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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Short time existence on nonlinear parabolic PDE

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fact, the number of ...
user avatar
7 votes
2 answers
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Uniform bound on the eigenfunctions of the Laplacian

Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty ...
John Zheng's user avatar
5 votes
2 answers
1k views

Analytic solution of a system of linear, hyperbolic, first order, partial differential equations

In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form $$\cos\left(t\right)\partial_{x}\mathbf{u}+\sin\left(t\right)\partial_{y}\mathbf{u}+...
FraSchelle's user avatar
4 votes
1 answer
323 views

To give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector function

I am consulting some papers (references below) about the Carleson's problem for the pointwise convergence of the Schrödinger group \begin{equation} S_t=e^{i t \Delta}. \end{equation} In this context ...
Giuseppe Negro's user avatar
3 votes
1 answer
2k views

Method of characteristics of a system of first order pdes

I asked the question on math.stackexchange.com, but didn't get any reply. So, I asked it again here. Any suggestion or hint is welcome, and thank you for your attention. Consider the system of first ...
William Wu's user avatar
14 votes
1 answer
2k views

Regularity of the Maxwell equations

As is well-known, the Maxwell equations can be phrased vectorially as, \begin{align} \nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\ \nabla \cdot \mathbf ...
JT_NL's user avatar
  • 455
11 votes
2 answers
687 views

Poincaré lemma for distributions

Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form $$ u=\sum_{1\le j\le n} u_j dx_j,\quad ...
Bazin's user avatar
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8 votes
2 answers
398 views

Bounded input Bounded output stability for heat equation

This is a cross-post from Computational Science. I am interested in proving or obtaining a counterexample to the following conjecture. Let $\Omega\subset\mathbb{R}^d$ be a bounded open domain. Let ...
fred's user avatar
  • 142
8 votes
3 answers
777 views

What does the flow of the principal symbol of the differential operator tell us about the PDE?

Disclaimer: Let me apologize in advance for asking this slightly vague question Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there'...
Saal Hardali's user avatar
  • 7,549
5 votes
2 answers
383 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 : \...
Jojo's user avatar
  • 333
5 votes
2 answers
347 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 ...
Pritam Bemis's user avatar
5 votes
3 answers
451 views

Structure of sign changes under the heat flow

Let $f$ be a smooth function on $R^2$, and define $N_f$ to be the set of points $p$ such that the nodal set of $f$ ($\{x\in R^2: f(x)=0\}$) divided every neighborhood of $p$ into four regions. Indeed, ...
Matchmaticians's user avatar
5 votes
1 answer
403 views

Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$: $$ \begin{cases} \partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\ u(0,x)=u_0(x). \end{cases} $$ ...
Juhana Siljander's user avatar
4 votes
2 answers
438 views

Heat equation and evolution of number of critical points

Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...
Matchmaticians's user avatar
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 {...
Alex M.'s user avatar
  • 5,207
4 votes
0 answers
141 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 $$\...
GJC20's user avatar
  • 1,220
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
1 answer
322 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.
Math_tourist's user avatar
3 votes
1 answer
374 views

Linear PDE, analytic continuation, Green's function and boundary conditions

I'm looking at the linear PDE in 3+1 dimensions, $$ \left[ -(\partial_t - \xi)^2 - \partial_k \partial_k \right] \phi(t,x) = 4\pi^2 \delta(t)\delta(x)\label{1} \tag{1} $$ Where $\xi$ is generally a ...
Fetchinson0234'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
2 votes
0 answers
166 views

Singularity of the solution of a PDE whose coefficients have zeros

The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post): $$\mathcal{A}p=0, \quad p\in C^2(\...
S.Surace's user avatar
  • 1,675
1 vote
2 answers
901 views

Solving a general, constant-coefficient, first-order, two-indep-variable system of PDEs

I have the following system of PDEs that I want to solve as "analytically" as possible: $$\left(\partial_t + A\partial_x + B\right)\mathbf{u}(t, x) = 0,$$ where $A$ and $B$ are constant, ...
Josh Burkart's user avatar
1 vote
0 answers
115 views

Poisson equation estimates near boundary

Fix $ \Omega$ a bounded smooth domain in $\mathbb R^N$ (take $N$ big) and let $ \frac{N+1}{2}<p<N$. We now consider nonnegative smooth functions $f$ such that $-\Delta u(x)=f(x) $ in $ \Omega$ ...
Math604's user avatar
  • 1,363
1 vote
3 answers
470 views

BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?

I asked the following question on math.SE (https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d) just over two months ago, and it ...
JahvedM's user avatar
  • 119