Questions tagged [ap.analysis-of-pdes]
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1,733
questions with no upvoted or accepted answers
1
vote
0
answers
79
views
SPDE via fixed point argument and Young's theorem
Let $(P_r)_{r\geq 0}$ be a strongly continuous semi-group (not necessarily the heat kernel).
It is well known that we can prove local well-posedness of a few SPDE using a fixed point argument: Young's ...
1
vote
0
answers
114
views
Are weak solutions and mild solutions for linear parabolic equations equivalent in $L^{q}([0,T],L^p(\Omega))$ with $1<q<\infty$, $1<p \leq 6/5$?
I have looked through some MO and ME posts, and the common opinion is that weak and mild solutions are equivalent for "many" cases of linear parabolic equation.
However, detailed proofs can ...
1
vote
0
answers
95
views
Computing a limit for the Weierstrass function
Let $a\in (0,1)$ and let $b$ be an odd positive integer such that $ab>1+\frac{3}{2}\pi$. Let $\alpha \in (0,1)$ be defined by $\alpha= -\frac{ln(a)}{ln(b)}$ and consider the well known Weierstrass ...
1
vote
0
answers
60
views
A question about semigroups in a Heisenberg group
I'm trying to understand if the regularity of solutions in Heisenberg groups works like in the Euclidean case. So far I haven't found any results, so I'm trying to check if the Regularity Theorems ...
1
vote
0
answers
75
views
What's the relation between viscosity solutions of infinity harmonic functions and normalized infinity harmonic functions?
The now famous infnity laplacian is the equations
$$
\langle D^2u Du,Du\rangle=0
$$
and the normalized infnity laplacian is
$$
\langle D^2u Du/|Du|,Du/|Du|\rangle=0.
$$
Is a viscosity solution of one ...
1
vote
0
answers
132
views
Riesz’s representation theorem in a weak form
Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^N$ $(N\geq 3)$, $\phi\in H_0^1(\Omega)$ is a solution of $$
\begin{cases}\Delta \phi+ \phi=h & \text { in } \Omega, \\ \phi=0 &...
1
vote
0
answers
102
views
PDE coupled with the pronic numbers (related to triangular numbers)
I am studying the linear PDE:
$$ t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\...
1
vote
0
answers
35
views
Studying the evolution of laplacian in NS equation
The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by:
\begin{equation}\label{Eq1}
\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...
1
vote
0
answers
34
views
Existence for a nonlinear evolution equation with a monotone operator that is not maximal
We consider the nonlinear evolution equation
$$
\dot{u}(t) + Bu(t) = 0, \quad u(0)=0
$$
with
$$
A: \mathcal{C}(\Omega)\to \mathcal{M}(\Omega),\; p \mapsto \arg\min_{\mu\in\partial\chi_{\{||\...
1
vote
0
answers
55
views
Reference request; fractional Laplacian; boundary regularity
Consider $B_2^+$ the half ball in $R^N$ and consider
$ (-\Delta)^s u = f(x) $ in $B_2^+$ with $ u=0$ outside. Is there any references where someone tries to use an odd extension of $u$ across $ x_N=...
1
vote
0
answers
23
views
Scalar nonlinear balance law with non-integrable source term on a bounded domain
I am considering the following PDE for $(t,x)\in\mathbb R_+\times[0,1]$:
$$
\begin{cases}
\partial_t u(t,x) + \partial_x[u(1-u)]=G(x,u),\\
u(0,\cdot)=u_0, \quad u(\cdot,0)=\alpha, \quad u(\cdot,1)=\...
1
vote
0
answers
75
views
Treating 2D NSE with an $L^4$ contraction mapping
For divergence-free initial data $u_0 \in L^2(\mathbb{T}^2)$, the two-dimensional Navier Stokes equation is known to have a global mild solution. This fact is classical. However, a written-out proof ...
1
vote
0
answers
38
views
Mixed boundary condition of parabolic equations
Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that
$$
\partial\Omega=\partial\Omega_D\cup\partial\Omega_N,
$$
where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...
