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

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General solution for this PDE?

let $f$ be a function maps $\mathbb{R}^2$ to $\mathbb{R}$. let: $u=f^{(1,0)}(x,y)$ $v=f^{(0,1)}(x,y)$ which are partial derivatives w.r.t the first & second argument of $f$. solve $f(h, \...
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1answer
40 views

Solving PDE using characteristics

I am having trouble solving $u_t+u^2u_x=0$ with initial value condition $u(x,0)=$\begin{cases} 1, & \text{if $x<0$} \\ 1-x, & \text{if $0<x<1$}\\ 0, & \text{if $x>1$} \end{...
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Is it true that $|∇u(x)|^2\chi_\Omega=|\nabla (u \chi_\Omega)|^2$

Let $u\in L^\infty(\Omega)\cap H^1(\Omega)$ with $\Omega$ open, bounded and regular (as you wish) domain of $\mathbb{R}^N$. Is it true that $$ \int_\Omega |\nabla u(x)|^2 \mathrm{d} x= \int_{\mathbb{...
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1answer
57 views

What is the regularity of the eigenfunctions for self-adjoint operators with non-smooth coefficients?

There are well know result about elliptic operators, $L$, that guarantee that an operator of order $2k$ generates a basis for $H^{k}$ on smooth domains.( See Do eigenfunctions of elliptic operator ...
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2answers
36 views

Solving the 1-D diffusion equation

For the equation $$u_t = Du_x$$ where $D$ is a diffusion constant, we can define the system $$u_x=v$$$$u_t=Dv_x$$ However, how does one solve for $v$? $\frac{\partial u}{\partial x}=v \iff {\partial ...
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1answer
33 views

Sturm–Liouville equation and the Eigenvalue general Problem (PDE)

As I am studying for my Partial Differential Equations exam, I came across Sturm–Liouville equation where it says that it's solutions $y(x)$ are the eigenfunctions of the general problem $Ly=λy$. I do ...
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Nonlinear PDE: regularity issues

I have this execrise I've been trying to solve Let $B=\{x \in \mathbb R^n: |x|<1\}, \quad a: \mathbb R^n \rightarrow \mathbb R , \quad a\in C^\infty(\mathbb R^n) \cap L^\infty(\mathbb R^n) $ $$\...
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14 views

Maxwell-type system: how to solve?

$\DeclareMathOperator{\curl}{curl}\renewcommand{\div}{\mathop{\mathrm{div}}}$ Consider the following system of equations for the unknown vector field $A$ in the unit 3d ball $B$ with boundary $S$: $$ ...
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3answers
94 views

Solution to $\frac{d^2f}{dr^2}+\frac{1}{r}\frac{df}{d r}=0$

I know that $f(r)=aln(r)+b$ where $a$ and $b$ are constants is a solution of $$\frac{d^2f}{dr^2}+\frac{1}{r}\frac{d f}{dr}=0$$ are there any other solutions to this, would appreciate it if someone ...
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1answer
18 views

Boundary Values and Initial conditions for Linear Stability analysis (Fluid Dynamics)

Lets say we have a system of partial differential equation $\Delta(x,y,t,u^{(n)})=0$ (Navier-Stokes Equations) with a given stationary solution $u_s(x,y)$ for a inviscid flow. Note that $u$ is a 2D-...
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28 views

PDE with Stochastic Coefficients

Does anyone have reference suggestions for pde's with stochastic coefficients? I've seen many papers on more advanced problems, but it would be great to have a reference discussing the basic theory ...
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0answers
10 views

PDE $Ru_{zzz}+u_{zzw}=0$: Solution verifcation

I tried to solve this PDE $$Ru_{zzz}+u_{zzw}=0$$ First i substituted $u_{zz}=v$: $$Rv_{z}+v_{w}=0$$ I solved this by the method of characteristics and integrated the result with respect to $z$ ...
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1answer
58 views

How to solve this system of PDEs?

