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

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Harnack inequality

Let $u(x)$ be an entire positive solution of the equation $$\Delta u - u = 0 \quad\hbox{on $\mathbb{R}^{n}$, $n>1$}$$ Assume the validity of Harnack inequality on any unit size balls for positive ...
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1answer
16 views

Weakly * continuous definition

What does it mean that $$t\to u(t,\cdot)$$ is waakly* continuous from $[0,T]$ to $L^\infty(\mathbb{R}^d)$? I guess that I have to see $L^\infty(\mathbb{R}^d)$ as the dual of $L^1(\mathbb{R}^d)$ and ...
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0answers
12 views

Hyperbolic Systems and Charactersitics

I am trying to follow the derivation of finding the characteristics of a hyperbolic system in the general case. I understand up till the equation, where I have highlighted the RHS in green. I ...
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1answer
33 views

Properties of weak derivatives in Sobolev spaces

In PDE Evans (chapter 5, Sobolev spaces), 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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25 views

Derivation of fundamental solution of heat equation by reduction to ODE - Question on integration factor

In the derivation of fundamental solution for heat equation ( as in PDE by L.Evans ), we come across the reduction to following ODE : $\alpha w + {1\over2}r w'+ w'' +{n-1\over{r}}w' = 0$ Set ...
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20 views

Solving an integral equation in general

I have an integral equation such that $$\int_t^Tf(s)g(s,t)~ds=h(t)$$ where $g$ and $h$ is given. we want to know function $f$ explicitly. As I know, this type of question is about the integral ...
2
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0answers
35 views

Eigenvalues problem for generalized Kuramoto-Sivashinsky equation

I been working on Kuramoto-Sivashinsky Equation. In the process of analysis, I need to solve the following eigenvalues problem $$ -u_{xxxx}-\lambda u_{xx}=\beta(\lambda)u $$ where $\lambda$ is a ...
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1answer
19 views

How rigorous is multiplying both sides of an eqaution for the differential of a function?

I have to solve this equation: $$ -C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A $$ where $C_0$ and $A$ are two real nonzero constant; $f:\mathcal{R}\to \mathcal{R}$ I have seen that the person who ...
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1answer
17 views

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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1answer
35 views

Definition of weak divergence

Can anyone give me the definition of the divergence of a vector field in the distributional sense?
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26 views

checking the solution of PDE

Let $u(x)$ be an entire solution of $\Delta u = 1$ on $R^{n}$, $n>1$. If $u(x)$ is also convex, that is $(D^{2}u(x))$ is non-negative definite for all $x$. Then $u(x)$ is given be a quadratic ...
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0answers
29 views

Mean value formula

Let $u(x)$ be an entire positive solution of the equation $$\Delta u - u = 0 ~~~on ~ R^{n} ~~n>1$$ (a) Can you find a mean value formula? (b) Let $\mu$ be a positive Radon Measure on the unit ...
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15 views

Partial Differential Equation with a flux term

I don't understand why $\phi=\frac{1}{2}u^2$ is the flux in this case?
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1answer
27 views

The range of the distributional laplacean, defined in $W_0^{1,1}(\Omega)$.

Let $\Omega\subset \mathbb{R}^N$ be a bounded, smooth domain. Assume that $u\in W_0^{1,1}(\Omega)$ and consider the distributional laplacean of $u$; $\Delta u$. My question is: when is $\Delta u\in ...
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1answer
8 views

Change of variables from intinite to bounded support.

I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in ...
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0answers
20 views

Maximum principle for linear elliptic operators of arbitrary order

What is known about maximum principles for strongly elliptic linear differential operators of even order (possibly higher than $2$)? By such an operator, I mean a linear differential operator with ...
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1answer
25 views

Weak solution of a parabolic equation when initial and boundary conditions are inconsistent

Consider the parabolic equation $$ u_t = u_{xx}, $$ $$ u(0,t)=u(1,t)=0, $$ $$ u(x,0) = 1. $$ Initial and boundary conditions are inconsistent but a weak solution exists. It belongs to ...
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1answer
35 views

When is separation of variables an acceptable assumption to solve a PDE?