1
vote
0
answers
87
views
$L^2(0,\infty;L^2(\Omega))$ estimate on solution of heat equation with Neumann boundary condition
Let $u$ be a solution of
$$u' - \Delta u = 0 \quad\text{on $\Omega$}$$
$$\partial_\nu u = 0\quad\text{in $\partial \Omega$}$$
$$u(t=0)=u_0\quad\text{on $\Omega$}$$
where $\Omega$ is a bounded ...
1
vote
0
answers
88
views
Notation for right hand side of local smoothing conjecture
In Tao's "Recent progress on the restriction conjecture"
On page 53, Tao introduced the local smoothing conjecture: let $u(t,x)$ be the solution to the wave equation $u_{tt}=\Delta u$, $u(0,...
1
vote
0
answers
76
views
Pointwise convergence of Schrodinger's equation with potential term
A famous problem of Carleson asks if $f\in H^s(\mathbb{R}^n)$, under what condition of $s$ do we have almost everywhere pointwise convergence of the solution to the Schrodinger's equation
$$iu_t-\...
1
vote
0
answers
46
views
Question on a mixed-norm estimate
I am currently reading the paper Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $\mathbb{R}$ by Colliander, Holmer, Visan, Zhang. In this article,...
1
vote
0
answers
78
views
Non-existence of classical solutions of Hardy PDE
On the paper "On the Cauchy Problem for Reaction-Diffusion Equations" Wang studies the Hardy-Hénon equation
$$
\begin{cases}
u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n ...
1
vote
0
answers
52
views
Comparison principle for porous medium equation in Fourier variables
Let $V:[0,\infty) \to[0,\infty)$ be convex, $C^2$ with $V(0)=0$. Define $F(u): = uV'(u)-V(u) $. Let $v\in L^1 (\Bbb R^d)$, $v\geq0$ so that $F\circ v\in L^1 (\Bbb R^d)$. For fixed $\varepsilon>0$, ...
1
vote
0
answers
112
views
Wave equation on 2D semi-infinite plane
I'm trying to understand what a correct procedure is for solving the following wave equation on a semi-infinite plane $-\infty < x < \infty$, $0 \le z < \infty$:
$$
\begin{cases}
\nabla^2 \...
1
vote
0
answers
64
views
A parametrix construction for heat boundary value problem using Fourier transformation
Let $\Omega$ be a smooth bounded open subset in $\mathcal{R}^{d}$, with $d \geqslant 3
$ and $T>0$. Consider the linear parabolic initial Dirichlet boundary value problem with $f\in H^{-1}(\Omega)$...
1
vote
0
answers
87
views
Friedrichs extension of the Laplacian from a smooth subspace and density of its eigenbasis in the Frechet topology of the subspace as well?
Let $C^\infty_\text{div}(\mathbb{T}^3)$ be the "real" Frechet space of periodic, divergence-free smooth vector fields on $\mathbb{R}^3$. That is, $\mathbb{T}^3$ is the $3$-dimensional torus.
...
1
vote
0
answers
108
views
When is there an inclusion between regular Orlicz Spaces?
It is a classical result that $L^p(\Omega) \subset L^q(\Omega)$ when $q<p$ and $|\Omega| < \infty$. I'd like to know if there is an Orlicz version of this fact. In other words, let $L^{G_1}$ and ...
1
vote
0
answers
50
views
On Designing Some Optimal Control Problems
In the context of a dynamical systems, some states may not be attainable with scalar controls from $L^1(0,T)$, but they may be reachable with feedback controls.
If we know that the system is null ...
1
vote
0
answers
41
views
The existence of an optimal distributed control problem
Consider $\Omega$ as a bounded interval of $\mathbb{R}$, and let $y\in L^{\infty}(\Omega \times (0,T))$ be a mild solution of the following parabolic partial differential equation:
\begin{equation}\...
1
vote
1
answer
249
views
Reverse estimate on the Riesz potential $I_\alpha : L^{n/\alpha}\to \mathrm{BMO}$
$\newcommand\BMO{\mathrm{BMO}}$Consider the Riesz potential on $\mathbb{R}^n$ given by
$$
I_\alpha f(x) = c_{n,\alpha} \int_{\mathbb{R}^n} \frac{f(y)}{\lvert x-y\rvert^{n-\alpha}} dy.
$$
It is known ...
