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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Parabolic theory for singular coefficients on bounded domains (Reference Request)

In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems. Is ...
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2 votes
1 answer
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Using Darboux's to solve 2D system of first order linear PDEs with variable coefficients

I've spent some time over the last few days looking at the references suggested in this question and this question and I think the information therein is my best shot at solving this system that arose ...
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Generalized functional for solution of PDEs

Asked this on Math Stack Exchange awhile ago but it got ignored then deleted. To solve a differential equation of one variable, you need constraints equal to the number of derivatives. For a partial ...
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4 votes
1 answer
169 views

Maximum principle and linear transport

Let us consider the linear transport equation $$ \partial_t u + \mathrm{div}(a(t,x)u)=0 $$ with initial data $u(0,\cdot) = u_0$ in $\mathbb R^N$. Here we consider a smooth Lipschitz vector field $a$. ...
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Spectral analysis for nonlocal elliptic operator

Suppose $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary. We note by $(-\Delta)^{-1}$ the inverse Laplacian i.e. $f\mapsto u$ where $u$ is the unique solution to $$-\Delta u=f,\...
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Liouville theorem for an elliptic equation with gradient perturbation

How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation? Let $u \in L^2(\mathbb R^n;\mathbb R)$ be a smooth solution of $$ -\Delta u + v \cdot \nabla u = 0 ...
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Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter

Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example $$ \left\{\begin{aligned} \Delta \Phi &=\sigma f(x, y) \...
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Decay of solution for linear system with damping

Let us consider the following linear system with damping: $$ \begin{cases} u_t - u_x = -\frac{1}{2} (u+v)\\ v_t + v_x = -\frac{1}{2} (u+v) \end{cases} $$ Let's write the solution as $w=(u,v)$ ...
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Bounds for solution of first-order linear pde system

In the paper "Stokes formula on the Wiener space..." they derive a system of first-order linear pdes for finding a density $\rho(\vec{x})$: $$\sum_{i=1}^{n}\frac{\partial\left(\beta_{i,k}(\...
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Solve a coupled PDE in a rectangle

We consider a coupled PDE in a rectangle $\Omega=(-1,1)\times(-1,1)$. For the simplicity, we assume that the functions are periodic in $x_{1}$ direction. \begin{cases} \nabla\cdot u=f_{1},\ & \...
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Regularity of solution to Cauchy problem given regular initial data

Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus) be a solution of a Cauchy problem $$\frac{d}{dt} f(t) = A f(t)$$ $$f(0)...
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Reference for global theory of Schrödinger operators

Question. What is a good reference to learn about the spectral properties of Schrödinger operators in $\mathbf{R}^n$? I am specifically interested in references that discuss examples where the ...
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A singular differential equation

In a neighbourhood of $0$ in $\mathbb{R}^n$ a smooth function $h=h(x)$, $h(0)=0$, is given. Take arbitrary real numbers $w,\lambda_1,\dots,\lambda_n\in\mathbb{R}$. The problem is to find a smooth ...
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Has this form of the heat equation been solved for the radiation boundary condition

Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...
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2 answers
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Density of traces of solutions to an elliptic equation

Let $D_1$ be a domain with smooth boundary and assume that $D_1$ is a proper subset of $D_2$ which is itself a bounded domain in $\mathbb R^n$ with a smooth boundary. Assume also that $D_2\setminus ...
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How to find a particular solution of a non-homogeneous parabolic partial differential equation

Consider the following non-homogeneous parabolic partial differential equation (PDE) \begin{align} \left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \...
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2 answers
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spaces of smooth functions for linear hyperbolic PDE

Which classes of (scalar or systems of) linear first or second order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:...
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2 votes
1 answer
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Existence of a global analytic solution to a linear first order PDE

Let $B=\lbrace \|z\|<1\rbrace$ be a unit ball in $\mathbb{C}^n, n\geq 2.$ Let $f_1,\cdots, f_n, f$ be holomorphic functions on $B.$ Now, consider the following first order, linear PDE: $$f_1\...
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6 votes
1 answer
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Space of solutions to a fourth order wave equation

I'm interested in finding solutions a fourth order version of the standard wave equation in $d$ dimensional Minkowski spacetime $\mathcal{M}^d$. Defining $\Box := \partial_0^2 - \sum_{i = 1}^{d-1} \...
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Linear hyperbolic PDE on compact two dimensional domain

Consider the equation $$ \begin{equation} \frac{\partial^2f}{\partial x\partial y}=f \end{equation} $$ on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is ...
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3 votes
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Linear transport equation with Lipschitz conditions

Given the equation here, I would like to ask the following relaxed question: Consider the PDE $$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$ with Schwartz initial data $f(0,x) = ...
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5 votes
2 answers
323 views

Linear transport equation with unbounded coefficients

Consider the PDE $$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$ with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$ I am wondering then if $q$ and all its ...
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3 votes
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Is this equation of hyperbolic type?

