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

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Partial differential equation

PDE: $u(x,t)$ satisfies $u_t=u_{xx}+x+t-2$ IC: $u(x,0)=2x$ BC: $u_x(0,t)=-2(u(0,t)-t)$ and $u_x(1,t)=2+t$ Question: How shall I find $u(0,t)$?
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Reference request: The compactness and compact embedding in Besov Space?

This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions. Thank you guys! Let $\Omega\subset \mathbb R^N$ be open bounded with ...
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1answer
17 views

Weak convergence in the Sobolev space and compact embeddedness

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sopolev space $u\in C^0\left(\overline\Omega\times [0,\infty)\right)\cap C^{2,1}\left(\Omega\times ...
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1answer
15 views

About the boundary conditions of the Black-Scholes-Merton PDE

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve. Let $c(t,x)$ be the value of the ...
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Long-time behavior of the solution of a parabolic partial differential equation

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain $f\in C^1(\overline\Omega\times\mathbb R)$ $u_0\in C^1(\overline\Omega)$ with $u_0=0$ on $\partial\Omega$ $u\in C^0(\overline\Omega\times ...
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13 views

Long-time behavior of semilinear autonomous systems

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain $f\in C^1(\overline\Omega\times\mathbb R)$ $u_0\in C^1(\overline\Omega)$ with $u_0=0$ on $\partial\Omega$ $u\in C^0(\overline\Omega\times ...
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1answer
32 views

How to numerically solve the Poisson equation given Neumann boundary conditions?

I want to solve the Poisson equation on a 2D domain given Neumann-type boundary conditions: The PDE: \begin{equation} \nabla^2 \; u(r,\theta) \;=\; f(r,\theta) \end{equation} The boundary ...
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2answers
434 views

Well-posedness of the Poisson problem with mixed boundary conditions

Let $\Omega \subset \mathbb R^n$ be a subdomain with Lipschitz boundary, i.e. locally any part of the boundary looks like the graph of a Lipschitz continuous function, after some affine coordinate ...
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9 views

Jacobian of Stabilized Eikonal Equation $| \nabla u| = 1$

I am trying to implement a Finite Element Solution for the Stabilized Eikonal Equation : $$ |\nabla u| = 1 + \Gamma \Delta u, \quad \text{ where } \quad u = \text{ distance function }$$ $$ ...
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1answer
18 views

Evaluating 'Constant' Term

suppose I have a pde $$u_{xt}(x,t)+u(x,t)u_{xx}(x,t)=h(t),\,\,\,\, x\in[0,\pi],\,\, t>0$$ for some unspecified function $h(t)$. This question is about finding what $h(t)$ is. Please, you may ...
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191 views

How to classify/ solve this PDE?

I am searching how to solve the PDE below but I can not seem to find a decent example online. My major did not focus much in solving PDEs so I feel very deficient. I know how to solve for the steady ...
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1answer
26 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\left( ...
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118 views
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Question about boundedness of a sequence in $ W^{3,q} $ for any $ 1\leq q < \frac{N}{N-1} $

I have asked this question several months ago, I have understood every thing and there are good comments and they have helped me , but only I have a question about tomas comment, how can ...
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106 views

Rearranging a spherical harmonics expansion

Referring to this article (click to enlarge): and How is it that they get from equation 2 to equation 3? Whenever I do it, I can't cancel the imaginary terms Is there some spherical harmonics ...
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A stochastic variant of the heat equation modulo $2\pi$ has weird unstable particle-antiparticle solutions. Does this equation have a name?

I implemented a discretization of a weird 2D heat equation "mod $2\pi$", $$\dot{f}(\mathbf{x},t)=\Delta^*f(\mathbf{x},t)$$ where (WARNING: handwavy, I'm not sure I understand it) ...
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1answer
435 views

Relationship between integral equations and partial differential equations

In my functional analysis class, we did a lot of problems involving integral equations such as proving existence & uniqueness using spectral theory and the Banach fixed point theorem. I've never ...
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1answer
38 views

Fundamental solution for a parabolic PDE with costant coefficents

as it is well known, the fundamental solution of the heat equation is the function $G(t,x)=\frac{1}{(4\pi t)^{n/2}}e^{\frac{|x|^2}{4t}}$, for all $t>0,x\in\mathbb{R}^n$. I wonder if exists (and ...
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2answers
36 views

how to solve the system of partial differential equations? [on hold]

I need to solve the following system: $$ \begin{cases} \frac {\partial K(x,y)}{\partial x}x + \frac {\partial K(x,y)}{\partial y}y+2k(x,y) = 0 \\ \frac {\partial K(x,y)}{\partial x}y - \frac {\partial ...
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2answers
70 views

Proof of an inequality involving gradient function

I'm reading ahead in my course, and I've encountered the following problem; Let $\Omega \subset \mathbb{R}^n$ be a bounded domain such that the divergence theorem holds. Assume that $u \in ...
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Question 13 in Taylor's PDE vol III section 16.1.

