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

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A generalization of holomorphic functions

Let's fix a matrix $A\in M_{2}(\mathbb{R})$. Assume that the following vector space of smooth functions is closed under complex multiplication: $$\mathcal{S}_{A}=\{f:\mathbb{C}\to \mathbb{C}\...
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0answers
43 views

Why isn't this a [PARTIAL] solution to Lewy's Example

I was considering the example of the equation from: https://people.maths.ox.ac.uk/trefethen/pdectb/lewy2.pdf $$ \frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y} - 2i(x + iy) \frac{\...
1
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1answer
54 views

Solving system of PDEs

I am stuck on the following problem: Solve for $f(x,y)$, where: $\frac{\partial f}{\partial y} = y$, $\frac{\partial f}{\partial x} = \frac{1}{2}xy$ My original strategy was to integrate the first ...
0
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1answer
47 views

Nodal lines of wave equation on rectangular membrane [closed]

I'd appreciate if someone could please let me know how to draw nodal lines. I don't really know how to do it and couldn't find much information with Google. I could find some info on nodal curves ...
1
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1answer
25 views

Gradient of the solution for Poisson equation

Let $f(x)=(\nabla N *g)(x)$, where $N(x)=\frac{1}{|x|^{n-2}}$ for $n\geq 3$ is the Newtonian kernel, and $g\in L^1(\mathbb{R}^n)\cap L^{\infty}(\mathbb{R}^n)$. Then we can have that $f\in L^{\infty}$ ...
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0answers
22 views

A $C^1$ function in Orlicz Sobolev space

How to prove that thi functional is $C^1$: $$ I(u)=\int_{\mathbb{R}^N} \Phi(|\nabla u|)+\Phi(|u|) dx-\int_{\mathbb{R}^N} F(u) dx $$ Where $\Phi$ is an N-function and $F(t)=\int_{0}^t f(s) ds$ where ...
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0answers
34 views

Complex version of Lax-Milgram Theorem

I'm trying to prove Lax-Milgram Theorem in the complex case, i.e. Let $X$ be complex Hilbert space and let $f\in X'$, its topological dual. If $a(\cdot,\cdot):X\times X\to \mathbb{C}$ is ...
1
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1answer
88 views

Can I have an analytical solution for $\frac{\partial \theta}{\partial t}=\frac{\partial^2\theta}{\partial {x}^2}+1$

Subjected to the following boundary conditions: $\left.\frac{\partial \theta}{\partial x}\right|_{x=0,t}=c_1\theta\bigg\vert_{x=0,t}$ $-\left.\frac{\partial \theta}{\partial x}\right|_{x=1,t}=c_2\...
2
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0answers
49 views

Conservation Laws: Difference and Reasonability Weak solutions, Integral solutions, distributional solutions

Suppose we have for $T\in\mathbb{R}_{>0}$ a conservation law of the following general form \begin{align} \dot{u}(t,x)+(a(t,x,u(t,x)))_{x}&=0 && (t,x)\in(0,T)\times\mathbb{R}\\ u(0,x)&...
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0answers
17 views

What operations can be performed on/with a differential operator vs partial differential operator?

What operations can be performed on or with a differential operator vs a partial differential operator? I ask because I see people "divide" or "multiply" with the differential operator but that same ...
2
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1answer
47 views

How to show that a Schwartz distribution is in a Lebesgue or Sobolev space?

It is known that all $L^p$ spaces (and, consequently, all $W^{s,p}$ spaces) can be embedded in the space of Schwartz distributions $\mathcal D '$. There is a problem, though: how do I check whether ...
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0answers
16 views

Testing heat equation with $\log(u)$?

On a bounded domain, let $u \in L^2\left(0,T;H^1\right) \cap H^1\left(0,T;H^{-1}\right)$ be weak solution to the heat equation $$u_t - \Delta u = 0$$ with the BC $\partial_\nu u = 0$ and some initial ...
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0answers
19 views

Proof of analyticity in Lewy's Example:

I was reading: https://people.maths.ox.ac.uk/trefethen/pdectb/lewy2.pdf where it is stated: The Lewy Operator on a function $f(x,y,t) : x,y,t \in \mathbb{C}$ is given by $$ \frac{\partial f }{\...
2
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1answer
35 views

Is Laplacian of product of functions (one is smooth) square integrable?