1
vote
0
answers
86
views
Nonlocal elliptic problem - what is its associated energy?
It is well known that for any smooth domain $\Omega\subset\mathbb{R}^N$ the energy functional (the one for which the Euler-Lagrange equation is our b.v.p.) associated to the following local problem:
$$...
1
vote
0
answers
176
views
For the solvability of the poisson equation $\Delta u = f$ on manifold with boundary
For poisson equation $\Delta u = f$ in bounded domain in $\mathbb{R}^n$, we can directly get the solution by Green function. For poisson equation $\Delta u = f$ on closed Riemannian manifold, the ...
1
vote
0
answers
158
views
Regularity up to the boundary of solutions of the heat equation
Given the heat problem:
$$\begin{cases}
\frac{d}{dt}u(x,t)=\Delta u(x,t) & \forall (x,t)\in \Omega\times(0,T) \\
u(x,0)=u_0(x) & \forall x\in\Omega \\
u(x,t)=0 & \forall x\in\partial\...
1
vote
0
answers
29
views
On spectral representation of solutions to wave equations with impulse initial data
Let $\Omega \subset \mathbb R^n$ be a bounded domain with a smooth boundary that contains the origin. Let us consider the following classical linear wave equation
$$
\begin{cases}
\partial^2_t u -\...
1
vote
0
answers
82
views
Representation formula for the continuity equation on a separable Hilbert space
The following is an informal question for which I'd like to (ideally) find a reference. I'm quite a novice in this area but would be happy to find a reference to a theorem along the following lines (...
1
vote
0
answers
84
views
Wave equation on $[0,1]$ with mixed boundary conditions
Consider the wave equation $u_{xx}-u_{tt}=0$ on the unit interval $x\in[0,1]$. Take mixed boundary conditions ($\alpha_{1,2}^2+\beta_{1,2}^2 \neq 0$)
\begin{align*}
\alpha_1 u(0,t) + \beta_1u_x(0,...
1
vote
0
answers
93
views
Oscillatory integrals and regularity
Let $U\subset\mathbb{R}^{d}$ be open and $N\in\mathbb{N}$. Furthermore, let $a\in\mathcal{S}^{m}_{\rho,\sigma}(U\times\mathbb{R}^{N})$ be a symbol and $\Phi\in C^{\infty}(U\times (\mathbb{R}^{N}\...
1
vote
0
answers
71
views
Are eigenfunctions of the Dirichlet problem for the Laplace equation uniformly bounded?
Let $Q\subset \mathbb R^n$ be a bounded domain with boundary $\partial Q\in C^\infty$ and $\varphi_1,\varphi_2,\ldots$ are eigenfunctions of the Dirichlet problem for the Laplace equation in $Q$ ...
1
vote
0
answers
41
views
Wellposdeness of some HJB equation
Consider the non-linear PDE for $u:[0,1]\times [-1,1]\to\mathbb R$ as follows:
$$u_t= \inf_{b\ge 1/e} \big(-b u_{xx} - \log b - 1\big), \quad \forall (t,x) \in (0,1) \times (-1,1),$$
together with the ...
1
vote
0
answers
59
views
Solution to $u_t = A(t)u + f(t)$ on bounded domain
I am dealing with the problem
\begin{align}u_t &= \nabla \cdot (a(x,t) \nabla u) + f(x,t) &\text{ on } \Omega \times (0,T)\\
\partial_{\nu} u &= 0 &\text{ on } \partial \Omega \...
1
vote
0
answers
82
views
Reference for an application of J. Simon "Compact sets in the space $L^p(0,T;B)$"
Theorem 5 p.84 of J. Simons paper "Compact sets in the space $L^p(0,T;B)$" states a generalization of the Aubin-Lions lemma which relaxes the required regularity in time to the existence of ...
