Questions tagged [linear-pde]
Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.
24
questions
12
votes
1
answer
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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 ...
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 ...
5
votes
2
answers
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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
$$\cos\left(t\right)\partial_{x}\mathbf{u}+\sin\left(t\right)\partial_{y}\mathbf{u}+...
4
votes
1
answer
323
views
To give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector function
I am consulting some papers (references below) about the Carleson's problem for the pointwise convergence of the Schrödinger group
\begin{equation}
S_t=e^{i t \Delta}.
\end{equation}
In this context ...
3
votes
1
answer
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Method of characteristics of a system of first order pdes
I asked the question on math.stackexchange.com, but didn't get any reply. So, I asked it again here. Any suggestion or hint is welcome, and thank you for your attention.
Consider the system of first ...
14
votes
1
answer
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Regularity of the Maxwell equations
As is well-known, the Maxwell equations can be phrased vectorially as,
\begin{align}
\nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\
\nabla \cdot \mathbf ...
11
votes
2
answers
687
views
Poincaré lemma for distributions
Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form
$$
u=\sum_{1\le j\le n} u_j dx_j,\quad ...
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 ...
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'...
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 : \...
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 ...
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, ...
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 $\mathbb R_+ \times\mathbb R^d$:
$$
\begin{cases}
\partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\
u(0,x)=u_0(x).
\end{cases}
$$
...
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 ...
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
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
$$\...
3
votes
0
answers
198
views
Analytic solution to two component, first order, linear PDE system
I would like to obtain analytic solutions to the following PDE system:
\begin{equation}
\rho_t + D(\lambda)\,\rho_\lambda = A(\lambda) \rho, \tag{1}
\end{equation}
with $\rho = (\rho_0,\rho_1)^T$, $D$ ...
3
votes
1
answer
322
views
Reference request: Schauder estimates for parabolic equations
Where can I find Schauder estimates for second order linear parabolic equations (in divergence form with potential)?
Any reference would be highly appreciated.
3
votes
1
answer
374
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Linear PDE, analytic continuation, Green's function and boundary conditions
I'm looking at the linear PDE in 3+1 dimensions,
$$
\left[ -(\partial_t - \xi)^2 - \partial_k \partial_k \right] \phi(t,x) = 4\pi^2 \delta(t)\delta(x)\label{1} \tag{1}
$$
Where $\xi$ is generally a ...
3
votes
0
answers
331
views
Method of characteristic for a system of first order PDEs
I am working with this system of first order PDEs:
\begin{equation}
\left\{
\begin{aligned}
%Suscettibili
&\frac{\partial{S}(a,t)}{\partial{t}} + \frac{\partial{S}(a,t)}{\partial{a}}= -\lambda(a,...
2
votes
0
answers
166
views
Singularity of the solution of a PDE whose coefficients have zeros
The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post):
$$\mathcal{A}p=0, \quad p\in C^2(\...
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, ...
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$ ...
1
vote
3
answers
470
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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 ...