We know that one of the classical methods for solving some PDEs is the method of separation of variables. It works for known types of PDEs and many examples of physical phenomena are successfully ...
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Non Linear general kinematic wave equation

I am rather confused by this section of my non-linear waves notes. In the parts I have underlined in green $c(u_0(\xi))$ is defined as a constant and then as a variable even though in both instances ...
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2answers
27 views

Fourier transform of a derivative

My PDE book gives the properties for the fourier transform of $u(x,t)$: $F(\frac{\partial^n}{\partial t^n}u(x,t))=\frac{\partial^n}{\partial t^n}\hat u(\xi,t)$, and $F(\frac{\partial^n}{\partial ...
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0answers
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How do I apply this maximum principle?

I have the maximum principle: $$\text{If } \psi\geq 0 \text{ on }\Gamma. \text{Then }L\psi\geq0\text{ implies } \psi\geq 0 \text{ in } \bar{D},$$ where $D=(0,1)\times (0,T], \Gamma$ is parabolic ...
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1answer
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Why are we discarding solutions to this heat equation?

Dirichlet problem on unit disc in polar: $u_{rr} + (1/r) + (1/r^2)u_{\theta\theta} = 0$ $u(1,\theta) = f$ Period in $\theta$ gives $u(r,\theta) = \sum R_n(r) e^{in\theta}$ Inserted into our PDE ...
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0answers
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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 ...
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1answer
54 views

Why isn't there a general comparison principle for higher order equations?

I am studying elliptic equations. For second order equations (linear and nonlinear), comparison principle has many applications, e.g. to show uniqueness of the (weak) solutions, to construct super- ...
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1answer
53 views

Function Approximation

I need to solve the following equation $$-\frac{\partial S(x,y,t)}{\partial t}=ax^2+bx\frac{\partial S(x,y,t)}{\partial x}+c\Big[\frac{\partial S(x,y,t)}{\partial ...
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Solving partial differential equation with mathematica.

I am trying to solve the heat equation in cylindrical coordinate. The object of inspection is a thick ring with height $\{z_0,z_L\}$ and radii $\{r_1,r_2\}$ within a heat generating environment. I ...
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1answer
15 views

Green identity for measures with compact support

Let $\Omega\subset\mathbb{R}^N$ be a bounded, smooth domain. Assume that $\mu \in \mathcal{M}(\Omega)$ has compact support in $\Omega.$ Let $u\in W_0^{1,1}(\Omega)$ be a solution of $$ \left\{ ...
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1answer
33 views

Function spaces for the 1-dim heat equation.

Consider the standard 1-dim heat equation: $u_t(x,t)-\alpha u_{xx}(x,t)=0$, where $u:\mathbb{R}\times\mathbb{R_+}\rightarrow \mathbb{R}$, with initial conditions $u(x,0)=g(x), x\in\mathbb{R}$ and ...
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1answer
18 views

Laplace Equation with non-const Dirichlet Boundary Conditions

I'm struggling to get a Laplace problem with inhomogeneous boundary conditions solved. My memories are very rusty, and it almost works out, but I've got my brain twisted in some way. So I'm kindly ...
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0answers
35 views

Problems with understanding a proof.

I have some problems with understanding the Div-Curl lemma's proof. More precisely I don't understand the first part of the proof of the Theorem 1.1 in the article below: ...
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1answer
32 views

Solve a system of equations.

I have a system of equations: \begin{align} & x_{21} (\frac{\partial}{\partial x_{11}}f_{1111})( x_{11} , x_{21}, y_{11} , y_{21} ) + \frac{y_{21}}{x_{11}^2} (\frac{\partial}{\partial ...
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Covariance Integral Operator and its inverse

can someone help me with the following question? I want to consider parameter estimation problems for a single parameter q using the likelihoodmethod, where the state $y$ is constrained by a pde. But ...
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0answers
19 views

Space-dependent diffusivity and finite-differences

I want to implement a finite difference code of this simple diffusion equation with space-dependent diffusivity: $$\partial_{t}u =D\partial_{x}^{2}u+\partial_{x}D\cdot\partial_{x}u$$ I go for a ...
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0answers
30 views

