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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Regularity of solutions to a linear degenerate parabolic pde

I've encountered the following problem which is causing me some trouble : Let $(M^2,g)$ be a smooth compact Riemannian surface (with constant curvature for instance) and consider a smooth function $u:...
Thomas Richard's user avatar
6 votes
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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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Linear PDE with non constant coefficients and properties of Green's Function

Lots of information is available about Poisson's PDE $\operatorname{div}(\operatorname{grad}(u(\vec{x}))))=f(\vec{x})$. However it is hard to find information about the more generalized case \begin{...
chloros2's user avatar
6 votes
1 answer
996 views

Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t =...
UPS's user avatar
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All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)

I have asked this question on MSE, but this is a better place. The heat equation and the heat kernel. Consider the heat equation on $\mathbb R$: $$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
Lorenzo Pompili's user avatar
5 votes
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196 views

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 \...
Jonas's user avatar
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Continuity of solutions of Elliptic PDE with respect to parameters

Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy $$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$ where $f$ is a fast decaying smooth function. I would like to know how the ...
Matchmaticians's user avatar
4 votes
0 answers
287 views

Pohozaev identity for linear equations

For $-\Delta u =0$, the Pohozaev identity on say $B_1$ says $$ \int_{S_1} |u_T|^2 \,d\sigma = \int_{S_1} |u_N|^2 \,d\sigma + (n-2) \int_{B_1} |\nabla u|^2 \ dx$$ Here $u_T$ are the tangential ...
Adi's user avatar
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Uniqueness of the "weak solution" to Fokker-Plank PDE

Let $C_b^2(\mathbb R_+)$ be the set of functions $f: \mathbb R_+\to\mathbb R$ s.t. $f, f' ,f''$ are bounded and $f(0)=0$. Consider a measurable function $p: \mathbb R_+^2\to\mathbb R_+$ satisfying $$\...
GJC20's user avatar
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Continuity of weak solutions to wave equation with time-dependent coefficients

Consider the following second-order wave equation $$ u_{tt} - div( a\cdot \nabla u) = f \quad \text{ in } (0,T)\times \Omega $$ with boundary conditions $$ u(0)=g, \ u_t(0)=h, \ u|_{\partial \Omega}=0....
daw's user avatar
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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 $...
J.E.M.S's user avatar
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The sum of linear partial differential operators of equal strength

If $P$ and $P'$ are linear partial differential operators with constant complex coefficients on $U = \mathring U \subseteq \Bbb R^m$, we say that $P \sim P'$ if and only if $\dfrac {\tilde P} {\tilde {...
Alex M.'s user avatar
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How can I can derive an explicit bound for the solution of the poisson's PDE?

i need some help on this question Let $\Omega$ be an open subset of $\mathbb{R}^{2}$ (say a square) with $\partial{\Omega} =\Gamma_{1} \cup \Gamma_{2} \cup\Gamma_{3} \cup\Gamma_{4}$. A structure ...
user106481's user avatar
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well-posedness of the transport equation

I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...
Kamil's user avatar
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Convex combination of cyclically monotone sets

I want to show the following statement, but I am not sure how. Proposition(?): Let $C \in \mathbb{R}^d$ be a compact convex set, and let $u, v : C \to \mathbb{R}$ be smooth convex functions. Suppose $$...
Paruru's user avatar
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Geometric properties of the unique solution of an elliptic BVP involving the Lie derivative of the metric by a vector field

Setting Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and let $\nabla$ be its associated Levi-Civita connection. Consider the following formally self-adjoint, second order linear ...
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Comparison principle for Elliptic PDE with exponential nonlinearity

Suppose $\varphi$ is a radial (and radially decreasing) solution of $$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$ with $ R>0$, and $\psi$ is a decreasing radial function ...
Matchmaticians's user avatar
3 votes
0 answers
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On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections

Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
Pelle Steffens's user avatar
3 votes
0 answers
93 views

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 \...
GilbertDu's user avatar
3 votes
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84 views

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 ...
Ivan's user avatar
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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 + ...
jfp's user avatar
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66 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 ...
utcyp's user avatar
  • 31
3 votes
0 answers
120 views

Partial regularity for transmission problem in corner domains

Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
PeteAgor's user avatar
  • 133
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0 answers
553 views

2D laplace equation with Robin boundary condition (Green function)

Let's say that I know a fundamental solution for the Laplace equation in the whole plane: $$\nabla^2u=\delta\quad \text{in the sense of distributions,}$$ and I need a solution for the laplace equation ...
Manuel Pena's user avatar
3 votes
0 answers
76 views

Smoothing inside the null space of a partial differential operator

Let $L$ be a linear partial differential operator with smooth coefficients in $U\subset\Bbb R^n$ and let $u\in W^{k,p}_{loc}(U)$ with $k\in\Bbb N$ and $p\in[1,\infty[$ satisfy $Lu=0$ in the ...
Lapasotka's user avatar
3 votes
0 answers
198 views

Analytic solution to two component, first order, linear PDE system

I would like to obtain analytic solutions to the following PDE system: \begin{equation} \rho_t + D(\lambda)\,\rho_\lambda = A(\lambda) \rho, \tag{1} \end{equation} with $\rho = (\rho_0,\rho_1)^T$, $D$ ...
Frits Veerman's user avatar
3 votes
0 answers
331 views

Method of characteristic for a system of first order PDEs

I am working with this system of first order PDEs: \begin{equation} \left\{ \begin{aligned} %Suscettibili &\frac{\partial{S}(a,t)}{\partial{t}} + \frac{\partial{S}(a,t)}{\partial{a}}= -\lambda(a,...
CrishaD's user avatar
  • 31
3 votes
0 answers
106 views

