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

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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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13 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
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solution of wave equations in odd dimension Evans PDE

Here I am looking at the proof of theorem 2 below In the last two lines, the exponent changes from $\frac{n-1}{2}$ to $\frac{n-3}{2}$, why? Could anyone explain?
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1answer
16 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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1answer
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Prove $u=\ln\ln\left(1+\frac{1}{|x|}\right)\in W^{1,n}(\Omega)$

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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1answer
22 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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0answers
21 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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0answers
23 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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8 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: ...
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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
34 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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0answers
19 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
69 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
54 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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0answers
15 views

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

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
42 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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0answers
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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50 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
48 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
92 views

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

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
19 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
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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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74 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
56 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 ...
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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
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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 ...
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1answer
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Example of an $H^{-1}$ function that isn't $L^2$

I'm going back over some PDE and Sobolev space theory, and the following is puzzling to me. Consider a nice domain $\Omega$ and the space $H^1_0(\Omega)$ of functions with $L^2$ first derivatives, ...
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1answer
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$u=\sum (u,\varphi_k)_{L^2}\varphi_k$ vs $u(x)=\sum (u,\varphi_k)_{L^2}\varphi_k(x)$ and related questions

I have some doubts after reading several threads. Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of ...
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1answer
26 views

Meaning of “$\triangledown$u*ñ=0 on the boundary”

I'm doing homework for my PDE class, I'm coming across this notation and I don't what the ñ means: $\triangledown$u* ñ=0. I have tried to google it, but unfortunately questions like this don't really ...
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Minimal boundary conditions for divergence theorem

I've noticed that some domain conditions of questions here were only supposed to be finite dimensional and bounded. And then the divergence theorem was applied in the answers. But if I'm not mistaken, ...
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1answer
33 views

how to calculate integral of product of exponential function and trigonometry function?

Let $x_0$ and $\sigma$ are constants. How to calculate this? $$ \int^{L}_{-L}e^{-\frac{(x-x_0)^2}{2\sigma^2}}\cos x dx $$ I think i can solve that with integration by parts. But I'm confused how to ...
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1answer
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Why can $D^ku(x) := \{D^\alpha u(x) \mid |\alpha| = k \}$ be regarded as a point in $\mathbb{R}^{n^k}$?

This is a comment made in the Appendix of Evans's Partial Differential Equations. He defines the set of all $k$ order partial derivatives as $D^ku(x):= \{D^\alpha u(x) \mid |\alpha| = k \}$ (using ...
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2answers
32 views

What do authors usually mean from a geometrical interpretation when “radially symmetric” is mentioned?

I was reading up on a text on an investigated case on heat flow in an annulus. The author mentioned "radially symmetric" and consequently proceeded to deduce that the variable 'theta' can be done away ...
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How to justify that the following system of PDE only admits linear solutions?

I have the following system of partial differential equations: $$ \frac{\partial\Lambda^{\mu}}{\partial x^{\nu}}= $$ $$ \small \begin{pmatrix} ...
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1answer
23 views

Absolute Continuity defined by Necas

I read a definition of the absolute continuity in Necas' book "Direct Methods in the Theory of Elliptic Equations": Let $\Omega$ be a domain in $\mathbb{R}^n$ , $P$ a line verifying ...
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1answer
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Eigenvalue of Laplacian with Robin boundary condition

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial \Omega$ and let $\nu$ denote the outer unit normal. Let $u$ be an eigenfunction of $-\Delta$ in $\Omega$ satisfying ...
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1answer
44 views

Conditions under which a function vanishing on the boundary belongs to $H_0^1$

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open set ,and $u \in C(\overline{\Omega}) \cap C^1(\Omega) \cap H^1(\Omega) $ be a function such that $u \big|_{\partial \Omega}=0 $.Prove that $ u ...
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0answers
61 views

How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary. My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that ...
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1answer
20 views

Derive diffusion coefficient for heat equation from random walk simulation

I want to simulate the underlying stochastic process of diffusion on a microscopic level and compare the result with the solution of the heat equation. However, I'm not able to match the solution of ...