My question comes from Taylor's PDE textbook, volume III. Consider a semilinear hyperbolic system, $u_t=Lu+g(u)$, $u(0)=f$, where $Lu=\sum_j A_j \partial_{x_j}u$, $g(0)=0, \ |g'(u)| \le C$, take ...
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1answer
58 views

How to prove the inequality in Burgers equation and what is the relationship to energy dissipation

Consider Burgers equation: \begin{equation} u_t+uu_x-\epsilon u_{xx}=0 \quad , \quad u(x,0)=g(x) \quad , \quad u = 0 \mbox{ when } |x| \mbox{ large}, \end{equation} where ...
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1answer
15 views

Higher accuracy of numerical derivative in 2D case

Recently, I face a problem about solving a PDE (2D in spatial direction) and I stuck on the discretization of the 1st order derivative. My stencil is as follow There are five points in my stencil. ...
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1answer
18 views

Numerical method for nonlinear wave equation

I need to solve the following nonlinear wave equation numerically $U_{tt}=(1+\epsilon U_{x}^2)U_{xx}$ with Initial conditions. what is the best method for solving it? I tried the finite elements ...
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1answer
298 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: ...
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3answers
74 views

Simple Partial Differential Equations

Solving Partial Differential Equation $x f_{y}+y f_{x} =0$ ? $f=f(x,y)$ is a real-valued function of two variables $x,y$. Thanks.
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Under some regularity assumptions to the boundary $\partial\Omega$, the first weak eigenfunction of $-\Delta$ in $\Omega$ is also a strong one

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $H:=W_0^{1,2}(\Omega)$ be the Sobolev space, $C^{2,\alpha}(\Omega)$ be the Hölder space for some $\alpha\in (0,1]$ and ...
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1answer
17 views

Convergence of a sequence of Hölder continuous functions with respect to the Sobolev norm

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $W_0^{1,2}(\Omega)$ be the Sobolev space and $C^{2,\alpha}$ be the Hölder space for some $\alpha\in (0,1]$. Suppose $(u_k)_{k\in\mathbb ...
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1answer
36 views

What theorem is this? (in PDE)

I'm confused because it's titled as "Gauss's theorem about heat flux" (not in English though, I'm translating), but instead of the heat equation there's Laplace's equation written above the theorem. ...
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2answers
76 views

Analytical Solution of a PDE

I need to solve a PDE which seems to be quite simple and to have an analytical solution. I tried the method of separation of variables, but could not complete the solution. Could you please let me ...
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12 views

Connection between two same form equation but with different independent variables [on hold]

I have two different nonlinear equations. Both of them are reducible to heat equation with different independent variables. Then all the analysis which are true for 1st equation will work out for the ...
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75 views

How do I solve first order non-linear system of PDE: $\partial f^i(x,y)/\partial z = F^i(f^1,f^2,…,f^n)$?

Suppose that I have a system of PDEs of the following form: \begin{eqnarray} \frac{\partial f^i(x,y)}{\partial z} = F^i(f^1,f^2,...,f^n), \qquad i = 1,..,n \end{eqnarray} Where $z = x + iy$, ...
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finite differences nonlinear least squares

I am facing to following nonlinear least-squares problem: $$\min_{u,\gamma} \frac{1}{1000} \int_{ \gamma(x,y)^2} + \int_{ [u(x,y)−u (x,y)]^2} + \int_{ [∆u(x,y)−\gamma(x,y)u(x,y)]^2}$$ where the ...
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1answer
24 views

Couple stress tensor reference.

