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

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1answer
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Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Hi this is problem 1 in Chapter 5 Evans PDE. I have trouble showing the second part, i.e. completeness of $C^{k,\gamma}$. I know there is some previous posts but I did not quite get the answers. And I ...
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0answers
5 views

Relationship between Faedo-Galerkin Method and Semigroup Method

I have multiple questions relating to Galerkin Method and the Semigroup Method of proving existence of solutions to PDEs. In the Galerkin Method, we decide on a function space, find eigenfunctions ...
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1answer
38 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
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0answers
18 views

Semigroup solution via Lumer-Phillips

Let $a$ be a coercive, bounded bilinear form on $H^1(\Omega)$, where $\Omega$ is some sufficiently "nice" region. I defined an operator $A:H^1(\Omega)\mapsto H^1(\Omega)^*$ by: $$ (Au)[v]=a(u,v)\quad ...
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1answer
23 views

Stepping backwards with Forward Euler?

Let us say I want to use Forward Euler scheme to solve the heat equation $$ \frac{\partial u}{\partial t} = -\frac{\partial^2 u}{\partial x^2} $$ in the domain $t \in (0, 1)$, $x \in (0,1)$ but ...
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0answers
15 views

principle maximum for the heat equation with Dirichlet conditions

Let us consider the Laplacian operator in a domain $\Omega\subset R^n$, with Dirichlet boundary conditions. For all $f\in L^2(\Omega),$ we denote by $S(t)f$ the solution of the equation $$ ...
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12 views

Noncharacteristic planes for nonzero PDO [on hold]

How to prove that for nonzero partial differential operator of degreem most planes are noncharacteristic?
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1answer
365 views

PDE: Greens-Function to 2D-Diffusion-Equation with time-dependent coefficient?

I have following problem: I want to construct a Greens Function for solving following inhomogeneous PDE: $$ \left(-D(t) \left( \frac{\partial ^2 }{\partial x^2} + \frac{\partial ^2 }{\partial y^2} ...
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1answer
21 views

Operator equation $Au = f$ for $-\Delta u(x)=f(x)$

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad ...
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3answers
55 views

prove that $u$ is equal a.e. to an absolutely continuous function

Prove that if $n=1$ and $u\in W^{1,p}(0,1) $ for some $1\leq p<\infty$, then $u$ is equal a.e. to an absolutely continuous function,and $u'$ (which exists a.e.) belongs to $L^{p}(0,1)$. My ...
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1answer
233 views

References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried the document Hamilton (1982) The Inverse Function Theorem of Nash and Moser, but the article is very encyclopedic. I have a ...
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0answers
22 views

Adi-method for Diffusion-reaction equation in 2d

i'm trying to solve this pde using an adi-method (alternating-direction-implicit). $\frac{d f}{d t}=D\nabla^2_{x,y} f+Q(x,y)f+C$ After discretizing, the equation looks like this. Implicit in ...
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1answer
45 views

Intuition behind eigenfunctions of the Laplacian operator

I'm reading about the notion of spectral dimension which is a measure of how particles diffuse in some space at different scales. An important aspect of spectral dimension is the ...
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1answer
32 views

How to evaluate this combination of sums and integrals?

I am reading a book on PDEs, and I am near the beginning where the author is talking about the heat equation and, specifically, solving the non-homogenous equation $u_t={\alpha}^2u_{xx}+f(x,t).$ The ...
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0answers
13 views

Non-linear hyperbolic systems

This question is about a naive approach to non-linear hyperbolic systems, thinking in the context of elasticity. To set up the problem suppose $\Omega\subset \mathbb{R}^n$ is open and bounded. ...
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0answers
17 views

Regularity of compactly supported solutions to the divergence equation: $\nabla\cdot \mathbf{v}=g$.

I have two questions, one specific and one general, on this result for compactly supported solutions to the divergence equation in star-like domains. The result which can be found in Galdi's "An ...
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1answer
24 views

Interchanging limits in $\lim\limits_{n \to \infty}\lim\limits_{j \to \infty}\int_0^T \langle u_n', w_j \rangle $ (weak time derivative)

Let $V$ be a Hilbert space which is separable. Let $u_n \in L^2(0,T;V)$ with $u_n(t,x) = \sum_{i=1}^n u_{in}(t)w_i(x)$ where $u_{in}$ are absolutely continuous on $(0,T)$ and $w_i$ are a smooth basis ...
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1answer
24 views

The meanings of some symbols in “Calculus of variations”

Could someone tell me the meanings of the "C" and its superscript "1" and subscript "0" in the equation which I have marked. Thank you very much!!!
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1answer
23 views

Does smoothness imply boundedness? Evans PDE chapter 2 Problem 18

In problem 18, enter link description here 1) I am having difficulty in extracting information in deciding the bounded for $g$ and $h$. In particular, to conclude $g$, $Dg$, $h$ $Dh$ are bounded by ...
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1answer
23 views

