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

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2
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1answer
30 views
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Rankine Hugoniot Jump Condition Derivation

I follow the majority of this derivation I just do not understand where the two terms highlighted in green come from.
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0answers
7 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
0
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1answer
33 views

Solve initial value problem (C.S.I.R)?

The initial value problem is $$ \frac{\partial u}{\partial t} +x\frac{\partial u}{\partial x} = x, \ \ 0 \leq x \leq 1, \ \ t > 0 \ \ and$$ $$ u(x,0) = 2x \ \ has$$ a unique solution $u(x,t$ ...
0
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1answer
15 views

About Sobolev-Poincare inequality on compact manifolds

Let $2^* = \frac{2n}{n-2}$ where $n$ is the dimension of a compact closed manifold $M$. We get from the Sobolev/Poincare inequality the identity $$\lVert u \rVert_{2^*} \leq C\lVert \nabla u ...
11
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1answer
427 views

Book recommendations for self-study at the level of 3rd-4th year undergraduate

I have only recently discovered an interested in mathematics and I could only take a year off work to be back at school. Needless to say, for financial reasons (couple of mortgages) I will need to ...
2
votes
1answer
21 views

Morrey's inequality

From PDE Evans, 2nd edition, page 281: Now \begin{align} \int_0^s \int_{\partial B(0,1)} |Du(x+tw)| \, dS(w) dt &=\int_0^s \int_{\partial B(x,t)} \frac{|Du(y)|}{t^{n-1}} \, dS(y) dt \\ ...
1
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1answer
56 views

Solving a master equation with linear coefficients

I have the following PDE: $$ \partial_t P(x,y,t)=x\partial_xP(x,y,t)+(y-1)\partial_yP(x,y,t)+2P(x,y,t). $$ Mathematica suggests that the solution is $$ ...
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0answers
15 views

viscous burgers equation physical meaning

The viscous Burgers' equation : $$ q_{t} + qq_{x} = vq_{xx}, \:\:\: \mbox{where} \:\:\: v > 0, $$ combines the nonlinear propagation of $q(x,t)$ and the diffusion. What is this equation for? (in ...
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0answers
20 views

Estimates of positive solutions of $\Delta u=u$ in $\mathbb{R}^n$, $n>2$.

Let $u$ be a positive solution to the problem $\Delta u=u$ in $\mathbb{R}^n$, prove that there are positive constants $C, C_1,C_2$ such that $$ Ce^{-C_1|x|}<u(x)<Ce^{C_2|x|}. $$ Hints are ...
0
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1answer
37 views

1st order Cauchy problem

Discuss the solution of $u_t+ uu_x= 0$ with the following Cauchy data: a) $u(x,0)= x ; 0\leq x\leq 1$ b)$u(x,0)=1/2; 0\leq x\leq 1$ Sketch the domains in the $x-t $ plane where the solutions are ...
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0answers
20 views

Regularity of semilinear heat equation

I'm facing following regularity issue and i wonder if anyone of you guys is able to help me. I'd like to show that the solution of a semilinear heat equation is classical, i.e. $C^2$ in space and ...
0
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0answers
44 views

Is it necessary to know all the details in proofs of theorems you study in PDE's?

I've been studying PDE from the book by L.Evans for some time now. I came across some statements in the proofs which I couldn't justify. But to complete the exercises I didn't need to know all these ...
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0answers
18 views

Limit/Integration in heat equation

While studying heat equation from PDE by L.Evans, I came across the following limit which I'm not able to prove. For $n>=1, \delta >0$ , $lim_{t \to 0+} \;\;{1 \over ...
1
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1answer
35 views

Electromagnetic fields and Laplace equations along a square

I'd like to solve Laplace equation satisfying the following BCs: $$\phi(x,y=0)=0$$ $$\phi(x=0,y)=0$$ $$\phi(x,y=1)=9\sin(2\pi x)+3x$$ $$\phi(x=1,y)=10\sin(\pi y)+3x$$ where $0\leq x,y\leq 1$. I have ...
0
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0answers
15 views

intuitive fact of a class of functions defined in $R^n$

I am reading an article and i have the following situation: Let $u: R^n \rightarrow R$ a continuous function in $R^n$. Supoose that u is nonnegative and that for all $t \geq 0$ the set $L_t = \{ x ...
0
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3answers
49 views

How to solve $(\partial_x + i\partial_y)u - c(\partial_x+i\partial_y)au=0$?