Let $\Omega\subset \mathbb{R}^d$: open, connected. Suppose $u\in L^2(\Omega)$ and $\Delta^n u\in L^2(\Omega)$ for $n=1,\dotsc,N$, where $\Delta$ is in the weak sense. Let $\zeta\in C_c^{\infty}(\...
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1answer
35 views

Problem from Evans PDE on $u$ and $v$ satisfying $u_t+u_x=d(v-u)$ and $v_t-v_x=d(u-v)$

This is a Problem from Evans PDE 2nd Edition,Chapter 2 Problem 22. Let $u$ denote the density of particles moving to the right with speed one along the real line and let $v$ denote the density of ...
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1answer
26 views

Leray projection on Besov spaces

I'm reading a book whose author claims it's very clear that $\mathbb P\nabla\cdot$ maps the Besov space $(B_{\infty,\infty}^{-N})^{d\times d}$ to $(B_{\infty,\infty}^{-N-1})^d$ where $\mathbb P$ is ...
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1answer
103 views

Intuitive explanation of a stochastic PDE

Lindgren et al 2011 connects Gaussian Markov Random Fields (which have fast calculation properties due to the Markov attribute) and Gaussian Processes (which can model many types of data). The ...
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1answer
25 views

Modeling - First Order vs Higher Order Differential Equations

Throughout my engineering education, I've only witnessed modeling using First Order differential equations. Almost all of my Higher Order knowledge was given either via a spring or heat equation or ...
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1answer
31 views

Continuity of the probability that a Brownian motion with drift hits an upper barrier before the lower barrier in the drift

Let $W$ be a Brownian motion and $u:\mathbb{R}_+ \to \mathbb{R}_+$ an upper barrier and $l:\mathbb{R}_+ \to \mathbb{R}_-$ a lower barrier. Let $$\tau_u(\mu) = \inf\{ t \colon \mu t + W_t \geq u(t)\}$$...
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0answers
15 views

Non-uniqueness of solutions to the steady-heat equation on the disk that do not converge uniformly to the boundary

According to exercise 18, chp. 2, of Stein & Shakarchi's Fourier analysis, $\frac{\partial P_r(\theta)}{\partial \theta}$ is a solution to the steady-heat equation that converges only pointwise to ...
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0answers
17 views

Estimate Sobolev function by its derivative on a ball

If $f$ is a smooth function and $f(0)=0$, it is clear that $\|f\|_{L^\infty(B)}$ can be estimated (for any Ball $B$) if $\|\nabla f\|_\infty$ is known. My question is whether something similar can be ...
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1answer
39 views

Method for solving collection of simple PDEs

How would you go about evaluating the following collection of simple PDEs: $$\frac{\partial A_3}{\partial y} - \frac{\partial A_2}{\partial z} = yz$$ $$\frac{\partial A_1}{\partial z} - \frac{\...
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0answers
83 views

Estakhr's relativistic correction & Navier-Stokes existence and smoothness problem

Why does this physical relativistic correction does not seem to have an affect in solving the mathematical problem of Navier-Stokes existence and smoothness? or it does!? $$\bbox[#AFA,5px,border:2px ...
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0answers
37 views

Why the use of a certain weak formulation?

For an open set $\Omega \subset\mathbb{R}^n$, consider the following problem: $$\left\{\begin{matrix} - \Delta u + u=f & \mbox{ in }\ \Omega \quad \quad \quad \quad \quad \\ \ \ \ \quad \...
0
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1answer
41 views

Why is the sum $\sum_{n=0}^{N-1}\left(e^{\frac{\pi i j n}{N}}\right)^2=0$?

