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

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1answer
87 views

The functional $I[w] = \int_U \frac{1}{2} |Dw|^2 - fw \, dx$ is weakly lower semicontinuous

I am studying calculus of variation, and I need to prove that $I[w] = \int_U \frac{1}{2} |Dw|^2 - fw \, dx$ with $f \in L^2(U)$ is weakly lower semicontinuous on $H_0^1(U)$. In classes, I only ...
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2answers
214 views

Random diffusion coefficient in the Fourier equation

I'm stuck on the following simple problem: It's given the Fourier equation: $$\partial_t{u(x,t)}=\partial_x[k(t)\partial_xu(x,t)]$$where the diffusion coefficient $k(t)$ is a random variable with a ...
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1answer
168 views

How to prove that $\iint\frac{|y|^2}{s^2}\,dy\,ds=4$?

Let $U\subset\mathbb{R}^n$ be an open set, $\Phi$ the fundamental solution of heat equation, $T>0$, $r>0$, $x\in\mathbb{R}^n$ and $t\in\mathbb{R}$. Defines $U_T=U\times(0,T]$ and $$E(x,t;r)=\{(y,...
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1answer
92 views

Method of charactersitics and second order PDE.

How may the method of characteristics be applied to solve a second order PDE? For instance, to solve the equation: $u_{tt}=u_{xx}-2u_t$.
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1answer
122 views

$(\partial_{tt}+\partial_t-\nabla^2)f(r,t)=0$

Hi I am trying to find the kernel of the linear differential operator $D$ $$ D\equiv\partial_{tt}+b\partial_t-a\nabla^2,\quad a,b>0. $$ We have $$ \nabla^2\equiv \frac{1}{r}\partial_r(r\partial_r)-...
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1answer
598 views

sobolev spaces - product of two functions

I am working in a exercise, to my solution works I need the following affirmation is true: Let $\varphi : R \rightarrow R$ a convex and smooth function. Let $u \in H^{1}(U)$ a bounded function and $v ...
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1answer
669 views

Evans PDE problem 9,Chapter 6

Let $U\subset \mathbb{R}^n$ be an bounded domain with smooth boundary. Assume $u$ is a smooth solution of $$ Lu=-\sum_{i,j=1}^na^{i,j}u_{x_ix_j}=f \ \text{in} \ U, \ \ u=0 \ \text{on} \ \partial U,...
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1answer
266 views

Euler-Lagrange equation [duplicate]

This is PDE Evans, 2nd edition: Chapter 8, Exercise 2: Find $L=L(p,z,x)$ so that the PDE $$-\Delta u + D\phi \cdot Du = f \quad \text{in }U$$ is the Euler-Lagrange equation corresponding to the ...
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1answer
46 views

Variation under constraint

I always can't compute right.$u=u(x),R=R(x)$ and $\tau$ is constant, and $M$ is compact manifold.If $u$ is the minimizer of $$ \inf\{\int_M [\tau(4|\nabla u|^2+Ru^2)-u^2\ln u^2-nu^2](4\pi\tau)^{-n/...
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1answer
448 views

PDE - Feynman-Kac vs. finite difference methods

I've heard that in greater than three dimensions, it's more efficient to solve a second-order parabolic PDE using a Monte-Carlo method based on the Feynman-Kac formula that it is to use finite ...
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1answer
2k views

Entropy Solution of the Burger's Equation

I am working on the following problem, which gives the Burgers' equation $u_t + uu_x=0$ with the initial data $g(x)=1, x < 0$, $g(x)=2, 0 < x < 1$, $g(x)=0, x > 1$. It then asks to ...
4
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1answer
84 views

Range of Influence of the Wave Equation?

Suppose $u$ is a solution of the two-dimensional wave equation $$u_{tt}-c^2 \Delta u=f(x)$$ with initial values $u(0),u_t(0)$ that have support on the disc $x_1^2+x_2^2 \le 1$. Up to what time can ...
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1answer
432 views

Solve $u_x+u_y=1$

I am asked to solve $$u_x+u_y=1$$ If is was homogeneous i.e., $u_x+u_y$ the answer would be $u(x,y)=f(y-x)$ where $f$ is an arbitrary function. I have found the following set of solutions: $$u(x,y)=\...
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2answers
570 views

Differentiating a boundary condition at infinity

A typical boundary condition for an initial boundary value problem is $$ \lim_{x\rightarrow\infty} T(x,t) = T_\infty.$$ For example, this might be the temperature at the end of a very long rod. ...
3
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2answers
145 views

Solve the pde $u_t(x,t)=u_{xx}(x,t)-bu(x,t)+q(t)$ for $u(x,t)$

I have the example pde $u_t(x,t)=u_{xx}(x,t)-b(t)u(x,t)+q_0$, where $b(t)$ is a function of only $t$ and $q_0$ is a constant, $0<x<\pi$, $t>0$. The subscripts denote derivatives. I also have ...
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1answer
91 views

Why is a differential equation a submanifold of a jet bundle?

