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

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4
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1answer
168 views

Solving Wave Equation with Initial Values

I am trying to solve the wave equation: $u_{tt}$ = $u_{xx}$ With initial values: $u(x,0) =\begin{cases} x^3 - x, &\text{for }|x|\le 1,\ \\0, &\text{for }|x|\ge1\end{cases}$ $u_t(x,0) ...
4
votes
2answers
176 views

Why do all solutions to this equation have the same form?

In this paper on page 45, the authors state that Let's assume we know that $$ w \times dw = d\varphi + \sum_{j=1}^3\alpha_jdx_j, \tag{1}$$ where $\varphi \in H^1(\mathbb{R}^3, \mathbb{R})$, ...
4
votes
1answer
137 views

Relation of the kernels of one bounded operator and its extension

Sorry for this long and formal post. The application in PDEs is mentioned just at the end. Let $$V \hookrightarrow H \text{ and } Q_H' \hookrightarrow Q',$$ where $V$ and $Q$ are Banach and $H$ ...
4
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1answer
1k views

Poincare Inequality

In page 290 of this book, Evans prove the Poincare inequality (Theorem 1) arguing by contradiction. Is there a direct proof of this theorem (Theorem 1) without arguing by contradiction?
3
votes
1answer
271 views

Interior $H^2$ regularity - applying “Cauchy's inequality with $\epsilon$”

This is from PDE Evans, 2nd edition, pages 327, 328, and 330. I have a question regarding one piece of the proof. The theorem concerned is THEOREM 1 (Interior $H^2$ regularity) which is stated on ...
3
votes
2answers
185 views

Fourier Transform of Poisson Equation

While trying to solve the Poisson Equation by using Green's Function I have to Fourier transform the equation i.e $$-\nabla^{2}\phi(r)=\rho(r).$$ In the book after Fourier transform, the solution ...
3
votes
2answers
553 views

A question concerning measurability of a function

Let $\Omega\subset \mathbb{R}^n$ be bounded and let $X:=H^1(\Omega)$. Let $a:\Omega\times \mathbb{R} \to \mathbb{R}, (x,z)\mapsto a(x,z)$ be a bounded function such that $a(x,.)$ is continuous on ...
3
votes
1answer
362 views

Exercise from Stein with partial differential operator

I have again something from Stein-Shakarchi I would really appreciate some help with. Any references are also welcome! Suppose $L$ is a linear partial differential operator with constant ...
2
votes
1answer
50 views

Existence of solution of $\frac{\partial f}{\partial t}=-\Delta f+|\nabla f|^2-R(x,t)$

When $t=t_0$, $f(x,t)=f_0(x)\in L^2(U)$. $t\in [0,t_0]$ and $U$ is a open subset of $R^n$.$R(x,t)$ is bounded and smooth about $x$ and $t$. I don't whether suitable the conditions is ,if not, please ...
2
votes
0answers
35 views

Linearization of PDE: $0$ is an eigenvalue since all translates of travelling waves are also travelling waves

Consider the following PDE: $$ u_t=u_{xx}+f(u)-w,~~~~~w_t=\varepsilon (u-\gamma w),~~~~~~~~~(1) $$ where $f(u)=u(u-a)(1-u), 0<a<\frac{1}{2}, \varepsilon,\gamma >0, \varepsilon\ll 1,\gamma\ll ...
2
votes
1answer
86 views

Products of distributions ill-defined

This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
2
votes
1answer
112 views

:How to find the general solution of $(y+ux)u_x+(x+yu)u_y=u^2-1$?

question : find the general solution of $(y+ux)u_x+(x+yu)u_y=u^2-1$ $\frac{dx}{dt}=y+ux,\frac{dy}{dt}=x+yu,\frac{du}{dt}=u^2-1$ I dont know how to start. is this quasilinear ? edit 1: tried ...
2
votes
2answers
120 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 $L$ ...
2
votes
1answer
637 views

Use Lax-Milgram theorem to prove the existence of weak solution for an elliptic equation

Let $\Omega$ an open bounded and regular domain to $\mathbb{R^n}$ and let $\{\overline{\Omega_1},\overline{\Omega_2}\}$ a partition of $\Omega.$ $\bar{\Omega} = \bar{\Omega_1} \cup \bar{\Omega_2}.$ ...
2
votes
1answer
153 views

Dimensions analysis in Differential equation

Differential equation of solitary wave oscillons is defined by, $$ \Delta S -S +S^3=0 $$ How can we write this equation as, \begin{equation} \langle(\vec{\nabla}S)^2\rangle+\langle S^2\rangle-\langle ...
2
votes
1answer
132 views