The background to this question is it's the proof that $\|\cos\left(\frac{jn\pi}{N}\right)\|=\sqrt{\frac{N}{2}}$ and $\|\sin\left(\frac{jn\pi}{N}\right)\|=\sqrt{\frac{N}{2}}$ asked by question 7 here....
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0answers
32 views

Finding Green's function

I have this problem. I am studying a physical problem and I came to this equation: $$ \frac{\partial}{\partial t} (R^{1/2} \Sigma) = \frac{12\nu}{s^2} \frac{{\partial}^2}{{\partial s}^2} (R^{1/2} \...
3
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1answer
29 views

Understanding Energy minimization and poisson equation

Let $M$ be a Riemmanian manifold and $X$ be a vector field thereon. My question is why are these two problems equivalent?: \begin{equation} \operatorname{argmin}_{\phi}\int_M |\nabla \phi - X|^2 \...
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0answers
41 views

Optimizing potential

Let $X$ be a vector field on a Riemmanian manfiold $M$. I recently read that solving: \begin{equation} \operatorname{argmin}_{\phi}\int_M (\nabla \phi - X)^2 d\mu, \end{equation} where $\mu$ is ...
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1answer
33 views

Eigensolver for Black-box matrix

$\DeclareMathOperator{\diag}{diag}$ Consider the generalized eigenproblem $A\mathbf{x}=\lambda B\mathbf{x}$. When solving PDEs numerically (specifically, I am interested on finding the Dirichlet ...
0
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1answer
45 views

Solutions to the Following PDE

I was wondering if there is any solutions for the following PDE: $$\frac{\partial \alpha}{\partial x}\frac{\partial \beta}{\partial y}=\frac{\partial \alpha}{\partial y}\frac{\partial \beta}{\partial ...
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1answer
41 views

Solving telegrapher's partial differential equation

Using the method of separation of variables and writing $u(x,t)=M(x)N(t)$, we can solve the equation $$u_{tt}-\gamma^2 u_{xx} + 2\alpha u_t=0$$ $$0<x<l$$ $$t\ge0, \alpha>0 \text{ (}{\alpha} \...
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1answer
32 views

Long time asymptotic of Fokker–Planck equation $\; \partial_tu-\nabla\!\!\cdot\!\left(\nabla u+xu\right) = 0$

Is it true that given a solution to the Fokker–Planck equation $$\partial_tu-\nabla\!\!\cdot\!\left(\nabla u+x\hspace{0.2ex}u\right) = 0,$$ then we have $$ \left\|\frac{u-\rho}{\rho}\right\|_{\...
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1answer
32 views

Trace Theorem and Neumann boundary.

I've been studying Trace Theorem. From PDE Evans, we have THEOREM 1 (Trace Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator $$T : W^{1,p}(U) \...
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0answers
51 views

A bound for a solution of a PDE

Let $u(t,x):\mathbb{R_+}\times\mathbb{R}\rightarrow\mathbb{R}$ be a very smooth function, which satisfies the equation: $$\dfrac{\partial u}{\partial t}+f(x,u)\dfrac{\partial u}{\partial x}=g(x,u),$$ ...
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0answers
22 views

Wavelet reference for PDE

Can anyone recommend a very readable introduction to wavelets for use in theoretical PDE/harmonic analysis? I'm frustrated with the account in Lemarie-Rieusset's Navier-Stokes book since he provides ...
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20 views

Prove this space is a Hilbert space

Let us consider a convex polygonal and bounded domain $\Omega$ in $\mathbb{R}^2$ containing two subdomains $\Omega_1, \Omega_2$ which satisfy $\overline{\Omega}=\overline{\Omega}_1\cup\overline{\Omega}...
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1answer
28 views

Finding $A$ such that $\nabla \times A = B$ for given $B$.

Let $B:U \rightarrow \mathbb{R}^3$ be a $C^\infty$ vector field, where $U = \mathbb{R}^3 \backslash \{(0,0,z):z \in \mathbb{R}\}$, defined by $$B(x,y,z)=\frac1{\rho^l} (-y,x,0)$$ where $\rho = \sqrt{...
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1answer
84 views

Weak solution for equation $-\Delta u = f$.