Constant in a trace Sobolev theorem for concave domains

I wonder is the following inequality is true/known: Let $\Omega\subset \mathbb{R}^n$ be a (locally) Lipschitz domain which is the complement of a convex set, then $$ \int_{\partial\Omega} |u|^2 ds \...
poupy's user avatar
  • 175
3 votes
0 answers
163 views

How can one do change of variables for solutions to a staochastic partial differential equation?

isHow can one do change of variables for solutions to a staochastic partial differential equation? For example, let us consider the following stochastic transport equation: $$ dy(t,x) + y_x(t,x) + y(t,...
user102730's user avatar
3 votes
0 answers
150 views

Quadratic forms over symmetric $3\times3$ matrices

This question is motivated by a technique called compensated compactness (CC), which was elaborated by L. Tartar and F. Murat for the analysis of PDEs. The foreword of CC is a problem about quadratic ...
Denis Serre's user avatar
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3 votes
0 answers
427 views

Boundary regularity of the solution to the Beltrami equation

Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference: Let us consider the orientation-preserving homeomorphic solutions $f: D \...
Analysis Now's user avatar
  • 1,451
2 votes
0 answers
120 views

Uniqueness of the solution to systems of first-order linear PDEs

Context: Let $\Omega \subset \mathbb{R}^p$ be an domain. For functions $A_{jk}^i : \Omega \to \mathbb{R}$ and $B_k^i : \Omega \to \mathbb{R}$ with some regularity, I am interested in the following ...
Paruru's user avatar
  • 51
2 votes
0 answers
60 views

Dirichlet's problem for Laplace's equation in the model domain

Let $$\Omega_\alpha=\left\{(\xi,\eta)\in \mathbb{R}^2 /\xi>\frac{1}{\alpha-1}a^{1-\alpha},0<\eta<1\right\},$$ $a>0$. we have $\Delta$ an isomorphism of $\mathbb{ w}^{2,p}\cap \mathbb{ w}_0^...
sidi mohamd deval's user avatar
2 votes
0 answers
143 views

Regularity of elliptic partial differential equation with mixed Dirichlet-Robin boundary condition, to prove $u\in H^{2}(\Omega)$

I have posted this problem on Math Stackexchange but got no reply. When I deal with the wave equation with dynamical boundary condition, I am confused by the regularity of the following elliptic ...
monotone operator's user avatar
2 votes
0 answers
103 views

A maximum principle in $\mathbb{R}^N$

Let $\delta > 0$ and define $$ H_\delta(x) = \prod_{j=1}^{N} \cosh(\delta x_j), \quad \forall x \in \mathbb{R}^N. $$ By straightforward calculations we get $\Delta H_{\delta} (x) = \delta^2 H_\...
Lucas Linhares's user avatar
2 votes
0 answers
120 views

On Fredholm alternative for Neumann conditions

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation $$\begin{cases} -\Delta ...
student's user avatar
  • 1,320
2 votes
0 answers
42 views

Polynomial solutions of differential equations vs smooth ones

Let $D_1,D_2$ be two linear differential operators with matrix valued constant (i.e. translation invariant) coefficients on $\mathbb{R}^n$. Assume $D_2\circ D_1=0$, in other words $$Im(D_1)\subset Ker(...
asv's user avatar
  • 21.1k
2 votes
0 answers
99 views

Question about the second order linear elliptic PDE on closed manifold

Recently I see a question linear second order PDE in which user Pedro post a reference in Gilbarg's book, which said that the solvability of the linear PDE $$ \Delta u +B^{i}(x)u_{i}+C(x)u=f $$ is ...
Elio Li's user avatar
  • 719
2 votes
0 answers
39 views

Estimates for higher order derivatives of the Airy Kernel

Consider the kdv equation (from here) $$\left\{\begin{array}{l} \partial_{t} v+\partial_{x}^{3} v=0 \\ v(x, 0)=v_{0}(x) \end{array}\right.$$ Its solution can be written as $v(t,x)=S_t*v_0(x),$ where $...
Student's user avatar
  • 653
2 votes
0 answers
58 views

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 $w=(u,v)$ ...
Riku's user avatar
  • 819
2 votes
0 answers
112 views

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},\ & \...
Kira Yamato's user avatar
2 votes
0 answers
94 views

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 ...
Desura's user avatar
  • 211
2 votes
0 answers
113 views

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})...
GaC's user avatar
  • 163
2 votes
0 answers
143 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 ...
kumquat's user avatar
  • 63
2 votes
0 answers
87 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 ...
Phil's user avatar
  • 123
2 votes
0 answers
95 views

Laplacian coupled with another equation over a two-dimensional rectangular region

I have the two-dimensional Laplacian $(\nabla^2 T(x,y)=0)$ coupled with another equation which is: $$\frac{\partial t}{\partial x}+\alpha(t-T)=0 \tag 1$$ where it is known that $t(x=0)=t_i$. The ...
Avrana's user avatar
  • 47
2 votes
0 answers
133 views

Isometric immersions of $\mathbb{S}^n$ into $\mathbb{S}^{2n}$

A major open problem in submanifold geometry is to determine whether there exists a nontotally geodesic isometric immersion of $\mathbb{S}^n$ into $\mathbb{S}^{2n}$. Every isometric immersion of $\...
SubGeo's user avatar
  • 89
2 votes
0 answers
200 views

Laplacian on a manifold with two boundary components

I am interested in the Laplace equation on knot complements. The full complement of a knot $K$ is in $S^3$, but for compactness, we delete an open tubular neighborhood around $K$. The Laplace PDE on $...
maxematician's user avatar
2 votes
0 answers
166 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|_{\...
MathLearner's user avatar
2 votes
0 answers
106 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, ...
Raphael J.F. Berger's user avatar