 4/a) without any problem, but part b for both problems I don't get it at all. I try to plug the boundaries in the ...
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1answer
25 views

Divergence structure equation

Consider Laplace's equation with potential function $c$: $$-\Delta u + cu = 0, \tag{$*$}$$ and the divergence structure equation $$-\operatorname{div}(aDv)=0, \tag{$**$}$$ where the function $a$ is ...
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1answer
22 views

Two density results, $C_{0}^{\infty}$ is dense in $L^{2}$

I really lack analysis background but I'm in a situation with PDE's (Finite elements specifically) where I'm trying to prove a couple intermediate results. Prove $C_{0}^{\infty}(\Omega)$ dense in ...
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1answer
32 views

Diffusion equation problem

I'm not sure what the question is trying to ask. Any help would be appreciated!
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37 views

Product rule in fractional Sobolev spaces

I have to prove the following inequality $$\Vert fg\Vert_{H^s(\mathbb{R}^3)}\leq C \Vert f\Vert_{H^\sigma(\mathbb{R}^3)}\Vert g\Vert_{H^{s+1}(\mathbb{R}^3)}$$ with $s\geq 0$, ...
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1answer
19 views

Eigenfunctions of the laplacian (1 dimension)

I have the following problem: $\frac{d^2 u}{dx^2}(x)+\lambda u(x)=0, x \in (a,b)$ and $u(a)=u(b)=0$. The general solution (for $\lambda>0$) is $u(x)=c_1\cos(\sqrt\lambda x)+c_2 \sin (\sqrt\lambda ...
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21 views

PDEs, infinite vector spaces, and functional analysis

Suppose I have a PDE for some quantity $q_{t}$:$= q(\boldsymbol{x},t)$ at some time $t$ on a 2D vector space $\boldsymbol{x} = (x_{1},x_{2})$, that looks, for the sake of the question, something like ...
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How to find first-order quasi-linear PDEs form second-order quasi-linear PDE?

Transform $u_{tt} u_{xx}-u^{2}_{tx} + uu_{tt} + 1=0 $ into first-order quasi-linear PDEs. Attempt: $u_{tt}(u_{xx}+u)=(u_{tx}-1)(u_{tx}+1)$ To get $u_{tt} = u_{tx}-1\Rightarrow u_t = u_x ...
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1answer
11 views

Question about Convergence Definition for Finite Difference Scheme

I have a question about the Convergence Definition for Finite Difference Scheme. The definition is given by Convergence: for one-step schemes approximating a IBVP to be convergent we compare ...
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Elliptic boundary value problems and elliptic partial differential equations

I am interested in the relation between the definition of an 'elliptic boundary value problem' and an 'elliptic partial differential equation'. From the wiki entries it seems that 'elliptic boundary ...
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monotonicity of a $C^2(\mathbb{R})$ function

Let $c>0$ and $u(\xi)\in C^2(\mathbb{R})$ be a solution of $$ (D(u)u')'+cu'+g(u)=0,\qquad '=\frac{d}{d\xi} $$ with $c$. The assumptions for $D$ and $g$ are respectively $$D\in C([0,1])\cap ...
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1answer
28 views

Euler equations

What's the relationship between the incompressible, free surface euler equations and the euler equations? Are the latter just the former when the free surface is identically zero?
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Local existence for semilinear wave equations

After having done some research, I could not find a reference for the following. Suppose I have a problem of the following type, on $(t,r) \in \mathbb{R} \times \mathbb{R}^2$: $$ \begin{array}{ll} ...
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Question on the local existence theory for the classical solution for the incompressible fluid dynamics equation

For the incompressible fluid dynamics system \begin{equation} \begin{split} &\nabla\cdot v=0,\\ &\dfrac{\partial v}{\partial t}+v\cdot \nabla v+\nabla p=A(S,D)v, \end{split} \end{equation} ...
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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, $$ ...
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1answer
23 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 ...
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32 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 $H^s(\mathbb{R^3})$?
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2answers
45 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 ...
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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 $\nabla\cdot v=0$. For any function, ...
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1answer
26 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 ...
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1answer
36 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 ...
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1answer
45 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 ...
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1answer
49 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} ...
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1answer
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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 ...
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0answers
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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 ...
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30 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 ...
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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?
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1answer
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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 ...
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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 ...
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1answer
31 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 ...
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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 ...
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3answers
81 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 ...
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1answer
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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 ...
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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 ...
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$\sup$ 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 ...
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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 $$u(t,x)=h(x-ta(u(t,x))).$$ ...
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1answer
25 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 ...
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1answer
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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 ...
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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)= ...
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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 $ ...
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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 ...
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1answer
31 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 ...
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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 ...
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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 ...
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1answer
35 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 $$||u||_{W^{2,2}(\Omega)} \le C ||\Delta u ||_{L^2(\Omega )} $$
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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 $M$, ...