Prove that $\left\{u\in W_0^{1,2}(\Omega):\int_\Omega|u|^{p+1}\;d\lambda^n=1\right\}$ is well-defined and closed

Let $\Omega\subseteq\mathbb{R}^n$ be a domain with a smooth boundary $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $p>1$ such that $$p<\begin{cases}\infty&\text{, if ...
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1answer
56 views

Operator theory curiosity

I'm not an expert in operator theory... but i was wandering if there's some practical applications. For example (the first one i came up with) compared to normal calculus techniques that usually the ...
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1answer
16 views

Perturbation of PDE and Green's function

If the Green's function of a second order differential operator $L$ is $G^{(0)}$, then if I add a small perturbation $\delta L$, a Green's function $G$ for the operator: $(L+ \delta L)$ should be: ...
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1answer
16 views

about convection term in the NSE

Let $u=(u_1(x, y), u_2(x, y))$ be two dimensional vector field and consider the convection term $(u\cdot\nabla) u$ in the NSE. In some books it is usually written in this form, but in some other books ...
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1answer
23 views

First Eigenfunction of Simple Equation

Consider the interval $[-a,a]$ and the following problem: $$\phi'' + \lambda\phi=0$$ $$ \phi(\pm a) = 0. $$ The obvious sequence of orthogonal eigenfunctions seems to be $\sin(\frac{\pi n}{a}x)$ ...
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3answers
25 views

solution of wave equations in odd dimension Evans PDE

Here I am looking at the proof of theorem 2 below Here I have the following difficulties: 1) In the last two lines, the exponent changes from $\frac{n-1}{2}$ to $\frac{n-3}{2}$, why? Could anyone ...
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0answers
44 views

PDE on a Damped Wave Equation

What are the eigenvalues and eigenfunctions of $$ X''+X'-\sigma X=0 \\ \text{ with boundary condition } X(0)=X(l)=0$$ I know that for $X''-\sigma X=0 $, the eigenvalues would be $ -\left ( \frac{n ...
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1answer
30 views

show that $uv\in W^{1,r}(\Omega)$

Let $\Omega\subset\mathbb{R}^{n}$ is bounded,with $\partial\Omega\in C^{1},p,q\geq 1$,$u\in W^{1,p}(\Omega),v\in W^{1,q}(\Omega)$,show that $ uv\in W^{1,r}(\Omega)$.here ...
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0answers
32 views

What math is needed for the inverse quantum problem?

What sort of mathematical background/familiarity is necessary and/or useful in tackling the inverse quantum problem? As an applied math major with a physics minor, I'm looking at different senior ...
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0answers
16 views

reversibility scalar conservation law

I am reading here and there (see for instance Denis Serre systems of conservation laws 1- p.36), the following for which I can't spot the mistake I am making that prevents me from arriving to the same ...
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0answers
65 views

Validity of approximating a difference equation with a differential equation

Consider the following two equations: $$ \begin{cases} A_k(n+1)-A_k(n)=\beta \displaystyle \frac{ A_{k-1}(n)- A_k(n)}{\alpha+ n} + \delta_{k, \beta} \\ \\ \displaystyle ...
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1answer
23 views

Unicity of solution for a parabolic problem?

How can I show that the parabolic problem $$ \begin{cases} \partial_xu- \Delta u=0 & \mathbb{R} \times (0,+ \infty)\\ u(x,0)=f(x) & \mathbb{R} \end{cases} $$ has a unique solution? Can ...
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0answers
26 views

An advection problem with weak diffusion in asymptotic analysis.

Consider the following advection problem with weak diffusion: $$ \varepsilon\partial_{x}^2 u=\partial_{t}u+\partial_{x}u, $$ for $−\infty < x < \infty$, and $t > 0$ where $u(x, 0) = ...
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1answer
33 views

Prove $u=\ln\ln\left(1+\frac{1}{|x|}\right)\in W^{1,n}(\Omega)$ [duplicate]

Prove that for $n>1$,the non-bounded function $u=\ln\ln\left(1+\frac{1}{|x|}\right)\in W^{1,n}(\Omega)$,Here $\Omega=B(0,1)\subset \mathbb{R}^{n}$ I think we have to prove that $$ ...
2
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0answers
34 views

Method of characteristics for a parabolic PDE

I am trying to solve a second order parabolic PDE using the method of characteristics. The PDE has the following form: $$\alpha\frac{\partial^2u}{\partial x^2}-\gamma\frac{\partial u}{\partial ...
2
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1answer
80 views

Show that $\frac{\int_\Omega|\nabla u|^2+\int_\Omega\alpha|u|^2}{\int_\Omega|u|^2}$ attains a minimum in $W_0^{1,2}(\Omega)$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $|\;\cdot\;|_p$ be the seminorm $$|u|_p^p:=\int_\Omega|\nabla u|^p\;d\lambda^n\;\;\;\text{for ...
3
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0answers
27 views

