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

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What is the “Enstrophy Miracle”?

I read that one of the main differences between the establishing global regularity for the Navier-Stokes equations for viscous and incompressible flow in 2D in 3D is that in 2D there is something ...
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32 views

Would a Counterexample to Navier-Stokes Problem be Sufficient?

Given the problem statement for the Navier-Stokes Existence & Smoothness problem from the Clay Institute website, wouldn't one need to show only one counterexample to the conjecture of global ...
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1answer
29 views

No non zero solution to E.V.P in $L^p$

Can you show that: If for some $1\leq p\leq \infty$ function $f\in L^p(\mathbb{R}^n)$ solves $\Delta f-\lambda^2 f=0$ then $f\equiv 0$. (This is essentially uniqueness of solution to homogenous ...
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2answers
50 views

heat equation with perfectly insulated end

In one of my tutorial question about $1$-dim heat equation,a question about heat equation with pefectly insulated end at $x=0 $ and $ x=l$ with $u(x,t)$ as temperature function,TAs used as perfectly ...
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24 views

Shooting method for PDE

There is a paper which I was going through (attached), where there are two equations which I would like to solve. Eq 16 and 17 based on 18 and 19 using shooting methods. Problem is the boundary ...
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21 views

Solving an Inhomogeneous Wave Equation with free end boundary conditions.

Solve D.E. $u_{tt}=4u_{xx}, 0 \leq x \leq \pi, -\infty < t < \infty$ B.C. $u_x(0,t) =-1, u_x(\pi, t) =1$ I.C $u(x,0) = \frac{x^2}{\pi}-x+2cos(3x), u_t(x,0)=cos(x)$ Attempt: My problem is ...
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39 views

Why aren't numerical methods sufficient to show existence and uniqueness?

Suppose somebody is modeling a solution (or set of solutions) to a particular partial differential equation (Navier-Stokes maybe) via some software that makes use of some numerical method(s) to solve ...
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19 views

Different Classifications of Solutions to Differential Equations

I've read online that there are solutions to differential equations that are weak and some that are strong. On top of that, I've read that there are certain types of weak solutions that are known as ...
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1answer
41 views

Conditions under which Laplace's equation inside rectangle has solution

Consider $u(x,y)$ satisfying Laplace's equation inside a rectangle $(0 < x < L)$ and $(0 < y < H)$, subject to the boundary conditions: $$\begin{cases} u_{x}(0,y)=0 \\ u_{x}(L,y)=0 \\ ...
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18 views

Monodromy Matrix of the Schrodinger operator

I have a quick question and I would be very grateful if someone can help. I know the Schrodinger operator is self adjoint and my intuition tells me its associated monodromy matrix is also self ...
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1answer
39 views

How to finish the computation of $u(x)=\int_{B_R(0)}\frac{1}{|y-x|}dy$?

Let $B_R(0)$ be a ball in $\mathbb R^3 $ and define $$u(x)=\int_{B_R(0)}\frac{1}{|y-x|}dy$$ Prove that $$ u(x) = \begin{cases} \frac{2}{3}\pi(3R^2-|x|^2) & \quad \text{for $0 \le|x|\le R$ ...
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48 views

Weak convergence of determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
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10 views

How do I show that the maximum principle holds for the PDE $u_t = u_{xx}$?

We have the PDE $u_t = u_{xx}$ with initial conditions $u(x, 0) = u_0(x)$ given. How does one show that $u(x, t) \leq \mathrm{max}_x\;u_0(x)$ for all $x$ and $t$? I later have to show that a maximum ...
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40 views

An interesting proof using Green's representation formula?

Let $B_R(0)$ be a ball in $\mathbb R^3 $ and define $$u(x)=\int_{B_R(0)}\frac{1}{|y-x|}dy$$ Prove that $$ u(x) = \begin{cases} \frac{2}{3}\pi(3R^2-|x|^2) & \quad \text{for $0 \le|x|\le R$ ...
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Inverse continuous

According Dehman, Gérard and Lebeau in "Stabilization and control for the nonlinear Schrödinger equation on a compact surface" (Math. Z. (2006) 254:729–749, DOI 10.1007/s00209-006-0005-3) is claimed ...
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13 views

Generalized D'Alembert's wave equation with Sitter's parameter

I am trying to solve the generalized D'Alembert's wave equation with Sitter's parameter, $u= u(x,t)$: $$ (1+\frac{x^2}{2})\frac{\partial^2u}{\partial x^2}+\frac{2xt}{R^2}\frac{\partial^2u}{\partial t ...
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35 views

Does the system of equations always have a nontrivial solution?

$f:[0,1]^2\to R_+$ is a continuous conditional density function. For $g,h\in C$ on $\{(x,y)\in [0,1]^2|x\geq y\}$, the system of equations is given by$$ \frac{\partial g}{\partial x}\leq ...
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16 views

Auxiliary Backward problems

I am reading a paper that deals with a certain finite element method. We have the weighted Sobolev space $\mathcal{W}_{\mu}$ that consists of functions for which the norm $\Vert u \Vert_{\mu}^{2} ...
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41 views

Why such strange separation of variables?

