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.

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 &...
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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}{\...
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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 \...
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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_{\{||\...
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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=...
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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)=\...
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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 ...
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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 ...
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$L^2(0,\infty;L^2(\Omega))$ 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 ...
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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,...
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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-\...
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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,...
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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 ...
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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 $\varepsilon>0$, ...
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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 \...
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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 $f\in H^{-1}(\Omega)$...
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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. ...
Isaac's user avatar
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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 ...
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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 ...
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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}\...
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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 ...
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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: $$...
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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 ...
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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\...
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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 -\...
Ali's user avatar
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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 (...
Gregor Samsa's user avatar
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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,...
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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}\...
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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 $Q$ ...
Andrew's user avatar
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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 ...
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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 \...
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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 ...
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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} -(\...
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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 ...
Rohan Didmishe's user avatar
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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{...
bluejyellow's user avatar
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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}...
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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....
Partha's user avatar
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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 $...
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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 ...
User1723's user avatar
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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 ...
Cézar Bezerr's user avatar
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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{...
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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{...
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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 \...
Ayman Moussa's user avatar
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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^...
Jakob Möller's user avatar
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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-...
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