The 1-D Euler’s equation for constant pressure can be written in terms of the two equations $$u_t + uu_x = 0, x\in\mathbb{R}, t> 0, u(x,0)=f(x)$$ $$\rho_t+\rho u_x+u\rho_x=0, x\in\mathbb{R}, t>0,...
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0answers
47 views

Integrating by Parts in Evan's book

How to perform Integrating by Parts here (Evan's PDE Book page 50-51) You have to do $3$ times integration by parts to move the operator from $f$ to $\Phi$, but then you have to get $4$ terms. I ...
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1answer
39 views

Green Function problem PDE

Someone can give me tips for solve this PDE, please! $\frac{d^4G}{dx^4}=\delta(x-x_0)$ with $G(0;x_0)=G(L;x_0)=\frac{dG}{dx}(0;x_0)=\frac{d^2G}{dx^2}(0;x_0)=0$ I do not know how to start this, ...
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0answers
27 views

A comparison principle for a nonlinear parabolic PDE

We know the following comparison principle holds for the diffusion equation: Suppose that $u(x,t)$ and $v(x,t)$ satisfy \begin{equation} \begin{cases} u_t\ge \Delta u+F(x,t,u), \ \ &x\in\Omega,\ \...
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2answers
49 views

Numerical solutions of partial differential equations

I'm studying mathematical physics and working on numerical solutions of partial differential equations. I am having trouble understanding the way we solve partial dif. equations, e.g., $\frac{\...
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0answers
27 views

solution for the Cauchy problem of an elastic bar forced vibrations

I am looking for the solution of the following problem: $\frac{\partial^2 w}{\partial t^2}+a^2\frac{\partial^4 w}{\partial x^4}=\Phi(x,t),\quad -\infty<x<\infty, t>0\\ w(t=0)=0,\quad \...
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1answer
27 views

Question concerning the proof of regularity for the Laplacian $f \in H^m(\Omega) \Rightarrow u \in H^{m+2}(\Omega) $

I am stuck at the proof of Theorem 9.25 in Haim Brezis' Sobolev Spaces, Functional Analysis and Partial Differential Equations. This theorem deals with the regularity for the Dirichlet Problem for ...
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1answer
24 views

How to linearize a pde system in a solution $u_0$?

Suppose we have $$ u_t=u_{xx}+\mu u+\lvert u\rvert^2u,~~u=u(x,t). $$ We can write this as the coupled system $$ u_t=u_{xx}+\mu u+\bar{u}u^2\\ \bar{u_t}=\bar{u_{xx}}+\mu\bar{u}+u\bar{u}^2. $$ Now, ...
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Homogeneous solution, Fourier mode, wavelength

Consider a system $$ \partial_t u=N(\sigma)u,~~u=(x,t)~~~(1) $$ where $N$ is a non-linear operator depending on some control parameter. Suppose that the system (1) admits a homogeneous ...
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Green functions

do you know some litterature about green functions for the heat equation ? in particular for the non-linear equation : $\frac{\partial u(x,y,t)}{\partial t}-\frac{\partial^2 \left[ f(u(x,y,t))u(x,y,t)...
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1answer
32 views

Poisson equation in a ring

I would like to solve: $-\Delta u = 1$ in $\Omega $ where $\Omega = \{ (x,y): 1 \leq x^2+y^2 \leq4 \}$ with $u=0$ in $\{ (x,y): x^2+y^2 = 4 \}$ and $\frac{\partial u}{\partial n}=0$ in $\{ (x,y): x^2+...
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1answer
38 views

Reference request: $W^{2,2}$ estimates of elliptic PDE with measurable coefficients

I have some questions on solvability of the following elliptic PDE: in $\mathbb{R}^2$, for $f\in L^2$, $$a^{ij} u_{x^i x^j} +b^i u_{x^i} + c u = f.$$ Here $\{a^{ij}(x)\}_{i,j=1,2}$ is symmetric ...
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0answers
89 views

Anti-derivative of Dirac distribution over $\mathbb{R}^{n}$.