I'm reading The geometry of jet bundles by D.J Saunders and struggle with the definition of a differential equation 6.2.23. on page 203. First of all, Saunders introduces a differential operator ...
3
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1answer
139 views

variational question

Let $\Omega$ a bounded domain, connexe and regular, and let $f \in L^2(\Omega).$ Let the variational problem: Find $u \in H^1(\Omega)$ such $$\int_{\Omega} \nabla u \nabla v dx + (\int_{\Omega} u dx)(...
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1answer
118 views

Harmonic Function with linear growth

We want to find harmonic functions $w$ in (say) $\mathbb{R}^2$ that are zero on $\{y=0\}$ with the linear growth bound \begin{equation} \sup_{\mathbb{B}_R} |w| \leq C(1+R) \end{equation} where $C>0$...
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1answer
194 views

Finding a value a for topologically conjugacy between two flows

Let A be a hyperbolic matrix such that all solutions of $\overrightarrow x' = A \overrightarrow x $ tend to the origin at t goes to infinity, and suppose B = $\begin{bmatrix}a-3 & 5 \\ -2 & a+...
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1answer
1k views

Solving a partial differential equation using method of characteristics

I keep getting stuck and have a hard time understanding my professor, so I'm hoping to get some help here. The question is: Solve the partial diff/eq: ${\partial u}\over{\partial t}$ + $c {{\partial ...
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1answer
691 views

Finding the uniquely determined region of a PDE

(a) Solve the equation $yu_x+xu_y=0$ with the condition $u(0,y) = e^{-y^2}$. (b) In which region of the xy plane is the solution uniquely determined? I did the first part but I don't understand ...
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2answers
126 views

Differential operators confussion

I want to solve this PDE: $$u_t-6uu_x+u_{xxx} = 0\,(1)$$ with the Inverse Scattering Method. This method is based on showing that the above equation can be expressed as $$L_t=LB-BL,\,(2)$$ where $L$ ...
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1answer
395 views

A problem about convergence…

I am studying an article of Berestychi-Caffarelli-Niremberg - Monotonicity for elliptic equations in unbounded Lipschitz domains, and I don't understand a convergence in the demonstration of the lemma ...
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1answer
53 views

Maximum principle of eigenvalue-like problem for harmonic equation

I am trying to identify for which $\lambda$ so that I can obtain the Maximum principle, that is, $$\sup_{\overline{\Omega}}u\leq \sup_{\partial \Omega}u$$ for equation $-\triangle u = \lambda u$, ...
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1answer
245 views

a vector calculus problem when coping with problem 9 chapter 5 evans

Hi I was trying to understand the solution given by Ray Yang in the post question 9 - chap 5 evans PDE It gets to the sort of things I am quite bad at... When I get to the point $$\int_{\Omega}|Du|^...
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1answer
38 views

general conditions for reverse poincare inequality

I'd like to know when the reverse Poincare inequality is true: Given a bounded domain $\Omega$, and $f \in L^2(\Omega)$, under what conditions on $f$ (neglecting the trivial constant case) and/or $\...
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1answer
43 views

Laplace Operator in $3D$

I am looking to find the radial part of Laplace's operator in three dimensions. I looked up Laplace's operator in spherical coordinates and from there I guess the radial part is: $\frac{\partial^2}{\...
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1answer
163 views

The differential equation $xu_{x}+yu_{y}=2u$ satisfying the initial condition $y=xg(x),u=f(x)$

I was thinking about the following problem : Find out which of the following option(s) is/are correct? The differential equation $xu_{x}+yu_{y}=2u$ satisfying the initial condition $y=xg(x),u=f(x)$, ...
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0answers
355 views

Integrating pde backwards in time

I need to solve a partial differential equation backwards in time. In other words, I have the discrete partial differential equation $$ \begin{align} p_\text{new} & = p_j + (dt/dx)q_jf'(u_j)...
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1answer
60 views

