Questions tagged [linear-pde]
Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.
152
questions with no upvoted or accepted answers
7
votes
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answers
433
views
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:...
6
votes
0
answers
139
views
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 $f$ ...
6
votes
0
answers
369
views
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{...
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 =...
5
votes
0
answers
308
views
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 $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
5
votes
0
answers
196
views
Where to locate $0\in \Omega$ to get $u_{\varepsilon}(0)\neq 0$ where $\Delta u_{\varepsilon} + (\lambda-\varepsilon) u_{\varepsilon} = \frac{1}{|x|}$
Let $\Omega \subset \mathbb{R}^3$ a smooth bounded domain with $0\in \Omega$ and $u_\varepsilon(x)$ the solution to
$$
\Delta u_\varepsilon + (\lambda-\varepsilon) u_\varepsilon = \frac{1}{|x|}\quad \...
4
votes
0
answers
105
views
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 ...
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 ...
4
votes
0
answers
143
views
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
$$\...
4
votes
0
answers
159
views
Continuity of weak solutions to wave equation with time-dependent coefficients
Consider the following second-order wave equation
$$
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....
4
votes
0
answers
234
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 $...
4
votes
0
answers
73
views
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 {...
4
votes
0
answers
89
views
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 ...
4
votes
0
answers
600
views
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 ...
3
votes
0
answers
57
views
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
$$...
3
votes
0
answers
59
views
Geometric properties of the unique solution of an elliptic BVP involving the Lie derivative of the metric by a vector field
Setting
Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and let $\nabla$ be its associated Levi-Civita connection. Consider the following formally self-adjoint, second order linear ...
3
votes
0
answers
92
views
Comparison principle for Elliptic PDE with exponential nonlinearity
Suppose $\varphi$ is a radial (and radially decreasing) solution of
$$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$
with $ R>0$, and $\psi$ is a decreasing radial function ...
3
votes
0
answers
93
views
On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections
Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
3
votes
0
answers
93
views
How to find a particular solution of a non-homogeneous parabolic partial differential equation
Consider the following non-homogeneous parabolic partial differential equation (PDE)
\begin{align}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \...
3
votes
0
answers
84
views
Semiclassical analysis and reflection law
I was curious about the following vague question. In pseudodifferential calculus and semi-classical analysis there are various theorems that relate geodesics of the underlying space to the behavior of ...
3
votes
0
answers
41
views
Reference request: existence of strong solutions to a linear parabolic problem with mixed boundary conditions
on a domain $\Omega \subset \mathbb{R}^d$ with smooth boundary $\partial\Omega$ subdivided into two parts $\Gamma_D$ and $\Gamma_N$ I am considering the parabolic problem
$$
\partial_t u = \Delta u + ...
3
votes
0
answers
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 ...
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 ...
3
votes
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 ...
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 ...
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 $\rho = (\rho_0,\rho_1)^T$, $D$ ...
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,...
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
$$
\int_{\partial\Omega} |u|^2 ds
\...
3
votes
0
answers
163
views
How can one do change of variables for solutions to a staochastic partial differential equation?
isHow can one do change of variables for solutions to a staochastic partial differential equation? For example, let us consider the following stochastic transport equation:
$$
dy(t,x) + y_x(t,x) + y(t,...
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 ...
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 \...
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 ...
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^...
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 ...
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_\...
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 $$\begin{cases}
-\Delta ...
2
votes
0
answers
42
views
Polynomial solutions of differential equations vs smooth ones
Let $D_1,D_2$ be two linear differential operators with matrix valued constant (i.e. translation invariant) coefficients on $\mathbb{R}^n$. Assume $D_2\circ D_1=0$, in other words
$$Im(D_1)\subset Ker(...
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 ...
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 $...
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 $w=(u,v)$ ...
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},\ & \...
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 ...
2
votes
0
answers
113
views
Divergence-free constraint for a boundary integral equation
Consider the system
$$
\begin{cases}
\operatorname{curl} \operatorname{curl} \mathbf{u} = 0 \qquad B \cup (\mathbb{R}^3 \setminus \overline{B}) \\
c_1 (\operatorname{curl} \mathbf{u} \times \mathbf{n})...
2
votes
0
answers
143
views
Explicit computation of a norm in context of operator-semigroups and differential equations
I am interested in the explicit calculation of the following norm $\vert \cdot \vert$.
Let $X$ a Banach space with norm $\Vert \cdot \Vert$ and $(T(t))_{t \geq 0}$ a strongly continuous one-parameter ...
2
votes
0
answers
87
views
$L^\infty$ Transport Equation Estimate: Characteristics of the Milne Problem on a Finite Slab
Cross-posted from MSE here.
I'm trying to justify equation (3.43) on page 18 of this paper by Lei Wu and Yan Guo on diffusion approximation of the radiative transport equation.
Consider the following ...
2
votes
0
answers
95
views
Laplacian coupled with another equation over a two-dimensional rectangular region
I have the two-dimensional Laplacian $(\nabla^2 T(x,y)=0)$ coupled with another equation which is:
$$\frac{\partial t}{\partial x}+\alpha(t-T)=0 \tag 1$$
where it is known that $t(x=0)=t_i$.
The ...
2
votes
0
answers
133
views
Isometric immersions of $\mathbb{S}^n$ into $\mathbb{S}^{2n}$
A major open problem in submanifold geometry is to determine whether there exists a nontotally geodesic isometric immersion of $\mathbb{S}^n$ into $\mathbb{S}^{2n}$. Every isometric immersion of $\...
2
votes
0
answers
200
views
Laplacian on a manifold with two boundary components
I am interested in the Laplace equation on knot complements. The full complement of a knot $K$ is in $S^3$, but for compactness, we delete an open tubular neighborhood around $K$. The Laplace PDE on $...
2
votes
0
answers
166
views
A basic question about the Spectral Theorem
Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
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, ...