I want to now whether this equation is of hyperbolic type: $$(1-\partial_{xx})y_{tt}+y_{xxxx}=0$$ with boundary conditions $$y(t,0)=y(t,1)=y_x(t,0)=y_x(t,1)=0.$$ I would say that the answer is yes. By ...
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Where to locate $0\in \Omega$ to get $u_{\varepsilon}(0)\neq 0$ where $\Delta u_{\varepsilon} + (\lambda-\varepsilon) u_{\varepsilon} = \frac{1}{|x|}$

Let $\Omega \subset \mathbb{R}^3$ a smooth bounded domain with $0\in \Omega$ and $u_\varepsilon(x)$ the solution to $$ \Delta u_\varepsilon + (\lambda-\varepsilon) u_\varepsilon = \frac{1}{|x|}\quad \...
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2 votes
1 answer
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Even and odd solutions for the Schrödinger equation

We consider $2a$ - periodic smooth solutions for \begin{eqnarray*} -\Delta u+V(x)\,u=0\qquad\hbox{in}\:[-a,a] \end{eqnarray*} We assume that $V$ is smooth and even (i.e. $V(-x)=V(x)$). We also assume ...
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Evolution PDE in dual space : Generalization of a result of Gelfand

The following result is proved in "Generalized functions", Volume 3, Chapter 2.2 by Gelfand : Let $\Phi$ be a Fréchet space with dual space $\Phi'$ endowed with the weak topology. For ...
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How can you make a PDE solution stabilize over time?

this is a problem I have been thinking about lately. I tried asking on stack exchange as well but did not find an answer. Suppose I have a simple linear first order PDE of the form: $$au_x+bu_y=0$$ I ...
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Does $f \in L^1([0,T]; S'(\mathbb R^n))$ define a $(1+n)$-dimensional distribution?

Let $f : [0,T] \rightarrow S'(\mathbb R^n)$ be a family of tempered distributions satisfying $$\langle f(t), \phi \rangle \in L^1([0,T])$$ for any Schwartz function $\phi \in S(\mathbb R^n)$. Does $f$ ...
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2 votes
1 answer
223 views

How to trap a particle without using potential field which is infinity at some point? (quantum physics) If impossible, how to prove it?

As we all know, the wave function of the stationary state a quantum particle trapped in a rigid box (with infinite potential outside the box) cannot have a non-zero value outside the box. So can we ...
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1 answer
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Relation between the solutions $v_t=Lv$ and $v_t=Av$ if $A$ is a relatively compact perturbation of the linear operator $L$

In a nutshell, here is my question. I read and know about the relation between the spectra of $L$ and $A$ if $A$ is a relatively compact perturbation of $L$. However, for my purpose, I am interested ...
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2 votes
1 answer
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What rotations are used as a reduction step in Kenig-Ruiz-Sogge's uniform Sobolev estimate?

I think I have understood the bulk of the paper [KRS], but one of the parts I cannot understand is when the authors reduce Theorem 2.1 (p.332) into Proposition 2.1 (p.335). I can understand all the ...
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0 votes
1 answer
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Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?

Consider the divergence form uniformly elliptic operator $\nabla \cdot a(x) \nabla$ where the coefficient $a$ are smooth and bounded and $D$ is a bounded and smooth domain of $\mathbb R^d$ $$ \begin{...
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5 votes
2 answers
301 views

General solution to an ultrahyperbolic PDE

$\DeclareMathOperator\SO{SO}$The following PDE defined on $\mathbb{R}^2$ $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}f(x,y) = 0,$$ has solution $$f(x,y) = g(x) + h(y),$$ where $g,h : \...
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6 votes
2 answers
301 views

Vacuum region with positive measure for the Schrödinger equation

Let us consider the free Schrödinger equation $(i\partial_t+\Delta_x)\psi=0$ in $\mathbb{R}_t\times\mathbb{R}_x^d$. I'm trying to understand the structure of the vacuum region $$\Omega(\psi):=\{(t,x)\...
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2 votes
1 answer
164 views

Existence of divergence-free unit vector field in conformally rescaled euclidean metric

