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

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variational analysis

Let $I$ be a functional over a Hilbert space, as in the Mountain pass theorem. Can the condition that there exists $v$such that $I(v)\leq 0$ for $||v||>r$ be replaced by $I(v)=0$?.
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1answer
22 views

Generality of a solution of a nonlinear PDE

Given the nonlinear PDE $$ \partial^2\phi+\phi^3=0 $$ I consider the characteristics $\xi=\kappa\cdot x$. Then I look for a solution in the form $$ \phi(x)=a\cdot\chi(\xi). $$ Provided ...
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1answer
14 views

Weak formulation of Poisson equation

I am learning about partial differential equations and I would like clarification on the weak formulation of the following 1D poission equation. Here is what I learned: $-u_{xx} = f(x)$ in $\Omega = ...
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1answer
12 views

Understanding Lipschitz domain

Here is the definition of Lipschitz domain given by Wikipedia. Let n ∈ N, and let Ω be an open subset of Rn. Let ∂Ω denote the boundary of Ω. Then Ω is said to have Lipschitz boundary, and is called ...
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Introduction to the boundary element method, with convergence analysis

I've looked through several textbooks on the BEM, but while they show how to set up the boundary integral formulation and how to discretize it, none give any indication of what the convergence ...
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1answer
20 views

Solution of a Partial Differential Equation

Problem statement Solve $\frac{\partial f}{\partial x}-x\frac{\partial f}{\partial y}=y$ using the change of variables $\left\{\begin{matrix} u=ax^2+y \\ v=x \end{matrix}\right.$ for a suitable ...
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Uniqueness for special elliptic-parabolic systems

I have a general question considering uniqueness results (or methods for establishing uniqueness) in the case of special elliptic-parabolic systems arising, for example, in fluid dynamics in porous ...
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19 views

Does the $C^1$ norm for the boundary of a ball depend on the radius?

Does the $C^1$ norm for the boundary of a ball with radius $r$ depends on the radius $r$? I use the classical definition of the $C^k$ boundary of a bounded domain: $\Omega$ is a bounded domain in ...
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Using Markov Property in solving PDE/SDE

I am solving the PDE I used Feynman-Kac and eventually arrived at $F(t,x)$ $ = E[X_T^2|X_t = x]$ $ = E[(X_t \pm \sigma (W_T -W_t))^2|X_t = x]$ (iirc) So, I try to evaluate $E[(X_t \pm \sigma ...
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1answer
26 views

Solve the PDE by the method of characteristics.

I am trying to figure out where my solution went wrong. I am off by a factor of two. $$ u_x + u_y + u = e^{x+2y}$$ I first found that the characteristic curves are determined by $$\frac{dy}{dx} = 1 ...
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1answer
10 views

Solution of a heat transport PDE

Solve the system of partial differential equations: $$(1)\space\space \frac{\partial g}{\partial t} + v\frac{\partial g}{\partial x} = -k_1\left(g-h\right)$$ $$(2)\space\space \frac{\partial ...
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equivalence of the characteristic curve and the associated ODE [on hold]

Show that $f=f(x,y)=c$ is a characteristic curve, i.e., $f$ satisfies the equation $$A(f_x)^2+Bf_xf_y+C(f_y)^2=0.$$ if and only if $f$ satisfies $A(y')^2-By'+C=0$, where $A,B$ and $C$ are the ...
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Problem about deformation theorem

I'm reading Evans PDE, on chapter 8.5 the proof of deformation theorem about the calculus of variation. On page 504 Evans wrote on the top: "we verify that the map $u\to dist(u,A)+dist(u,B)$ is ...
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$f$ is a characteristic of PDE if and only if $f$ satisfies $A(y')^2-By'+C=0$ [on hold]

How can I show that the following statements are equivalent? i. $f=f(x,y)=c$ is a characteristic curve of the general form of PDE, i.e., $f$ satisfies the equation $$A(f_x)^2+Bf_xf_y+C(f_y)^2=0.$$ ...
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1answer
21 views

References for mean curvature flow in Riemannian manifolds

I am interested in mean curvature flows (MCFs) in Riemannian manifolds. But most textbooks about MCF seem to treat mainly MCFs of hypersurfaces in $\mathbb{R}^{n+1}$. Should I study them before ...
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15 views

Comparison theorem for parabolic partial differential equations

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $J\subseteq\mathbb{R}$ be an intervall $T\in(0,\infty)$ and $f\in C^0\left(\overline{\Omega}\times[0,T]\times J\right)$ be locally Lipschitz ...
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Second order linear ODE arising from Euclidean heat kernel

When solving for the Euclidean heat kernel $H(t,x,y) \in C^{\infty}((0,\infty) \times \mathbb{R}^n \times \mathbb{R}^n)$, one way to proceed is to look for a solution in the form $H(t,x,y) = ...
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Solving a PDE to Yield Determining Equations

I'm going through an example in Peter Hydon's book "Symmetry Methods for Differential Equations" which finds the basis for the Lie Algebra of the point symmetry generators for Burgers' equations. ...
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35 views

Diffusion of a chemical species inside a Y-shaped tube

I'm trying to model diffusion of a chemical species X inside a Y-shaped tube, whose diameter (thickness) is constant everywhere. The diffusion constant of X is $D$ ($\mu$m$^2$/s), so the concentration ...
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8 views

Solve 1D wave equation on half-line using method of images

I'm trying to solve $\theta_t - D\theta_{xx} = f(x,t)$ on the half-line $0 < x < \infty$ for $0< t < \infty$ given boundary and initial conditions $\theta(0,t) = h(t)$, $\theta(x,0) = ...
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1answer
31 views

Partial differentiation or normal differentiation

Consider the function $$ f(x,y) = \begin{cases}\frac{xy(x^2-y^2)}{x^2+y^2}, & (x,y)\neq(0,0)\\ 0, & \text{otherwise.}\end{cases} $$ Compute $$\frac{d^2f}{dxdy}(0,0)$$ and ...
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34 views

$-\Delta u = f$ in $L^2(0,T;H^{-1}(\Omega))$ (as opposed to $H^{-1}(\Omega)$)

Why does nobody consider the equation $-\Delta u = f$ in the space $L^2(0,T;H^{-1}(\Omega))$? Eg. given $f \in L^2(0,T;L^2(\Omega))$ find a solution $u \in L^2(0,T;H^1_0(\Omega))$ such that ...
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1answer
22 views

Differentiation and PDE Theory

I have been given the following two definitions: 1) $D^ku$ is the set of all derivatives of order k of u 2) Let $\Omega$ be a non-empty subset of Euclidean space $\mathbb{R}^N.$ An expression of the ...
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what is the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE?

I want to know the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE when right hand side of the elliptic operator is $L^1.$ Also how does right hand side with ...
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29 views

Weak solution of elliptic equation depends continuously on parameter

Suppose I have a weak formulation of the form: find $u \in H^1_0(\Omega)$ such that $$\int_\Omega b(p)(\nabla u \nabla v + \lambda uv)=0$$ holds for all $v \in H^1_0(\Omega)$ where $b:[a,b] \to ...
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27 views

Essay about PDEs

I've taken an introductory course in PDEs and I have to write an essay of 4-8 pages on a topic in partial differential equations. The topics we touched on are: First order linear partial ...
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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 being forced to be linearly related to others), then the condition that ...
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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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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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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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2answers
39 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
53 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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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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56 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
21 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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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
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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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1answer
37 views

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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23 views

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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29 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
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 ...
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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
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 ...
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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 $\nabla\cdot h=0$. For any function ...
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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 ...