I want to determine an $f\in C(\mathbb{R}^{n})$ and a multi-index $\alpha$ such that $\partial^{\alpha}f=\delta$. I found an example in some lecture notes which claimed that for $$f(x)=\frac{1}{\...
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1answer
54 views

Bound related to Schwartz space

If $u \in \mathcal{S}'(\mathbb{R}^n)$, then is there an integer $m \ge 0$ and $C>0$ such that for all $\phi \in \mathcal{S}(\mathbb{R}^n)$,$$|u(\phi)| \le C\|\phi\|_m,$$where$$\|\phi\|_m = \sum_{|\...
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1answer
63 views

Why is the Black-Scholes PDE called degenerate

I am working in Mathematical Finance and know that the Black-Scholes PDE is degenerate at $x=0$ (I assumed that this was because at 0 the convection and diffusion terms vanish and one is left $V_{t} = ...
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18 views

Linear system of advection diffusion equations

I am trying to find the eigenvalues and eigenfunctions of the coupled PDE system $$ \partial_t \vec{u} = - \stackrel{\leftrightarrow}{A} \partial_x \vec{u} + \stackrel{\leftrightarrow}{D} \partial_x^2 ...
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0answers
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Determine if a function belongs to the sobolev space $W^{1,p}(\mathbb{R})$ and not to $L^q(\mathbb{R})$

I don't understand the first conclusion of the user Tomas in the exercise Properties of function $f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R}$ with $0 < \alpha < 1$....
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A question from Murdock's Pertubations: Theory and Methods.

I have the following two set of PDEs: $$u_{0_{xx}}+u_{0_{yy}} = 0 , u_0|_{x^2+y^2=1}=1$$ $$u_{1_{xx}}+u_{1_{yy}} = u_0^2 , u_1|_{x^2+y^2=1}=0$$ For the first PDE I know that the solution is $u_0(x,...
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1answer
23 views

Smoothness of the solutions of the Cauchy problem for quasi linear pde (Burgers' equation) [duplicate]

Consider the cauchy problem of finding u=u(x,t) such that $u_{t}+uu_{x}=0$ for $x \in \mathbb R$ , $t \gt 0$ with $ u(x,0)= u_0(x)$ for $x \in \mathbb R$ then, which choices of the following ...
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1answer
89 views

Applications of PDEs in many variables

One reason that solving systems of partial differential equations is so important is the many applications of PDEs in science and engineering (eg. the heat equation, the wave equation, etc.). Often ...
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29 views

If we know Spec($M_1$) and Spec($M_2$), what could we say about Spec($M_1 \cup M_2$)?

Let two domains $M_1$ and $M_2$ (Dirichlet conditions). If we know the spectrum of the Laplacian on $M_1$ and $M_2$, what could we say about Spect($M_1 \cup M_2$)? Is there a theorem that might give ...
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What is/are the boundary condition for the steady-state temperature distribution in the half space , to use Fourier-Bessel series?

For, temperature distribution $u=0$ when r=a, we can write $J_n(Ka)=0$ so that we can use Fourier-Bessel series, and for half space, as $r$ goes to infinity $u$ will go to $0$, but then i can not use ...
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Infinitely differentiability of $\frac{1}{\sqrt{4\pi t}}\int_{\mathbf R}\exp\left({i\frac{(x-y)^2}{4t}}\right)f(y)dy$

Showing infinitely differentiability of $u(x,t):=\frac{1}{\sqrt{4\pi t}}\int_{\mathbf R}\exp\left({i\frac{(x-y)^2}{4t}}\right)f(y)dy$ in both $x$ and $t$ in $\mathbf R\times \mathbf R_{>0}$, where $...
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128 views

Intuitive explanation of Tao's work on Navier-Stokes equations needed [closed]

In February 2014 mathematician Terence Tao posted his work on a partial solution to the Navier-Stokes existence and smoothness problem. He proved that a "finite time blowup" exist "for an averaged ...
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1answer
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How should one approach this PDE's?