Question. Let $\Omega \subset \mathbf{R}^2$ be a convex polygonal domain, equipped with a Riemannian metric $g$. Under which conditions on $g$ is there a vector field $X$ in $\Omega$ with $\mathrm{div}...
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2 votes
0 answers
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Divergence-free constraint for a boundary integral equation

Consider the system $$ \begin{cases} \operatorname{curl} \operatorname{curl} \mathbf{u} = 0 \qquad B \cup (\mathbb{R}^3 \setminus \overline{B}) \\ c_1 (\operatorname{curl} \mathbf{u} \times \mathbf{n})...
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1 answer
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References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$

I am looking for a reference for basic properties of the Green function for a symmetric, uniformly elliptic operator $\nabla \cdot a \nabla$ where the coefficients $a_{ij}= a_{ji}$ are only assumed to ...
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2 votes
0 answers
111 views

Explicit computation of a norm in context of operator-semigroups and differential equations

I am interested in the explicit calculation of the following norm $\vert \cdot \vert$. Let $X$ a Banach space with norm $\Vert \cdot \Vert$ and $(T(t))_{t \geq 0}$ a strongly continuous one-parameter ...
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2 votes
1 answer
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The only rotation fields satisfying this PDE are constant

$\newcommand{\div}{\operatorname{div}}$$\newcommand{\SO}{\operatorname{SO(2)}}$$\newcommand{\R}{\operatorname{\mathbb{R}}}$$\newcommand{\bdx}{\partial_x}$$\newcommand{\bdy}{\partial_y}$$\newcommand{\...
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1 vote
1 answer
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Heat flow derivative of entropy

In a 1966 paper (Speed of Approach to Equilibrium for Kac's Caricature of a Maxwellian Gas, Arch. Rational Mech. Anal., Vol. 21), McKean seems to suggest that the successive derivatives of entropy $H (...
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3 votes
1 answer
165 views

Neumann/Robin Laplacian semigroup well-known estimate

Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on: $$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \...
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1 vote
0 answers
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Fourier Transform; half space baby problem (new)

This question is related to a prior question i asked, see Fourier Transform ; half space elliptic baby problem. Essentially I am asking the same question now but taking a lot more care. So lets ...
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3 votes
2 answers
166 views

Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight

Let $\zeta, u_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded. \begin{equation}\label{Star-3.7} \begin{cases} \partial_t u -\Delta u + \zeta u=0 &\mbox{ in }\...
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Solutions of constant coefficients differential operator on $\mathbb{R}^n$

Let $D$ be a constant coefficients linear differential operator on complex valued functions on $\mathbb{R}^n$. Let $\tilde D=\mathbb{F}^{-1}\circ D\circ \mathbb{F}$ be its conjugation with the Fourier ...
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1 vote
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Calculating frequency of sound of ringing metal coin

I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...
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3 votes
1 answer
194 views

Is there a metric on Euclidean space that turns the Helmholtz equation into the Laplace equation?

Is there a Riemannian metric $\tilde g$ on $\mathbb R^d$ such that $$\tag{1} \Delta_{\tilde g}=e^f(\Delta +1),$$ for some $f\in C^\infty(\mathbb R^d)$? Here $\Delta=\partial_{x_1}^2+\ldots+\partial_{...
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3 votes
0 answers
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Semiclassical analysis and reflection law

I was curious about the following vague question. In pseudodifferential calculus and semi-classical analysis there are various theorems that relate geodesics of the underlying space to the behavior of ...
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2 votes
1 answer
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Reconstructing the metric on $CP^2$ with special one forms

I know that $(z_1,z_2)$ are the affine\inhomogeneous coordinates on the complex projective space $CP^2$. Now I have four one forms $(Y_1,Y_2, Y_3, Y_4)$. I want to rewrite the Fubini Study metric on $...
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1 vote
0 answers
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Perturbation in the equation $u_t=\epsilon Pu$, where $P$ is an elliptic partial differential operator

Let $Pu=\sum_{ij} \partial_j(a_{ij}(x) \partial_{i} u)$ be an elliptic operator. Consider the equation $$ (u=u_\epsilon)\\ \partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\ u(0,x)=u_0(...
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  • 1,457
2 votes
0 answers
70 views

$L^\infty$ Transport Equation Estimate: Characteristics of the Milne Problem on a Finite Slab

Cross-posted from MSE here. I'm trying to justify equation (3.43) on page 18 of this paper by Lei Wu and Yan Guo on diffusion approximation of the radiative transport equation. Consider the following ...
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