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$H^s$-mild solution for Navier–Stokes : why do we restrict attention to the function spaces "without Fourier zero mode"? (Related to Terence Tao blog)

This question has been triggered by the Definition 32 and Remark 33 in the blog of Terence Tao. There, every function space is restricted to ones without the Fourier zeroth mode. And the Remark 33 ...
Isaac's user avatar
  • 3,477
2 votes
0 answers
120 views

Closure of Laplacian

Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator $$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$ There are two ...
B.Hueber's user avatar
  • 1,171
2 votes
0 answers
329 views

Conditions for an existence of smooth solution to a parabolic PDE

I'm interested to know the conditions of when the parabolic PDE ($U \subset \mathbb{R}^n$ is some bounded open subset): \begin{equation*} u_t - \sum_{i,j=1}^n(a^{ij}(x,t)u_{x_i})_{x_j} + \sum_{i=1}^nb^...
user113988's user avatar
2 votes
0 answers
180 views

Approximating $L^p$ functions by eigenfunctions of Laplacian

I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932. In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
ze min jiang's user avatar
2 votes
0 answers
126 views

Differential equations: trying to connect a nonlinear equation to a linear one

The following is motivated by taking a product space $\Omega$ and splitting it into two parts via projections, whose subspaces, $T$ and $X$, are home to functions which satisfy a nonlinear PDE and a ...
John McManus's user avatar
2 votes
0 answers
94 views

Existence of Green function for some perturbation of Laplace operator

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ $(N\geq2)$ and $\lambda>0$ is a small parameter. I wonder if there exists a Green function such that $$(\Delta+\lambda) G(x,y)=\delta_x\...
Davidi Cone's user avatar
2 votes
0 answers
160 views

Understanding the Bochner space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ in terms of the Fréchet derivative

In the context of linear parabolic equations, the Sobolev space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ appears all the time. Here, $U$ is some bounded region of $\mathbb{R}^n$ and $1<p,q<\infty$. ...
Isaac's user avatar
  • 3,477
2 votes
0 answers
153 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 &...
Davidi Cone's user avatar
2 votes
0 answers
79 views

Does this variant coincide with the usual Hölder space?

$\newcommand{\NN}{\mathbb N} \newcommand{\RR}{\mathbb R}$ Let $\alpha \in (0, 1]$ and $d, j \in \NN^*$. The usual Hölder space $C^{j, \alpha} := C^{j,\alpha} (\RR^d; \RR)$ is defined as the space of ...
Akira's user avatar
  • 825
2 votes
0 answers
206 views

Failure of Calderón–Zygmund inequality at the endpoints

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'd like to prove that the famous Calderón–Zygmund elliptic estimate $$ \norm{ \partial_{ij}u }_{L^p} \leq C \norm{\Delta u }_{L^...
Marc's user avatar
  • 457
2 votes
0 answers
114 views

A maximum principle in $\mathbb{R}^N$

Let $\delta > 0$ and define $$ H_\delta(x) = \prod_{j=1}^{N} \cosh(\delta x_j), \quad \forall x \in \mathbb{R}^N. $$ By straightforward calculations we get $\Delta H_{\delta} (x) = \delta^2 H_\...
Lucas Linhares's user avatar
2 votes
0 answers
259 views

Research in analysis of PDEs

I am currently figuring out what topic to work on for my undergraduate thesis and was able to narrow it down to mathematical analysis. As of now, I have two main options: a thesis working on Sobolev ...
Gab Romero's user avatar
2 votes
0 answers
160 views

On Fredholm alternative for Neumann conditions

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and $f \in L^2(\Omega)$. It is well known that if $\lambda$ is a Dirichlet Laplacian eigenvalue, then the equation $$\begin{cases} -\Delta ...
student's user avatar
  • 1,350
2 votes
0 answers
92 views

Linearization stability condition

The following is a theorem from Fischer and Marsden's 1975's paper: Linearization stability of nonlinear PDEs. Theorem. Let $X, Y$ be Banach manifolds and $\Phi: X \rightarrow Y$ be $C^1$. Let $x_0 \...
Gordhob Brain's user avatar
2 votes
0 answers
120 views