In this article the authors give the following expansion of wavefunction of three-body system (equation $(16)$ in text): $$\Psi(\textbf{x},\textbf{y})=\sum_{q=\lambda}^l\psi_q(\textbf x^2,\textbf ...
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use energy method to show that $C^{2}$ solution depends uniquely on Cauchy data

Consider the initial value problem $u_{tt}-c^{2}u_{xx}+\alpha u_{t}=0$ for $0<x<1$ and $t>0$ $u(0,t)=u(1,t)=0$ for $t>0$ $u(x,0)=g(x), u_{t}(x,0)=h(x)$ for $0<x<1$ where $c, ...
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1answer
21 views

PDE question, weak derivative equals 0 then it's constant

Suppose $\Omega$ is open, bounded, with $C^1$ boundary. Suppose $u\in W^{1,p}(\Omega)$ for some $p\in(1,\infty)$. Suppose $Du=0$ a.e. in $\Omega$. Prove that $u$ is a constant a.e. in $\Omega$.
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44 views

Relationship between the diffusion equation and the heat equation

In physics we have the heat equation which describes the propagation of heat $$\dfrac{\partial u}{\partial t} = \kappa \dfrac{\partial^2 u}{\partial x^2},$$ while in biomathematics we have the ...
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58 views

Why is Existence and Uniqueness for Navier-Stokes Easier in 2-D than in 3-D?

I know that existence and uniqueness for incompressible viscous flow in the 2-D case has already been established$^1$, and that doing the same for the 3-D case has yet to be shown. Not only that, but ...
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1answer
60 views

About the optimality of a capacitor

I'm studying Control of PDEs at the University from this site and these notes and there are some questions I am not capable of resolve. I've never studied EDPs... So, there are two questions related ...
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63 views

How does perturbation method guarantee its solution for the perturbed pde $\Delta u + \epsilon u^2 =0$

My question is quite simple: Suppose we are given a PDE of with a boundary condition $$ \Delta u + u^2 =0 $$ where $u=u(r,\theta), 0<r<1$ and $u(1,\theta) = \cos\theta$ with $0 \leq \theta \leq ...
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22 views

What is a function in $f\in L^n(\mathbb{R}^n)$ but $g(x)=\int_{|y|<1} \frac{|f(y)|}{|x-y|^{n-1}}dy$ is not in $L^\infty$

What is a function in $f\in L^n(\mathbb{R}^n)$ but $$g(x)=\int_{|y|<1} \frac{|f(y)|}{|x-y|^{n-1}}dy$$ is not in $L^\infty$. I have no idea where to start. Apparantly this is related to the ...
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Expressing solution of this wave equation problem in different Fourier expansion

I have managed to solve the wave equation $u_{tt} = c^{2} u_{xx}$ on the interval $[0,L]$ for $t > 0$, and subject to initial conditions $u(x,0) = f(x)$ and boundary conditions $u_{t}(x,0) = g(x)$. ...
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Reverse 1-d heat equation

I'm interested in solving the partial differential equation : $$\frac{\partial f(t,x)}{\partial t}+\frac{\partial^2 f(t,x)}{\partial x^2}=0$$ and $f(0,x)=f_{in}(x)$ with ...
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38 views

Confused about weak derivatives in Evans

I'm a bit confused about how Evans refers to derivatives at some points and if he means weak derivative. In particular on page 301 he gives the definition that if $\textbf{u} \in L^1(0,T;X)$ and ...
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1answer
25 views

Estimate for functions in Sobolev space $H^s$

Can anyone help me with this question Use the Fourier transform to prove that if $u\in H^s(\mathbb{R}^n)$ for an integer $s$ such that $s>n/2$ then $u\in L^\infty (\mathbb{R}^n)$.
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1answer
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Method of characteristics (quasilinear pde- nonlinear transport )

I want to solve the following pde ( nonlinear transport - I guess ?) $\phantom{}$ $ u_t - a(x) u_x = 0 $ with $a(x) > 0 $ and $u(x,0) = u_0(x)$ $\phantom{}$ I tried the method of ...
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31 views

Dependence of solutions on parameters in partial differential equations

In the standard homogenization problem $$-\nabla.\left(A\left(x,\frac{x}{\epsilon}\right)\nabla u^{\epsilon}(x)\right)=f\ \mbox{in } \Omega,$$ the homogenized matrix $A_0$ is given in terms of ...
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115 views