1
vote
0
answers
79
views
positivity of solution for Baouendi-Grushin operator
Let $(x,y)\in\mathbb{R}^N = \mathbb{R}^m \times \mathbb{R}^n$ with $m \geq 1$, $n\geq 0$, $\alpha \geq 0$ and $ Q=m+ n(\alpha +1)$.
Consider the following eqution
\begin{equation}\label{critical}
-(\...
1
vote
0
answers
98
views
Schrödinger equation approximation – continuity of eigenvalues with respect to potential
The question has been crossposted from Stackexchange after receiving no answers.
Setup: the time-independent Schrödinger equation (eigenvalue problem):
$(-\frac{\hbar^2}{2m}\Delta +V)\psi = E\psi$
(On ...
1
vote
0
answers
150
views
About Agmon-Douglis-Nirenberg complementing boundary condition
Let's consider the following Poisson equation with Neumann boundary condition
\begin{align*}
-\Delta u &= f \quad \text{in } \Omega \\
\partial_n u&=0 \quad \text{on } \partial \Omega,
\end{...
1
vote
1
answer
229
views
Integral identity for critical points of the Ginzburg-Landau functional
I am reading a paper of Comte and Mironescu [CM96], where they discuss critical points $v = v_{\epsilon}: G \to \mathbf{C}$ of the (non-magnetic) Ginzburg–Landau functional
$E_\epsilon(v) = \frac{1}{2}...
1
vote
0
answers
69
views
Limiting behavior of Kazdan-Warner equations
It's a well known result by Kazdan and Warner that on a closed Riemannian manifold the pde:
\begin{align*}
\Delta f+ge^f=c
\end{align*}
has a unique solution for $g\geq 0,$ and $c$ a positive constant....
1
vote
0
answers
107
views
Recovering phase function using Fourier decomposition
I have a function $\phi(x): \mathbb{R} \to [0, 2 \pi)$, which describes phase of another function
$$f = e^{i \phi(x)}. $$
I am interested in the following problem. If I know the function/distribution $...
1
vote
0
answers
64
views
Reference regarding the discrepancy of Sobolev inequality in manifolds
In the paper [1] by Bianchi and Egnell, they proved the following discrepancy result for the classical Sobolev embedding
$$
\|\nabla\phi\|_{2}^2-S^2\|\phi\|_{2^*}^2\geq\alpha d(\phi,M)^2.
$$
This ...
1
vote
0
answers
84
views
Hölder regularity in a quantitative manner
Why the investigation pertaining to sharp estimates in diffusion pde is important? In what scenarios the quantitative way reveals important nuances of a problem and play a decisive role in a finer ...
1
vote
0
answers
53
views
Dirichlet-to-Neumann estimate for minimal graphs
Let $\Omega \subset \mathbf{R}^n$ be a smooth, bounded domain. The Dirichlet problem for the minimal surface equation
\begin{equation}
(1 + \lvert Du \rvert^2) \Delta u - D_i u D_j u D_{ij} u = 0
\end{...
1
vote
0
answers
118
views
Maximal domain of an unbounded linear operator in a weighted Hilbert-space
Let's consider the following (unbounded) linear operator. (So called transport operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
1
vote
0
answers
233
views
Maximal regularity heat equation
Considering the heat equation on the flat torus $\mathbf{T}^d$, we have the maximal regularity estimate
\begin{align*}
\forall \varphi\in\mathscr{D}(\mathbf{R}\times\mathbf{T}^d),\quad \|\Delta \...
1
vote
0
answers
117
views
Estimate of a product of functions where $1/p + 1/q >1$
I have the following problem. I need to estimate the following quantity:
$$|\nabla|^{-1}A u \nabla u \label{1}\tag{$*$}$$
I know that $A\in L^6(B_R)$, $u \in L^{14/5}\cap L^2(B_R)$ and $\nabla u \in L^...
1
vote
0
answers
65
views
Parabolic PDE: Zero now means zero anytime before
Studying some mathematical models I came across a simple-looking question that I do not know how to handle.
If we have the following problem:
$$\begin{cases} \dfrac{\partial Z}{\partial t}-\Delta Z=aZ-...