Questions tagged [linear-pde]

Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.

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28 views

Finite element method reference, from the perspective of the finite elements themselves

I found the finite element chapters in The Finite Element Method of Elliptic Problems especially enlightening and would like to learn more about the theory behind the base components of a general ...
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1answer
81 views

Space of holomorphic functions multiplied by smooth functions taking real values

Suppose we have a fixed function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (sufficiently regular, say $C^\infty$ ). The question is: for which $f$ there exists a scalar function $g: \mathbb{R}^2 \...
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44 views

Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)

I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it: ...
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0answers
25 views

Solution of the Mason-Weaver equation in cylindrical coordinates

I'd like to know if there is some litterature about the cylindrical Mason-Weaver equation. The basic Mason-Weaver equation is treated in Wikipedia : $$\partial_t f(t,x)=\partial_x f(t,x)+\partial_{...
3
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1answer
137 views

Ancient Heat equation and Liouville's theorem

I encounter a difficulty when proving the bounded solution of ancient heat equation implying constant function. Suppose $u(t,x)$ is the solution of ancient heat equation: \begin{equation} u_{t} = \...
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0answers
58 views

Weak maximum principle for a perturbation of the Laplacian

This question is motivated by similar considerations for the Kohn-Laplacian in the Heisenberg group, but it seems that I cannot even give an answer in the Euclidean case, so here we go. Suppose that ...
3
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2answers
253 views

Gradient $L^p$ estimates for heat equation

I'm looking for a proof of the gradient estimate associated to the heat equation with Dirichlet boundary conditions, to see if I can express the constant $\color{red}{C}$. $$\|e^{t\Delta_d}f\|_{W^{1,...
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1answer
201 views

Regularity on the boundary for the heat equation with linear source

This is probably a known problem but I was not able to find exactly what I am looking for. I have the following linear heat equation with zero-flux boundary conditions: \begin{equation} \begin{cases}...
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154 views

Equivalent definitions of differential operator

This puzzles me from some time and is in parts connected to the questions Symmetrized derivatives version and Symmetrized derivatives version II. For me the linear DO between vector bundles $E$ and $...
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0answers
157 views

A basic question about the Spectral Theorem

Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e. $$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
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1answer
345 views

Existence for an overdetermined system of PDEs

I am interested in the existence of a vector valued solution $y = y(x, t) \in\mathbb{R}^n$ to a system of $2n$ equations: there are twice more equations than unknowns. More precisely: Let $A$ and $...
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266 views

System of linear pde with non constant coefficients

I'm a physicist and during my research work, I found a system of linear pde with non constant coefficients that I have to study, since I have totally no experience about systems of pde and I have even ...
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0answers
88 views

Justification for uniqueness of solutions to dispersive PDE

For the sake of concreteness, we consider the linear Schrödinger equation $$ \partial_t u = i\Delta u, \ \ \ \ u(0, x) = u_0(x). $$ The solution is typically obtained by taking the Fourier transform ...
4
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1answer
283 views

Lack of exponential $L^2_{t,x}$ decay for a heat equation with growing coefficients

Edit: I have changed the nature of the question, but in order to have a better idea of what I can expect for the original problem (see below). Given $T>0$ and $n \in \bf Z$, consider the following ...
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3answers
374 views

Solutions of $\Delta \phi + \phi =0$ on $\mathbb{R}^2$

I know a few solutions of the equation $\Delta \phi + \phi =0$ on $\mathbb{R}^2$. They can be described as the Fourier transform of any finite measure $\mu$ on $S^1$. In particular take $\mu$ a finite,...
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1answer
98 views

Anti-diagonal matrix operator

Let $(A,\mathcal{D}(A))$ be an infinitesimal generator of a strongly continuous semigroup $(T(t))_{t\ge 0}$ on a Banach space $X$ and define on $\mathcal{X} := X \times X$ the operator matrix $$\...
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35 views

Reference request: existence of strong solutions to a linear parabolic problem with mixed boundary conditions

on a domain $\Omega \subset \mathbb{R}^d$ with smooth boundary $\partial\Omega$ subdivided into two parts $\Gamma_D$ and $\Gamma_N$ I am considering the parabolic problem $$ \partial_t u = \Delta u + ...
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105 views

