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

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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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1answer
40 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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19 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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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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16 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
21 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
31 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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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
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
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
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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42 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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31 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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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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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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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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1answer
29 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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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 $$ ...
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3answers
50 views
+50

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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33 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 ...
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1answer
18 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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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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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: ...
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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
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
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
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 ...
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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 $\lambda_1$ ...
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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 ...
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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 ...
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16 views

Using Laplace transform to solve the pde resulting from solving simple birth process differential equations using generating function method [closed]

Using Laplace transform to solve the pde resulting from solving simple birth process differential equations using generating function method $$\frac\partial{\partial t} G(z,t)+\lambda z(1-z)\; ...
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1answer
30 views

Solving $u_y + (1-2u)\cdot u_x = 0$ using characteristic equations

I need to solve the following partial differential equation: $$F(x, y, u, p, q) = u_y + (1-2u)\cdot u_x = 0$$ with $$u(x, 0) = \left\{\begin{array}{cc} \frac{1}{4} & x < 0 \\ \frac{3}{4} & ...
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1answer
58 views

Runge-Kutta method for PDE

I consider certain partial differential equation (PDE). For example, let it be heat equation $$u_t = u_{xx}$$ I want to apply numerical Runge-Kutta method for solving it. As a first step I ...
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33 views

Using Multipule Scale Analysis to solve a non-linear differential equation

I would like to know if there are other methods to solve equations such as this one below. I don't really understand the theory behind the multiple scale analysis and why it works I understand some of ...
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52 views

Integration by parts for general measure?

Let $\mu$ be a general measure, suppose $f,g$ has compact support on $\mathbb{R}$, when does the integration by parts formula hold $$\int f'g d\mu = - \int g'fd\mu?$$ I know in general this is false, ...
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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.
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1answer
51 views

2 to 1 dimension in linear PDE with non-constant coefficients

I have a question that can majorly help in my physics. Problem Say, we have a linear PDE \begin{equation} \hat{D}~F(x,y)=0, \end{equation} with $\hat{D}$ being a (second order) differential ...
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2answers
50 views

Prove that a classical solution of $-\langle\nabla,A\nabla u\rangle=f$ is also a weak one

Let $\Omega\subseteq\mathbb{R}^n$ a domain $f\in L^2(\Omega)$ $A:\Omega\to\mathbb{R}^{n\times n}$ be Borel-measurable and $A(x)$ be symmetric, for all $x\in\Omega$ $u\in C^2(\Omega)$ with $A\nabla ...
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1answer
26 views

$L^2(a_1,b_1;H_0^1(a_2,b_2))\subset L^2(a_1,b_1;L^2(a_2,b_2))$ and Convergence

Let $[a_1,b_1]\times[a_2,b_2]\subset\mathbb{R}^2$. Suppose $$u_n\rightharpoonup u\,\,\,\text{ weakly in } L^2(a_1,b_1;L^2(a_2,b_2))$$ and $$\{u_n\}\text{ is bounded in }L^2(a_1,b_1;H_0^1(a_2,b_2)).$$ ...
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Develop a concept of weak solvability for $-\langle\nabla,A\nabla u\rangle=f$

Let $\Omega\subseteq\mathbb{R}^n$ a domain $f\in L^2(\Omega)$ $A:\Omega\to\mathbb{R}^{n\times n}$ be Borel-measurable with $A(x)$ is symmetric, for all $x\in\Omega$ $\exists c_1,c_2>0$ with ...
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1answer
94 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 ...
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1answer
36 views

How to derive the weak form of the PDE?

I have some difficulties solving the weak form of the PDE: The proof of the preceding statement is elementary. The weak form of the PDE $\nabla \cdot (A(x) \nabla u) + \omega^2 q(x) u = 0$ for all ...
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1answer
20 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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39 views

How to “uniformly” mollify a sequence without knowing the higher derivative.

The motivation and previous discussion of some related ideas can be seen here. My question: given a sequence $u_n\in u_0+H^1_0(\Omega)$ where $u_0\in H^1(\Omega)$ and $\Omega\subset \mathbb R^N$ is ...
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1answer
30 views

Evans PDE, Problem 8 Chapter 2 clarification on $|x-y|$

Hi I am attempting problem 8 (Chapt2 Evans PDE). Again I found the solution on the internet. enter link description here I understood much of everything of the proof except for one line. " Since ...
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93 views
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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
57 views

How do you read a partial differential equation?

In calculus we can read the "normal derivative", $\frac {df}{dx}$, as the rate of change of our function $f$ with respect to $x$. With partial derivatives of multivariate functions it is very much the ...
4
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67 views

Find the general solution of $u_{ttxx}(x,t)=(u_{tt}(t,x))^2$

Find the general solution of the equation $$u_{ttxx}(x,t)=(u_{tt}(t,x))^2$$ Let set $v(x,t)=u_{tt}(x,t)$. Then $$v_{xx}(x,t)=(v(x,t))^2$$ What should I do next? Any help would be greatly appreciated.
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1answer
44 views

analytic solution poisson equation spherical coordinates

I'm trying to analitically solve a poisson equation $\nabla^{2}p(r,\theta,\phi)=f(r,\theta)$ in spherical coordinates; the boundary condition is $p(\infty,\theta,\phi)$=0. I'm quite used to the ...
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0answers
21 views

If $\partial\Omega\in C^{2+\alpha}$ and $-\Delta\Theta=f\text{ in }\Omega$ with $f\in C_0^\infty(\Omega)$, then $\Theta\in C^{2+\alpha}$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with $\partial\Omega\in C^{2+\alpha}$ for some $\alpha>0$ $f\in C_0^\infty(\Omega)$ $\Theta\in C^0(\overline{\Omega})\cap C^2(\Omega)$ be the ...