I have got two tasks with PDE's and I am not really familiar with them, so I have no idea, how to approach this kind of problems. First: Solve the following PDE:$$\frac{\partial u}{\partial t} = \...
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1answer
28 views

The sign of elliptic operator (divergence form)

The elliptic operator (in divergence form) in (Evans, and many texts) is defined as $$Lu=-D_j(a^{ij}D_i u)+b^i D_i u+cu$$ $D_i u$ denotes $u_{x_i}$,assuming the summation convention is understood. ...
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1answer
31 views

checking ellipticity of a 2nd order elliptic PDE (In Evans)

320 Evan PDE 2nd edition. I have trouble verifying (ii) of theorem 2, i.e. ellipticity for the following elliptic operator $$ Lu=-D_j(a^{ij}D_i u)+cu$$ Multiply by test function then do integration ...
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Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for $W^{s,p}(\Omega)$ when $s$ is an integer

Take $W^{1,2} = H^1$ for example. If we still use Slobodeckij norm (which is normally defined for a fractional Sobolev space) as follows for a $u\in H^1(\Omega)$ with the exponent being in integer: $$ ...
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1answer
31 views

Examples of functions zero on the boundary

What are examples of a function $f(x)$ satisfying $$f(0)=f(l)=f'(0)=f'(l)=0?$$ One example is $f(x)=x^{\beta_1}(l-x)^{\beta_2}$ with $\beta_1, \beta_2>1$. Edit Not of the form $f\cdot g$ or etc.
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Dimensionless Equations

Let us consider a famous problem in finance: American put option with strike price $X$ that expires at time $T$. Let $V(S,t)$ denote the value of an American put option, $S$ the price of the ...
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1answer
41 views

When expanding a function in a sphere, why is the complex conjugate of the spherical harmonic function used to calculate the coefficients?

When expanding a function on a sphere $f(r,\theta, \phi)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n}=A_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta,\phi)$. Since what I'm asking involves the ...
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1answer
32 views

Prove that this Cauchy problem has at most one solution.

Problem: $$ \begin{cases} u_t = u_{xx} +5 u_{x} &\text{ in } R_+\times(0,1)\\ u(0,x)=g(x) &\text{ for } x\in(0,1)\\ u_x(t,0)=\alpha(t) &\text{ for } t>1\\ u_x(t,1)=\beta(t) &\text{ ...
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1answer
29 views

Energy estimates (Evans PDE Chapter 6): Sobolev norm is controlled by an elliptic operator

I am looking at the energy estimates of 2nd-order elliptic PDE in Evans p.318. In the following, everything follows, except that I cannot understand how the final line obtained. When I apply poincare ...
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2answers
37 views

If $A(x)$ is 1-periodic and $\frac{A''(x)}{A(x)} = C$, then $C=-4\pi^2 n^2$?

This might be a trivial question but I forgot my differential equation. Anyway, I am trying to solve the heat equation on circle. Given that $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\...
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0answers
29 views

assigning boundary values to a weakly differentiable function

There is a sentence in Evans I cannot justify. The claim made is $u=g$ on $\partial U$ in the trace sense. Why? I understand that $u\in H^1$ implies $u\in W^{1,p}(U)$. But we also need the assumption ...
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0answers
56 views

Schwartz functions dense?

I want to show that the Schwartz functions are dense in $$\left\{f \in L^2; \int |x|^2 \left|f(x)\right|^2 dx + \int |\xi|^2 \left|\hat{f}(\xi)\right|^2 d \xi < \infty\right\}$$ where the norm ...
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1answer
25 views

Weak solutions of Navier-Stokes are square integrable distributions?

I'm reading Lemarie-Rieusset's book Recent developments in the Navier-Stokes problem and have the following issue: he defines a weak solution to the Navier-Stokes equations on $(0,T)\times\mathbb R^d$ ...
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1answer
26 views

Quasi-linear PDE - solution process looks correct, yet solution isn't

This is a quasi-linear PDE $$u_t+uu_x=-ku^2$$ where $k>0$, $t>0$, $u(x,0)=1$, and $x\in\mathbb{R}$. And here's how I'm solving it: Take parametrization of the surface $u(x,t)$ as $x=x(s), t=t(...