Geometric interpretation of Lagrange's linear equation and its solution [closed]

What is the geometric meaning of the Lagrange's linear equation, $$P\dfrac{\partial z}{\partial x} + Q \dfrac {\partial z}{\partial y}= R \\Pp+ Qq= R$$ where $P$,$Q$,and $R$ are functions of $x\,$,$\,...
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44 views

A linear first order PDE with boundary condition

I want to solve the following first order PDE $$ (\star)\quad\begin{cases} \nabla u\cdot \nabla\xi=f \quad\text{in}\,\Omega, \\ u\mid_{\partial \Omega}=0 \end{cases} $$ where $\xi\in C^2(\overline{\...
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0answers
60 views

Solution of equation on vector field

I have a vector field function $\vec{J}: {\bf R}^3\to {\bf R}^3$ looking like: $$ \vec{J}(\vec{r}) = (\vec{B} \times \vec{v}(\vec{r}))\rho(\vec{r}) $$ with a (very well behaved) real, positive, ...
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1answer
637 views

Research topics in Microlocal Analysis

Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution ...
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1answer
150 views

Reference request: Schauder estimates for parabolic equations

Where can I find Schauder estimates for second order linear parabolic equations (in divergence form with potential)? Any reference would be highly appreciated.
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2answers
169 views

$W^{k,1}$ regularity for elliptic equations

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and assume $u$ is a solutions of $\nabla \cdot (a \nabla u)=f$ with $a>c>0$ in $\Omega$, where $a\in C^k(\bar{\Omega})$. Is the following ...
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122 views

6 linear PDE for only 3 unknowns?

Let $x \in (0,L)$, $t \in (0,T)$, and let $u_0 = u_0(x) \in \mathbb{R}^3$, $g=g(t) \in \mathbb{R}^3$, $P = P(x,t) \in \mathbb{R}^3$ and $Q = Q(x,t) \in \mathbb{R}^3$ be continuously differentiable ...
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1answer
113 views

First order partial differential equation [closed]

I know there is a solution to this pde $$\partial_{t} f(t,x)= \partial_{x}(v(x)f(t,x))$$ $$ f(0,x)=g(x)$$ ( Where $v$ and $g$ are known functions) which is given by $$ f(t,x)=\frac{1}{v(x)} h(t+\...
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1answer
176 views

Eigenfunctions of elliptic equations

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $a_1, a_2$ be smooth positive functions such that $a_1-a_2$ is compactly supported in $\Omega$, and $a_i>c>0$, for some constant $c$. ...
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1answer
158 views

Global solutions of the wave equation with bounded initial condition

Let $f,g$ be bounded compactly supported smooth functions, and assume $u$ is the solutions of the wave equation $$u_{tt}-c^2(x)\Delta u=0 \ \ \hbox{on} \ \ \mathbb{R}^n \times (0,\infty)$$ $$u(x,0)=f, ...
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1answer
104 views

Two PDE for one unknown?

Let $x \in (0,L)$, $t \in (0,T)$, and let $f_1 = f_1(x,t) \in \mathbb{R}$, $f_2 = f_2(x,t) \in \mathbb{R}$, $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions. My ...
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68 views

Wave equation regularity

I have an equation of the type $$\hat{\rho} u_{tt}-\hat{E}u_{xx}=f(x,t)$$ for $x\in (0,1)$ and $t>0$, where $\hat{\rho}$ and $\hat{E}$ are constants, $u(0,t)=u(1,t)=0$, $u(x,0)=p(x)$, $u_t(x,0)=q(...
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100 views

Harnack Inequality for uniformly elliptic PDE via constructing a singularity

I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
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159 views

Replacing the initial conditions for a PDE

The problem The PDE I am working with is given by $\left(\partial_a^b \leftrightarrow\frac{\partial^b}{\partial a^b}\right)$ $$\partial_t \psi = i \partial_x^2 \psi$$ $$\psi(x,t=0) = \psi_0(x)$$ $$\...
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1answer
64 views

Exact solution of two coupled transport equations

I want to solve the following system $$\eqalign{ & y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr & z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr & y(0,x) = y_0,\,\,z(...
5
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3answers
667 views

fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix

I am looking for the fundamental solution of the following PDE $$\partial_i (a^{ij}\partial_j u)=f$$ where $a^{ij}(x)$ is a non-symmetric matrix with possibly non-constant coefficients. I could ...
4
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1answer
258 views

Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$

Consider the following PDE: \begin{equation} p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$} \end{equation} where $g$ is a flat function at the point (...
2
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0answers
73 views

Wave equation with 'spring' integral boundary condition

I am really stuck with this small toy problem. I am trying to solve a wave equation on unit interval $x \in [0,1]$ with a specific boundary condition which I can't see how to tackle. The problem is: ...
3
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3answers
232 views

ODE with Bessel decay

This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily. I would like to estimate the asymptotic behaviour of the ...
2
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0answers
55 views

Bessel decay for nonhomogeneous PDE

I'm interested in the following nonhomogeneous PDE $$ (\Delta-k^{2})u=-g $$ on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...
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1answer
608 views

Moduli space of linear partial differential equations

Is there a way to view "the space of all possible linear PDE's" as an algebraic variety with singularities? This is in connection with a quote from someone on the web that I saw a long time ago. At ...
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0answers
65 views

Nodal domains on a surface

What is it known about the topology of nodal domains of eigenfunctions of self-adjoint operators? In particular I'm interested in self-adjoint operators on a complete, non-compact, surface $\Sigma \...
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0answers
54 views

Uniqueness of solution of linear PDE of first order

Let $\vec u\colon \mathbb{R}^n\times \mathbb{R}\to \mathbb{R}^k$ be at least $C^1$- smooth vector valued function. Assume they satisfy a first order linear equation $$\partial_t \vec u(x,t)=\sum_{j=1}^...
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1answer
135 views

Existence of unique critical points to second order elliptic PDEs

Let $\Omega\subset\mathbb{R}^2$ is bounded and convex and $\partial\Omega$ be smooth. Consider the second order elliptic PDE (1) $$ \begin{cases} Lu = f &\text{ on } \Omega\,,\\ u=0 &\text{ ...
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0answers
61 views

Rankine Hugoniot Condition for non piecewise smooth solution

I studied the following theorem: (Rankine-Hugoniot condition) Let $u:\mathbb{R} \times [0,+\infty) \to \mathbb{R}$ be a piecewise $C^1$ function. Then $u$ is a weak solution of the conservation law ...
4
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1answer
147 views

Method of characteristics beyond the Lipschitz setting

I have come across the following easy-looking problem that is driving me mad. I have a family of measures (on the real line $\mathbb R$) $\{\mu_t\}_{t>0}$ which is uniformly bounded (the measures ...
3
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0answers
54 views

How to solve this linear Cauchy Problem

within my thesis, I am struggeling with the following PDE: $u_t+a(x,y)u_{xx}+b(x,y)u_{xy}+c(x,y)u_{yy}+d(x,y)u_{x}+e(x,y)u_{y}+f(x,y)u=0$ $u(T,x,y)=1,$ where $a,b,c,d,e,f$ are polynomials and the ...
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0answers
38 views

Random solute transport equation

After reading the article Probability density functions for solute transport in random fields, by M. Shvidler, K. Karasaki, see https://pubarchive.lbl.gov/islandora/object/ir%3A116072/datastream/PDF/...
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1answer
94 views

Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting. I want to verify and compare different Discretizations of the ...
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2answers
356 views

Solving the Poisson equation using a random walk on $\mathbb Z ^d$

How do I solve the Poisson equation with the help of a discrete random walk on $\mathbb Z ^d$?
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0answers
111 views

Observability inequality for the heat equation

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions: \begin{array}{c} \varphi _{t}+\...
2
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0answers
129 views

Underdetermined PDE

Let $\Omega\subset\mathbb{R}^d$, $d>1$ be an open and bounded Lipschitz domain. It is well-known, that for a given function $f\in L^q(\Omega)$ with $1<q<\infty$ and and such that $\int_\Omega ...
0
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1answer
134 views

Unique continuation for the wave equation

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $\Gamma \subset \subset \partial \Omega$, assume $u$ solves the wave equation $$u_{tt}-c(x)\Delta u=0, \ \ u(x,0)=f(x),$$ where $f$ and $1-c$ ...

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