Meaning of “Canonical System of the First Order”

I am learning about PDEs and came across the following. "Convert a partial differential equation of higher order into a canonical system of the first order" What does the above statement mean/imply? ...
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1answer
49 views

Green's representation on a compact domain

This is from page 17-18 of Trudinger and Gilbarg Let $\Omega$ be a domain for which the divergence theorem holds. Let $\Gamma(x-y)$ be the normalised fundamental solution of the Laplace's equation, ...
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1answer
41 views

Applications of PDE and laplace equation

The edge r = a of a circular plate is kept at temperature f(theta). The plate is insulated so that there is no loss of heat from either surface. Find the temperature distribution in steady state. I'm ...
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0answers
33 views

Solving PDE via Fourier Transform & Uniqueness

When a PDE is solved via Fourier transform, is there already a uniqueness assertion that comes for free? For example, if we Fourier the heat equation \begin{align} \partial_t u(x,t) &= \Delta ...
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1answer
34 views

References on estimating capacities (Newton, Martin etc) for sets & alternative formulations.

By G-capacity for capacitable set K I mean: $Cap(K)=[inf\{\int\int G(x,y)d\mu(y)d\mu(x):\mu$ probability measure on K$\}]^{-1}$. where G(x,y) is any kernel eg. the Green kernel. Q1:We've calculated ...
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Solutions of a PDE in two dimensions

I should solve the following PDE: $$\dfrac{\partial u(x,y,t)}{\partial t}=a\nabla^2u(x,y,t)+b\nabla u(x,y,t)+g(x,y,t)-c u(x,y,t)$$ where $(a,b,c)$ are constants, $$u(x,y,0)=u_0$$ $$u(x,y,t)=0$$ for ...
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Is there a software to solve variational problem using finite difference? [closed]

I am looking for a software to solve the variational problem. I have formulated it using PDES. I want to use finite difference method. Preferably academic or commercial software for this. ...
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0answers
18 views

Local Mollification of a $L^1$ function

Let $f$ be a $L^1(\Omega)$ function, where $\Omega\subseteq \mathbb{R}^2$ is a smooth and bounded domain. Then can we find a fuction $\phi\in W_0^{2,2}$ suct that $\phi>0$ in $\Omega$ and ...
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2answers
74 views

Intuitive interpretation of $\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$

I'm trying to visualize what the following equation is saying: $$\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$$ where $S$ is a probability-density, but I think you can ...
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34 views

On an interesting boundary condition

So I am tackling an interesting boundary condition, where $B(Du)=0$, for $x\in\Omega$, where $B$ is the signed distance function to $\Omega^*$ (where $\Omega,\Omega^*$ are convex domains in $\Bbb ...
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3answers
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Pair of PDEs to be solved together

I have the following pair of equations to be solved together to find the functions $H_{x}$ and $H_{y}$, which are the components of a vector $\bar{H}(x,y)=H_{x}(x,y)\hat{x}+H_{y}(x,y)\hat{y}$ in ...
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1answer
91 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
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Second degree partial differential equation with variable-change

Edit: @Etienne mentioned that I did a typo, writing $u_y' = -xye^{-y}$ instead of $u_y' = -xe^{-y}$. I've corrected that in the calculations and now it's closer to being correct! Though I still miss ...
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1answer
43 views

Elliptic partial differential equations and elliptic operators

I'm starting to study elliptic partial differential equations and I just want to know if there are any connections between the following concepts: An elliptic partial differential equation is given ...
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13 views

Deriving lower bound for eigenvalues of laplace operator

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 \quad \mbox{ on } \quad D, \quad u = 0 \quad \mbox{ on } \partial D $$ and let $w$ be a function such that $\Delta w + \beta w < 0$ ...
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normalized mean curvature flow with convex initial hypersurface has finite velocity

I can't understand the prove in [Xi-Ping Zhu] Lectures on mean curvature flows. The statement as follow. Lemma 3.5 (page 32) There exists a positive constant $C$ such that ...
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Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...