The following equation, $$(\partial_x + i\partial_y)u - c(\partial_x+i\partial_y)au=0$$ ($a=a(x,y)$ and $\partial_x=\frac{\partial}{\partial x}$) with solution, $$u=\exp(ca)f(x+iy)$$ where $f$ and ...
0
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1answer
22 views

Understanding Distributional Meanings and Test Functions for PDEs

thank you for taking the time to read my question. My question is about distributional meanings in PDEs. My specific question is at the bottom, but I'd be interested in a bit of general theory (even ...
3
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0answers
21 views

Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
2
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2answers
55 views

Separation of variables: when to have exponential solution and when sinusoidal?

In separation of variables, one can assume a solution of V(x,y) = X(x)Y(y) and after plugging this into Laplace's equation which is: ${{\partial^2 V} \over {\partial x^2}}$ + ${{\partial^2 V} \over ...
0
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0answers
20 views

How to find the solution in $x$ when the differential equation is in $\partial_y$ such that $\partial_y = \partial_x + f(x)$

For my problem, the differential equation can be simplified by defining a new operator like $\partial_y = \partial_x + f(x)$. But at the end I need a solution in terms $x$. How can I do that? As an ...
2
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1answer
14 views

Weak convergence in some space.

I have the sequence $\{u_{k}\}_{k}$ weak convergent in the space $L^{2}(0,T; W^{1,2}(\Omega))$. What exactly does it mean? Do it imply weak convergence $\{u_{k}\}_{k}$ in $L^{2}(0,T;\Omega)$ or $\{ ...
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0answers
13 views

obtain the continuous form of a master equation

I have the following master equation: $$ \partial_tP(x,y,t)=-[x+(1-x-y)]P(x,y,t)\\ +(x+\epsilon)P(x+\epsilon,y-\epsilon,t)\\ +(1-x-y+\epsilon)P(x,y-\epsilon,t) $$ where $\epsilon$ is a small number, ...
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0answers
45 views
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Partial Differential equations and applications- Reference request

I will be taking up a PDEs course next semester and would like to find some good references. The topics covered in the syllabus is given below. Partial differential equations: Conservation laws, ...
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0answers
55 views

heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...
1
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1answer
24 views

Equivalence of Localized Fourier Restriction Estimates

I'm reading Tao's Park City notes on the restriction conjecture http://arxiv.org/abs/math/0311181. He says at some point that the estimate: there is a constant $C > 0$ such that, for any $R \ge ...
1
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1answer
26 views

Don't understand a $L^\infty$ bound argument involving measure of set

I'm trying to understand the proof of Proposition 2.2, part 2 of this paper. this is where I am stuck. For any $k > 0$, we have $$k^{\frac{2(N+1)}{N}}|\{|u|^m > k\}| \leq ...
10
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1answer
469 views

Hille Yosida theorem application

Disclaimer: pretty long and specific (contraction semi groups involved). I have fourth order parabolic equation $$ u_t + \Delta^2 u = 0 $$ on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
1
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1answer
71 views

Can Dirichlet and Neumann eigenfunctions coincide for the Helmholtz equation?

We consider the null space (corresponding to the possible eigenvalue zero) of the linear indefinite elliptic PDE $\Delta u+k^2(x)u=0$ in $\Omega$ with $u=\partial_{\nu}u=0$ on the boundary. If the ...
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0answers
24 views

Di Perna-Lions theory for transport equation

Does someone know if some notes on the topic mentioned in the title are available online? I'm reading the paper "Ordinary differential equations, transport theory and Sobolev spaces" by Di Perna and ...
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0answers
15 views

existence of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
1
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0answers
25 views

About a property of weighted Sobolev spaces

If we define the weighted Sobolev norm as $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ where $f(x):\mathbb R^n \to \mathbb R$ and $\Delta$ is ...
2
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0answers
57 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
0
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1answer
41 views

