Questions tagged [linear-pde]
Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.
287
questions
0
votes
0answers
36 views
General solution to a second order linear PDE
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 $g,h : \mathbb{R} \mapsto \mathbb{R}$ are ...
5
votes
2answers
256 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
$$\Omega(\psi):=\{(t,x)\...
2
votes
1answer
139 views
Existence of divergence-free unit vector field in conformally rescaled euclidean metric
Question. Let $\Omega \subset \mathbf{R}^2$ be a convex polygonal domain, equipped with a Riemannian metric $g$. Under which conditions on $g$ is there a vector field $X$ in $\Omega$ with $\mathrm{div}...
1
vote
1answer
53 views
Divergence-free constraint for a boundary integral equation
Consider the system
$$
\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})...
1
vote
1answer
64 views
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 ...
2
votes
0answers
89 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
1answer
150 views
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}$$\newcommand{\...
1
vote
1answer
106 views
Heat flow derivative of entropy
In a 1966 paper (Speed of Approach to Equilibrium for Kac's Caricature of a Maxwellian Gas, Arch. Rational Mech. Anal., Vol. 21), McKean seems to suggest that the successive derivatives of entropy $H (...
3
votes
1answer
109 views
Neumann/Robin Laplacian semigroup well-known estimate
Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:
$$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}\ \...
0
votes
0answers
44 views
Where does the order of conormality comes from?
Let $X^n$ be a closed manifold, and let $Y^k \subseteq X$ be a closed embedded submanifold. A distribution $ u \in C^{-\infty} \left(X\right)$ is said to be conormal of order $m$ along $Y$ if its ...
1
vote
0answers
76 views
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 ...
3
votes
2answers
134 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 }\...
1
vote
0answers
66 views
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 ...
1
vote
0answers
91 views
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 ...
3
votes
1answer
173 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_{...
3
votes
0answers
44 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 ...
2
votes
1answer
164 views
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 $...
1
vote
0answers
73 views
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
$$
(u=u_\epsilon)\\
\partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\
u(0,x)=u_0(...
0
votes
0answers
107 views
Engineering mathematics course and the order of the teaching of its topics
I am going to teach a course names "Engineering Mathematics" and the topics in it, are:
Fourier series and integrals; including the motivation and computational aspects and their ...
2
votes
0answers
57 views
$L^\infty$ Transport Equation Estimate: Characteristics of the Milne Problem on a Finite Slab
Cross-posted from MSE here.
I'm trying to justify equation (3.43) on page 18 of this paper by Lei Wu and Yan Guo on diffusion approximation of the radiative transport equation.
Consider the following ...
3
votes
1answer
102 views
Maximum principle for an elliptic like operator
I am trying to prove some monotonicity of a solution of a given pde; after considering a quantity like $ \phi(x) = x \cdot \nabla v(x)$ ($v$ is the solution of a given pde) I arrive at something ...
2
votes
0answers
89 views
Laplacian coupled with another equation over a two-dimensional rectangular region
I have the two-dimensional Laplacian $(\nabla^2 T(x,y)=0)$ coupled with another equation which is:
$$\frac{\partial t}{\partial x}+\alpha(t-T)=0 \tag 1$$
where it is known that $t(x=0)=t_i$.
The ...
4
votes
0answers
75 views
Continuity of weak solutions to wave equation with time-dependent coefficients
Consider the following second-order wave equation
$$
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....
2
votes
0answers
117 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 $\...
1
vote
2answers
124 views
Difference between semilinear and fully nonlinear
I'm confused why the Hamilton Jacobi Bellman equation:
$$\frac{\partial u}{\partial t}(t,x)+\Delta u(t,x) -\lambda||\nabla u(t,x) ||^{2}=0$$
is considered fully nonlinear, but not semilinear.
By ...
2
votes
0answers
181 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 $...
1
vote
0answers
58 views
How to proceed in this Boundary value problem where Eigen values are calculated numerically?
While solving a boundary value problem (background provided in the Context section) I reach the following variable separated two equations ($F(x)$ and $G(y)$)
\begin{eqnarray}
\lambda_h F''' - 2 \...
0
votes
0answers
46 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 ...