Can someone give me a good mathematical reference for couple stress tensor in its most basic form. Thank you.
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27 views

Compact Imbedding into $L^{1} (\Omega)$ [on hold]

Let $\Omega$ be a bounded subset of $\mathbb R^{d}$ with a Lipschitz continuous boundary.Prove that: the canonical imbedding of $BV(\Omega)$ into $L^{1}(\Omega)$ is COMPACT .
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Regularity of a weak solution

The problem is formulated as follows: Let $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. For $u\in L_2(\Omega)$ one can define the one dimensional weak dirivative $\partial_1 u$. Now define ...
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Regularity of solutions of Parabolic PDE

Consider the equation: $u_{t} - a \Delta u + \lambda u = g $ in $\mathbb R$ X $(0,T)$ $u(x, 0) = u_{0}(x)$ Where $a \gt 0$ & $\lambda \in \mathbb R$ are given constants!! Now, assume that for ...
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1answer
34 views

Uniqueness of harmonic function with Mixed Dirichlet Neumann condition

Let $u \colon \{\mbox{Im } z>0\}\subset\mathbb{C}\to \mathbb{R}$ be a positive harmonic function in the upper half plane, i.e $$ \Delta u=0,\,\, \mbox{for}\,\mbox{ Im } z>0. $$ Consider now the ...
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23 views

Existence and uniqueness of a pde solution

I have the PDE system: $\frac{\delta}{\delta t}u(t,r)=-\int_0^1 H(|r-r'|)v(t,r')dr'u(t,r)$ $\frac{\delta}{\delta t}v(t,r)=\int_0^1 H(|r-r'|)v(t,r')dr'u(t,r)-v(t,r)$ $x(0,r)=\rho(r), ...
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33 views

The best constant in Poincare-liked inequality in BV space

Let $\Omega\subset \mathbb R^N$ open bounded smooth boundary. Then we could prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E $, where $\mathcal E$ denotes the ...
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Trace zero not needed for $H^2$ regularity if $V_N\subset H^2$ is finite dim?

Reading Evans and this note after asking this question, I have been thinking about the estimates for interior/global regularity in Evans, 6.3.1, theorem 1, and thoerem 4 in 6.3.2, of the form ...
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1answer
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Existence of the first weak eigenvalue of the Laplacian in a bounded domain

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $$H:=W_0^{1,2}(\Omega):=\left\{u\in L^2(\Omega):\nabla u\in L^2(\Omega)\right\}$$ be the Sobolev space, where $\nabla u$ denotes the weak ...
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1answer
478 views

A question about Fisher's Equation and the Traveling Wave Equation

I am dealing with the Fisher's Equation: Consider the PDE: $u_t=u_{xx}+u(1-u)$, where $-\infty<x<\infty$ and $t>0$.Prove that there exists $c^*>0$ such that for each $c>c^*$ there ...
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1answer
21 views

Maximum Principle for the PDE $\Delta u - a^2u=a^2$

I have this Dirichlet probem \begin{align*} \Delta u - a^2u&=a^2\quad\text{on}\;\, \Omega \subset \mathbb{R}^n \\ u&\equiv 0\quad \, \text{ on }\partial \Omega, \end{align*} where $a^2$ is ...
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1answer
88 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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Nonlinear Solution to PDE (sine-Gordon Equation)

So I have this nonlinear PDE, the sine-Gordon Equation, $u_{tt}-c^{2}u_{xx}+\omega_{p}^{2}\sin u=0$ whose linearized solution is given by $u_0$. ($c$ and $\omega_p$ are constant.) My reference tells ...
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1answer
27 views

PDE model of metal rod at temperature=1 plunged into a bath of temperature=0

Consider a metal rod (0 < x < l), insulated along its sides but not at its ends, which is initially at temperature=1. Suddenly both ends are plunged into a bath of temperature=0. Write the PDE, ...
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Are there existence results for the heat equation on unbounded Lipschitz domain?

I am looking for a reference/ ideas on the following problem. Let $\Omega\subset\Bbb R^2$ be a Lipschitz domain (if it helps, the domain can be piecewise smooth with only one "kink", for example the ...
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48 views

Orthonormal basis of $L^2$ and it's impact on the solution to the heat equation.

If we consider the homogeneous Dirichlet eigenvalue problem on a bounded domain $\Omega\subset\Bbb R^n$ - (one part of my question is if we can assume $\Omega$ to be a Lipschitz domain and still ...
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24 views

Stability in partial differential equations

I have the following PDE, with parameters $a$ and $b$: $$ \frac{\partial c}{\partial t} = \frac{\partial}{\partial z} \left( a c + b \frac{\partial c}{\partial z}\right) $$ with, for now, just one ...
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1answer
51 views

What is the difference between single and double layer potential

I want to know the difference between single layer and double layer potentials. Is there a link between the choice of single/double layer potential and the boundary condition of a PDE or an ...