Regularity of the heat kernel

Let $(M,g)$ be a compact Riemannian manifold. Let $H:M\times M\times\mathbb{R}_{>0}\to\mathbb{R}$ be the heat kernel. i.e. $H\in C^0(M\times M\times\mathbb{R}_{>0})$ is the unique continuous ...
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1answer
26 views

two solutions in $2^{nd}$ order linear differential equations

Could you please explain why we need two solutions $y_1$ and $y_2$ (fundamental set of solutions) for determine the general solution $y=cy_1+c_2y_2$ for a $2^{nd}$ order linear differential equation ...
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40 views

As I can put the Neumann boundary conditions? the Crank Nicholson scheme, the Heat Equation

How i can put the Neumann BC in my code? I tested but I get error, because the arrays are not the same size $$U_t=U_{xx},\quad 0<x<1$$ $$u_x(0,t)=0$$ $$u_x(1,t)=0$$ $$u(x,0)=f(x)$$ I have my ...
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23 views

Two definition of Fourier's transformation agrees? [duplicate]

Definition 1: If $f\in L^1(R^n)$, $\hat{f} (s)=\int _{R^n} e^{-isx}f(x)dx$ Definition 2: If $f\in L^2(R^n)$, let $f_i \in$ {Schwartz functions} such that $f_i$ converges to $f$ in $L^2$, then ...
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14 views

Existence of Self-Similar Solutions

I was going over these notes which illustrates how to find the self-similar solutions for the heat equation. I know that self-similar solutions exist for many cases of boundary layer problems in fluid ...
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2answers
69 views

Interesting Gradient problem? Don't know how to write it?

if $p=q+\varepsilon v$, where $v=\dfrac{\nabla f(q)}{\|\nabla f(q)\|}$. $f(q)=0$. $\nabla f(x) $ is not $0$ on the domain. How to prove there exists $\varepsilon_1>0$ such that if ...
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Confusion about superposition principle of the PDE and Boundary Condition of an ODE.

I want to solve a PDE like this: $\frac{\partial y}{\partial t}=a\frac{\partial ^2y}{\partial x^2}-b\frac{\partial y}{\partial x}-c y,(a,b,c\in \mathbb{R})\tag{1}$ with the boundary conditions: $ ...
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1answer
21 views

Multivariate Taylor's theorem, prove remainder is small?

Suppose $f\in C^2$ and using the second taylor expansion at $q$. We have $f(x)=f(q)+<\bigtriangledown f(q),x-q>+R_2(x)$ for $x\in \overline {B\epsilon (p)}$ How to prove that $|R_2(x)|\le ...
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19 views

Inhomogeneous heat equation

I'm trying to find inhomogeneous heat equation with neumann boundary condition.We use subtraction method but i can't use it in my problem.So where can i find more examples? Can you suggest some books ...
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16 views

Closed-form Green's Function for a given problem

I'm having a bit of trouble getting out the closed-form of the Green's function for a certain problem. I've been asked to find an answer to the following problem; $$\ddot u - u = f(x), u(0) = ...
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1answer
35 views

PDE Transport Equation(?) with Decay and Forcing Term

So I am kind of lost on how to solve this PDE IVP. $${\mathrm du \over \mathrm dt}+2{\mathrm du \over \mathrm dx}+4u=x$$ where $t>0$ and $x$ is in $R$, with the initial condition ...
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1answer
31 views

Mass Continuity Equation for Fluid - Running Into a Problem

I'm running into a problem when trying to show the mass continuity equation for a fluid, which says $$\frac{\partial \rho}{\partial t} + \left(\nabla \cdot \rho \textbf{u}\right) = 0$$ Where ...
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19 views

Maximum principle for a “modified” laplacian

Let $\Omega \in \mathbb{R}^{n}$ be bounded dmain, given $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a solution of $$\Delta u+c(x)u=0$$ where $c(x)\leq 0 \in \Omega.$ Shown that $u=0$ on $\partial ...
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1answer
41 views

Leray-Schauder fixed point theorem

I know the proof of the Schauder fixed point theorem which states Schauder fixed point theorem : If $D$ is a non-empty , convex and compact subset of Banach space $B$ and $T:D \to D$ a ...
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25 views

Method of Characteristics: trouble with ODES

I am given the PDE that follows with $x_1,x_2 > 0$ : $$x_1\partial_{x_2} u - x_2 \partial_{x_1} u = u(x_1,x_2)\\u(x_1,x_2)=g(x_1,x_2),\:with \: \: x_2 = 0 $$ Next, with the boundary conditions we ...
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21 views

Harmonic function with Neumann BC

I have a problem where I denote by $D = B(0, 2)$ the disk in the plane with radius 2 centered at the origin. I have to find a harmonic function $u(r, \theta)$ in $D$ which satisfies the additional ...
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15 views

Can you help me on this Sturm-Liouville Problem?

Here is the problem: If $L$ is the following first order linear differential operator, $L = p(x) \frac{\partial}{\partial x} $, then determine the adjoint operator, $L^*$ such that $\int_{a}^{b} ...
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1answer
47 views

interesting boundary problem of PDE

Consider the following PDE \begin{equation} \frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0 \end{equation} with ...