Suppose $f \in L^{2}$. I know that $$\left\{\begin{array}{c} −\Delta u = f(x) & \text{on }\Omega \\ u(x)=0, \partial\Omega \end{array}\right.,$$ where $\Omega \subset \mathbb{R}^{N}$ is open ...
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0answers
22 views

Solution route to solve coupled nonlinear PDEs?!

When I have a nonlinear coupled non-steady PDEs, what are the steps to get a solution. My background is for single linear PDE in which I distretize space and time terms, get system of linear equations,...
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1answer
68 views

Is $\Delta C_c^\infty$ a dense subset of $L^p(\mathbb{R}^d)$?

I'm struggling to obtain some density result. It is well known that $C^\infty_c(\mathbb{R}^d)$ is dense in $L^p (\mathbb{R}^d)$ for $1\leq p<\infty$. It is well known that for $\lambda>0$, $(\...
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0answers
36 views

Maple: Symmetries of strange modified heat-equation

I thought about the following PDE (the $u(x=0,t)$ is not a boundary condition! It really is a part of the PDE): $$\dfrac{\partial u(x,t)}{\partial t}=\alpha \dfrac{\partial^2 u(x,t)}{\partial x^2}+u(...
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1answer
30 views

How to solve this equation: $v_1\partial_xu+v_2\partial_y u=0$

I have the following equation: $$ v_1\dfrac{\partial u}{\partial x}+v_2\dfrac{\partial u}{\partial y}=0 \tag{1}$$ $u(x,y)$ is the unknown function (a scalar-valued function), $v_1$ and $v_2$ are two ...
2
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1answer
47 views

A non-strict inequality on skew symmetric matrices

As we know that skew-symmetricity means $A=-A^\top$ where $A\in\mathbb{R}^{n\times n }$. But recently I came across an inequality that states, $A+A^\top\preceq0$ can also be considered as an ...
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1answer
39 views

Solve Kolmogorov differential equations for birth-death process with constant rates

I need to solve the Kolmogorov forward equations for a birth-death process whose birth/death rates $\lambda_k,k=0,\ldots$ and $\mu_k,k=1,\ldots $ are constant, i.e., $\lambda_k=\lambda$ and $\mu_k=\mu$...
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1answer
41 views

Vainberg Theorem in measure theory

In a lecture notes about Variational Methdos, I found the following theorem: THEOREM: Let $(f_n)$ a sequence in $L^{p}(\Omega)$ and $f \in L^{p}(\Omega)$, such that $f_{n} \rightarrow f$ in $L^{p}(\...
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1answer
27 views

Solving quasilinear PDE - 1D, time-dependant, convection

I have a task to solve the following quasilinear PDE (find $c(x,t)$): $$ c_x v + c_t = - v_x c $$ $c \in (0,20) , t \in (0, \infty)$ where I know function $v(x)$ to be $v(x) = \frac{3}{40}(1+\cos(\...
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0answers
18 views

Type of boundary conditions of this differential equation (neumann or Dirichlet) and if its homogeneous or inhomogeneous.

The equation is: $$S\frac{\delta h}{\delta t}= K\left(\frac{\delta^2}{\delta x^2}+\frac{\delta^2}{\delta y^2}\right)h+R$$ being S, K, and R constants and h the potential. And the boundaries are: $h(...
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0answers
22 views

Expressing spherical harmonics as a combination of other spherical harmonics

Spherical harmonics are a useful tool in physics, particularly in classic electrostatics and electrodynamics. Given an integer $l$, the spherical harmonic $Y_{l,m}$, where $-l\leq m\leq l$, solves the ...
2
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0answers
25 views

Wave equation in a cube

Is it possible find a computable solution to the following homogeneous wave equation problem: Let $\mathcal{C}=\{(x,y,z)\in \mathbb{R}^3, 0< x,y,z < 1\} $ be the open unit cube. Find $u$ such ...
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1answer
19 views

“Hessian” differential equation

In my homework, I'm given the following problem: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a twice differentiable function. For an $\alpha \geq 2$, let: $$f(\lambda x) = \lambda^a f(x)$$ for ...