3
votes
1answer
83 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 \...
1
vote
0answers
52 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:
...
1
vote
0answers
49 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 :
$$\partial_t f(t,x)=\partial_x f(t,x)+\partial_{...
3
votes
1answer
203 views
Ancient Heat equation and Liouville's theorem
I encounter a difficulty when proving the bounded solution of ancient heat equation implying constant function. Suppose $u(t,x)$ is the solution of ancient heat equation:
\begin{equation}
u_{t} = \...
1
vote
0answers
64 views
Weak maximum principle for a perturbation of the Laplacian
This question is motivated by similar considerations for the Kohn-Laplacian in the Heisenberg group, but it seems that I cannot even give an answer in the Euclidean case, so here we go.
Suppose that ...
3
votes
2answers
417 views
Gradient $L^p$ estimates for heat equation
I'm looking for a proof of the gradient estimate associated to the heat equation with Dirichlet boundary conditions, to see if I can express the constant $\color{red}{C}$.
$$\|e^{t\Delta_d}f\|_{W^{1,...
2
votes
1answer
231 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}...
4
votes
0answers
187 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 $...
2
votes
0answers
159 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.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
1
vote
1answer
367 views
Existence for an overdetermined system of PDEs
I am interested in the existence of a vector valued solution $y = y(x, t) \in\mathbb{R}^n$ to a system of $2n$ equations: there are twice more equations than unknowns. More precisely:
Let $A$ and $...
7
votes
2answers
339 views
System of linear pde with non constant coefficients
I'm a physicist and during my research work, I found a system of linear pde with non constant coefficients that I have to study, since I have totally no experience about systems of pde and I have even ...
2
votes
0answers
100 views
Justification for uniqueness of solutions to dispersive PDE
For the sake of concreteness, we consider the linear Schrödinger equation
$$
\partial_t u = i\Delta u, \ \ \ \ u(0, x) = u_0(x).
$$
The solution is typically obtained by taking the Fourier transform ...
5
votes
1answer
353 views
Lack of exponential $L^2_{t,x}$ decay for a heat equation with growing coefficients
Edit: I have changed the nature of the question, but in order to have a better idea of what I can expect for the original problem (see below).
Given $T>0$ and $n \in \bf Z$, consider the following ...
4
votes
3answers
402 views
Solutions of $\Delta \phi + \phi =0$ on $\mathbb{R}^2$
I know a few solutions of the equation $\Delta \phi + \phi =0$ on $\mathbb{R}^2$. They can be described as the Fourier transform of any finite measure $\mu$ on $S^1$. In particular take $\mu$ a finite,...
2
votes
1answer
102 views
Anti-diagonal matrix operator
Let $(A,\mathcal{D}(A))$ be an infinitesimal generator of a strongly continuous semigroup $(T(t))_{t\ge 0}$ on a Banach space $X$ and define on $\mathcal{X} := X \times X$ the
operator matrix
$$\...
3
votes
0answers
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
$$
\partial_t u = \Delta u + ...
1
vote
0answers
177 views
Geometric interpretation of Lagrange's linear equation and its solution [closed]
What is the geometric meaning of the Lagrange's linear equation, $$P\dfrac{\partial z}{\partial x} + Q \dfrac {\partial z}{\partial y}= R \\Pp+ Qq= R$$ where $P$,$Q$,and $R$ are functions of $x\,$,$\,...
1
vote
0answers
47 views
A linear first order PDE with boundary condition
I want to solve the following first order PDE
$$
(\star)\quad\begin{cases}
\nabla u\cdot \nabla\xi=f \quad\text{in}\,\Omega, \\
u\mid_{\partial \Omega}=0
\end{cases}
$$
where $\xi\in C^2(\overline{\...
2
votes
0answers
65 views
Solution of equation on vector field
I have a vector field function $\vec{J}: {\bf R}^3\to {\bf R}^3$ looking like:
$$ \vec{J}(\vec{r}) = (\vec{B} \times \vec{v}(\vec{r}))\rho(\vec{r}) $$
with a (very well behaved) real, positive, ...
7
votes
1answer
755 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 ...
2
votes
1answer
165 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.
5
votes
2answers
179 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 ...