Bounding the solution of a wave equation in 3 dimensions

Let $u:{\mathbb{R}^ + } \times {\mathbb{R}^3} \to \mathbb{R}$ be a solution of the Cauchy problem $\left\{ \begin{gathered} {u_{tt}} - \Delta u = 0 \\ u\left( {0,x} \right) = {u_0}\left( x ...
1
vote
1answer
48 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 $-\triangle u = \lambda u$, ...
1
vote
2answers
68 views

Interior $H^2$ regularity - inequalities over regions $U$ and $W \subset U$

This question is a direct continuation of my previous question. However, this one is requesting only for a relatively simple explanation. Note: $W \subset U \subset \mathbb{R}^n$. On page 330 of PDE ...
1
vote
1answer
92 views

Extension Theorem

From PDE Evans, 2nd edition, pages 268-270. My question is at the bottom of this post. THEOREM 1 (Extension Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Select a bounded open set $V$ ...
1
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1answer
85 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: $u_{tt}=u_{xx}-2u_t$.
0
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1answer
43 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 ...
0
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1answer
167 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 ...
0
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1answer
387 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 ...
0
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1answer
301 views

Show $au_x+bu_y=f(x,y)$ gives $u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds +g(bx-ay)$ if $a\neq 0$.

For my homework I am asked to do the following: Solve $au_x+bu_y=f(x,y)$, where $f(x,y)$ is a given function. If $a\neq 0$ write the solution in the form $$u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds ...
8
votes
1answer
428 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 ...
5
votes
1answer
1k 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 ...
5
votes
2answers
271 views

How to prove that $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ and $\lVert u \rVert_{H^2}$ are equivalent norms?

How to prove that $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ and $\lVert u \rVert_{H^2}$ are equivalent norms on a bounded domain? I hear there is a way to do it by RRT but any other way ...
4
votes
1answer
184 views

Nonlinear equation (oscillon) comparison

Lagrangian for a spherically-symmetric, real scalar field in d spatial dimensions, $$L=c_d \int r^{d-1}dr\left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
3
votes
1answer
163 views

Construction bump function with positive Fourier transform

I am looking for the construction of a smooth bump function, $f$, mapping the real line to itself which has two special properties: (1) $f$ is constant on some interval in its support (for instance ...
3
votes
2answers
144 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 ...
3
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1answer
133 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 ...
3
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2answers
514 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. ...
2
votes
2answers
169 views

How can I solve these pde's?

Three different problem I got: 1.. $xu_x+2x^2u_y-u=x^2e^x$ and $u(x,x^2+x)=xe^x+x^2$ 2.. $yu_{xx}+(x+y)u_{xy}+xu_{yy}=0, \quad x\neq y$ 3.. $(y+xu)u_x+(x+yu)u_y=u^2-1$ Couldnt even start. Could ...
2
votes
1answer
161 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 & ...
2
votes
1answer
952 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 ...
2
votes
1answer
344 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 ...
1
vote
1answer
174 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 ...
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1answer
36 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: ...
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2answers
113 views

about weak convergence in $L^2(0,T;H)$.

Exercise Suppose $H$ is Hilbert space and $u_k$ converges weakly to $u$ in $L^2(0,T;H)$. Suppose further we have the uniform bounds $\mathrm{esssup}_{0≤t≤T} ||u_k(t)||≤C$. Then ...
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0answers
43 views

Inequality $\Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le C\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$

For complex-valued functions $f_1, f_2, f_3:\mathbb R\to\mathbb C$, I want to know that the following inequality holds: $$ \Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le ...
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0answers
199 views

Separation of variables - Facing difficulty in application of Orthogonality Condition

I am trying to model transient cooling of two concentric cylinders sharing an interface along the length. Heat will flow in radial direction only. Radius of inner cylinder (or inner radius of outer ...
1
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1answer
2k views

Solving Laplace's equation using finite differences

I having coding in MatLab to approximate solutions to Laplace's equation in 2D using finite differences. I was able to do it without much problem. I learnt about how to implement this using this: ...
1
vote
1answer
156 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
330 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 + ...
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votes
2answers
83 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) ...
0
votes
1answer
980 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 ...
0
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1answer
81 views

I need to find a specific maximum principle

I need a maximum principle that says: If $L$ is an elliptic operator and $u$ is a positive function ($u\in C^2(\Omega)$, with $\Omega\subset\mathbb{R}^n$ an unbounded domain) such that $Lu\geq0$ ...
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1answer
124 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 ...
-1
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1answer
119 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 ...
117
votes
5answers
8k 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 ...