Question using Young inequality

In fact i am looking for $\alpha_1,\alpha_2,\alpha_3$ such that the following inequality is true:$a^\frac{1}{80}\le \alpha_1\frac{a^{\frac{1}{4}}}{b^{\frac{3}{4}}c^6}+\alpha_2b^{\frac{1}{16}}+\alpha_3 ...
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1answer
1k views

Solve the partial differential equation $u_t + uu_x=0$ [duplicate]

Solve the following partial differential equation $u_t + uu_x=0$ with $u=u(x,t)$ and $u(x,0)=x$. I am having trouble in applying the SIDE CONDITION. The Characteristics are $dx/dt$=$u$, here u is ...
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1answer
55 views

Sequence of functions: Convergence

For each of the following, give an example of a sequence of functions $f_n$ that converges to f A. uniformly but not in the mean square sense. B. in the mean square sense but pointwise nowhere. I ...
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2answers
84 views

Does the following have a solution for f(x,y)?

I have the following equations: \begin{equation} {1\over f(x,y)} {\partial f(x,y) \over \partial x} \alpha(x,y) + {1\over f(x,y)} {\partial f(x,y) \over \partial y} \beta(x,y) = \gamma(x,y) \end{...
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1answer
58 views

$C^2 $embedded in holder space?

I have a question on Holder space.Is the space $C^2(\mathbb{R}^n)$ embedded in $C^{1,\alpha}(\mathbb{R}^n)$? I think in the bounded domain case, this should be true. But what if in the case where the ...
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1answer
138 views

Diffusion equation problem

Heat conduction is described by the diffusion equation $$u_t = k u_{xx},$$ where $u(x,t)$ is the temperature at $x$ at time $t$ and $k$ is some constant. A homogeneous thin pipe occupying the region $...
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1answer
123 views

Proving $\frac{200}{\pi}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)\cosh\left(\frac{\pi}{2}(2n+1)\right)}=25$

I solve a partial differential equation (Laplace equation) with specific boundary conditions and I finally found the answer: $$U(x,y)=\frac{400}{\pi}\sum_{n=0}^{\infty}\frac{\sin\left((2n+1)\pi x\...
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8answers
48k views

Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations? [closed]

Moderator Notice: I am unilaterally closing this question for three reasons. The discussion here has turned too chatty and not suitable for the MSE framework. Given the recent pre-print of ...
123
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5answers
9k views

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
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5answers
3k views

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and ...
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2answers
3k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
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5answers
4k views

Blow up of a solution

What exactly does blow up mean, when people say, for example, that a solution (to a pde (say)) blows up. Thanks.
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2answers
1k views

Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
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1answer
460 views

Dual Bases: Finite VS Infinite Dimensional Spaces

Motivation I know that in finite dimensional spaces like a $n$ dimensional Euclidean space $\mathbb{E}^n$, for every basis $G=\{g_1,g_2,...,g_n\}$ we can define a dual basis $G'=\{g^1,g^2,...,g^n\}$ ...
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2answers
2k views

A Problem in Evans' PDE

Problem 7 in §6.6 states as follows: Let $u\in H^1(\mathbb{R}^n)$ have compact support and be a weak solution of the semilinear PDE $$-\Delta u+c(u)=f\,\,\text{ in } \mathbb{R}^n,$$ where $f\...
9
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1answer
2k views

Elliptic Regularity Theorem

I want to collect some results on elliptic regularity. The problem I consider is \begin{align} Lu&=f,&in \quad U,\\ u&=g,&on \quad \partial U.\tag{1} ...
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4answers
532 views

Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...
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2answers
785 views

Relation of Brownian Motion to Helmholtz Equation

one can obtain solutions to the Laplace equation $$\Delta\psi(x) = 0$$ or even for the Poisson equation $\Delta\psi(x)=\varphi(x)$ in a Dirichlet boundary value problem using a random-walk approach, ...
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3answers
1k views

Is Lipschitz's condition necessary for existence of unique solution of an I.V.P.?

Is Lipschitz's condition necessary condition or sufficient condition for existence of unique solution of an Initial Value Problem ? I saw in a book that it is sufficient condition. But I want an ...
8
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2answers
672 views

what is separation of variables

I was pushing my way through a physics book when the author separated the variables of the Schrödinger equation and I lost the plot: $$\Psi (x, t) = \psi (x) T(t)$$ can someone please explain how ...
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1answer
379 views

Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...