Trace-zero functions in $W^{1,p}$

This is an excerpt of a textbook's proof for a theorem (Trace-zero functions in $W^{1,p}$), from PDE Evans, 2nd edition, page 275. Next let $\zeta \in C^\infty(\mathbb{R}_+)$ satisfy $$\zeta ...
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0answers
15 views

Finite difference method for elliptic pde stability condition

I am trying to implement a finite difference scheme for the elliptic pde $\nabla(a(x_1,x_2)\nabla\phi(x_1,x_2))=0$ where $(x_1,x_2) \in (0,l_1)\times(0,l_2)$ with $l_1 \neq l_2$ necessarily. I know ...
0
votes
1answer
33 views

Separation of variables, when possible?

$$ \Delta \Psi(x, y, z) + V(x, y, z)\Psi(x, y, z) = E \Psi(x, y, z) $$ For which $V(x, y, z)$ can this partial differential equation (eigenproblem) be solved by separation of variables?
3
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2answers
327 views

Best numerical method to integrate a differential equation like $\frac{\partial y}{\partial t}=f\left(t,\frac{\partial y}{\partial r}\right)$?

What is the best numerical method to solve an equation like this: $$\frac{\partial y}{\partial t}=f\left(t,\frac{\partial y}{\partial r}\right)\quad?$$ Can somebody give, at least, a reference?
1
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3answers
45 views

Solving $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0$ by changing variables

Transform the differential equation $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0 $ by introducing new variables $x = u+v$ and $y=u-v$. then solve it. I which I could show ...
3
votes
1answer
40 views

Trace Theorem question

From PDE Evans, page 272. My question is towards the bootom of this post. THEOREM 1 (Trace Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator ...
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0answers
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Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
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0answers
18 views

Existence and Uniqueness of Solutions of PDEs

I have been looking into the Cauchy-Kovalevskaya Theorem where one can "establish the local existence of analytic solutions to a system of PDEs". I wanted to see an application (for example, see ...
0
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1answer
21 views

How to get comparison principle from contraction principle for PDE

let $u$ and $v$ be two solutions to some PDE with initial data $u_0$ and $v_0$. Is $$|u(t)-v(t)|_{L^1} \leq |u_0-v_0|_{L^1}$$ a contraction principle? I read that "contraction principle gives ...
1
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1answer
41 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 $V$ ...
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0answers
28 views

Basic exercise of real analysis and of p harmonic functions

I am studying the following definition of an article: Definition (blow up): Let $u$ a function (assuming real values) in the open ball $B(x_0,1)$. For $r>0$ define the function $u_r(x)$ in $B(0, ...
3
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1answer
67 views

Properties of weak derivatives in Sobolev spaces

In PDE Evans 2nd edition, pages 261-263, there is a theorem and its proof which concerns the four properties of weak derivatives. Unfortunately, I do not understand the fourth property, which I will ...
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1answer
150 views

Evans PDE example 4, p.260 clarification

I am having trouble understanding the following example from Evans' PDE book. It is as follows (Example 4, p.260, Evans, Partial Differential Equations, 2nd ed.,) Let $\{r_k\}_1^\infty$ be a ...
0
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1answer
23 views

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$?

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$, where $H^1(R;H^1(R^n))=\{f\colon R\to H^1(R^n)\colon \int \|f(t,\cdot)\|_{H^1}^2\,dt <\infty \quad \text{and} \int \|f'(t,\cdot)\|_{H^1}^2\,dt <\infty \}$ ...
2
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1answer
34 views

Third Order PDE written as a System of (Linear) First Order PDEs

I need to rewrite the PDE $$f_{y}+ff_{x}+f_{xxx}=0,$$ where $f=f(x,y)$ as a system of first order quasi-linear PDEs. I have no idea how to tackle this problem. Any form of help will be appreciated. ...
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0answers
37 views

Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in ...
0
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1answer
12 views

About the weighted Sobolev norm

I'm wondering that the Sobolev norm with weight $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ is equivalent to the norm $$ ...
1
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1answer
26 views

Well-Posedness PDE of the Form $\partial_t u = P(\partial_x) u$ for a Polynomial $P$

My question is to determine whether the PDE $\partial_t u = P(\partial_x) u$, with $2\pi$-periodic boundary conditions, for a polynomial $P$, is well-posed; this depends on the polynomial, and my ...