Interpretation of a certain transform

I'm having troubles with understanding the physical meaning of a certain transform. If $u$ is a solution to the wave equation $$\partial_t^2u-\Delta u=0\ \mathrm{in}\ ...
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96 views
+50

Integrating an infinite sum (statement from paper)

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Define $$v(x,y) = \sum_{k ...
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1answer
19 views

Convolution with standard mollifier

Let $\Omega \subset \mathbb{R}$ open and $f \in L^p(\Omega).$ Now, we define $$\eta(x):=\chi_{[-1,1]}(x) e^{\frac{-1}{1-x^2}}.$$ Then we define $$\eta_h(x):=\frac{1}{h} \eta( \frac{x}{h}).$$ This ...
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0answers
10 views

How to construct a fundamental solutions of a PDE from well-posedness?

A fundamental solution of a linear operator $P$ on a manifold $M$ is a distribution $G$ such that: $$P(G)=\delta(x-y)$$ In formal terms this is stated as given a test function $\phi$ then: ...
2
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1answer
567 views

Numerical techniques for solving systems of first-order semi-linear hyperbolic PDEs in two variables

I need to solve such systems of PDEs numerically and I'm wondering what the "standard" method (if there is one) is. The systems I'm interested in have the form: $\frac{\partial F_i}{\partial t} + ...
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1answer
44 views

Schauder estimates of weak solutions of a elliptic PDEs of 2nd order

Let $\Omega$ be a bounded domain in $\mathbb{R}^d$, $0<\sigma<1$, $L$ be a 2nd order strictly elliptic linear differential operator of divergence form: \begin{eqnarray} ...
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1answer
35 views

Estimate for a weak solution to a PDE

Let $f \in L^2(B_R(0))$ and let $u \in W^{1,2}(B_R(0))$ be a weak solution of the equation $$Lu = - \sum_{i,j=1}^{n} D_i(a_{ij}D_ju)+ \sum_{i=1}^{n} b_i D_i u + cu =f.$$ There are constants $0 \le ...
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1answer
95 views

Is $T$ a compact mapping from $W_{0}^{1,2}\left(\Omega\right)$ into itself? [closed]

Let $\Omega$ be an open bounded subset in $\mathbb{R}^{6}$ and $f$ be in $L^{8}\left(\Omega\right)$. For any $w$ in $W_{0}^{1,2}\left(\Omega\right)$, define $T\left(w\right)$ be in ...
1
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2answers
36 views

Neumann boundary conditions for Laplace equation with Raviart-Thomas elements

I am working on creating a finite element model for the Darcy equation using Raviart-Thomas elements and the mixed hybrid formulation. The problem in mixed form is this: $\mathbb{K}\nabla p = ...
9
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3answers
375 views

Does $\operatorname{div}\left(\nabla G +xG\right)=0\Longleftrightarrow \nabla G +xG=0$?

Let $G$ be a smooth function defined on $\textbf{R}^d$. What are the assumptions I should use to assume that $$\operatorname{div}\left(\nabla G(x) +xG(x)\right)=0 \quad (\forall x\in \textbf{R}^d)$$ ...
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1answer
55 views

integration by part and a limit- Evans PDE Chapt2 problem 13

1) I am having a hard time in seeing how the integration by part done in this problem (page 11) enter link description here Could anyone help explaining? I cannot see how he got 3 terms instead of ...
3
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3answers
347 views

Introduction to Pseudodifferential operators

I'm interested in elementary introduction to pseduodifferential operators and its application to hyperbolic PDE's. I know measure theory, Fourier analysis and some elementary(linear) hyperbolic PDE's ...
3
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1answer
34 views

Is it possible, that the fist two weak eigenvalues of $-\Delta$ in a bounded domain are equal?

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $\lambda_1$ be the first weak eigenvalue of $-\Delta$ in $\Omega$ $\varphi_1$ be the weak eigenfunction associated with $\lambda_1$ ...
2
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1answer
28 views

If $l_i$ is the first weak eigenvalue of $-\Delta$ in a domain $G_i$ and $G_1\subseteq G_2$, then $l_1\ge l_2$ and equality is possible

Let $\Omega_i\subseteq\mathbb{R}^n$ be a domain $\lambda_i$ be the first weak eigenvalue of $-\Delta$ in $\Omega_i$ It's easy to verify that $\Omega_1\subseteq\Omega_2$ implies $\lambda_1\ge ...
2
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1answer
38 views

Riemann problem of Burgers equation with source term

How to solve $u_t+uu_x=u$ with initial condition $u(x,0)=ul$ if $x<0$ and $u(x,0)=ur$ if $x>0$ where $ul$ and $ur$ being constant.