Noether's theorem in the critical heat equation

I initially posted this question on Stackexchange Mathematics (see here) but as I got no answer, maybe someone here will be able to help me. I am watching a serie of lectures on "Blow up solution ...
Falcon's user avatar
  • 452
2 votes
0 answers
201 views

Green function of a 2D exterior domain

Consider solutions of the laplace equation \begin{equation} \begin{split} -\Delta u=f, \ \ u|_{\partial D}=0, \end{split} \end{equation} where the domain $D\subset \mathbb{R}^2$. If $D$ is bounded ...
W.J.'s user avatar
  • 379
2 votes
0 answers
72 views

Semilinear elliptic equations in complex plane

Let $D$ denote the closed unit disk centered at the origin in the complex plane. Let $F: D \times \mathbb C \to \mathbb C$ be a smooth function. Is there any theory for well-posedness (in the sense of ...
Ali's user avatar
  • 4,153
2 votes
0 answers
64 views

Scaling limit of ODE with double-well potential

Let us consider the ODE $$ \frac{d}{dt}x_\epsilon(t) = -\frac{1}{\epsilon} W'(x_\epsilon(t)) $$ where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the ODE reads $$...
Riku's user avatar
  • 839
2 votes
0 answers
117 views

Bounding integral expression with BV norm of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$ for $\epsilon>0$, $f \in L^\...
user avatar
2 votes
0 answers
94 views

Existing work on $\Delta u=c-h e^{u}$ on compact manifold with dimension n, I have read J.Kazdan's work, the condition $c > 0, n \ge 3$ is not solved

I'm reading Prof. Kazdan's lectures At page 69, Prof. Kazdan describes the research on the $\Delta u=c-h e^{u}$ PDE on a compact $n$-dimensional manifold before 1983. (Here $c$ is a constant while $h$ ...
Elio Li's user avatar
  • 809
2 votes
0 answers
104 views

Relations between different "propagation of chaos" type results?

My questions come from the paper Logarithmic Sobolev inequalities for some nonlinear PDE’s written by F. Malrieu (May 2001). The basic set-up is that we have a $N$-particle system $(X^{i,N}_t)_{1\leq ...
Fei Cao's user avatar
  • 730
2 votes
0 answers
150 views

Reference for weighted Sobolev spaces

I'm looking for a comprehensive reference illustrating, from the ground up, the basics of weighted Sobolev Spaces on Lipschitz domains (this case should be included, but I don't need less than it). ...
Lilla's user avatar
  • 235
2 votes
0 answers
66 views

Existence of saddle points under a $C^0$-perturbation of a continuous function

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function and has a strict maximum point $a$ and strict minimum point $b$. Define $g(x,y)=f(x)+f(y)$ and $h_\varepsilon(x,y)$ be a family of continuous ...
W.J.'s user avatar
  • 379
2 votes
0 answers
195 views

Trouble understanding Lax method for KDV equation for inverse scattering method

I am trying to learn the Lax pair condition on my own so that I can eventually learn the inverse scattering method. I am following a paper by Tuncay Aktosun ("Inverse scattering transform and the ...
Will_Phys4's user avatar
2 votes
0 answers
166 views

Green's function for elliptic PDE with potential

$\newcommand{\div}{\operatorname{div}}$Suppose I have an elliptic operator $\mathcal{L} u = -\div (A \nabla u) $ on some open set $\Omega \subseteq \mathbb{R}^d$ where here $A$ is uniformly elliptic ...
Joshua Isralowitz's user avatar
2 votes
0 answers
73 views

Question about Gidas-Ni-Nirenberg result

Background: So I know that the Euler Lagrange equation associated with the Sobolev inequality takes the following form, $$-\Delta u = u^p$$ where $p=2^*-1$ and here we assume that $u>0$ on $\mathbb{...
Student's user avatar
  • 537
2 votes
0 answers
60 views

Decay of solution for linear system with damping

Let us consider the following linear system with damping: $$ \begin{cases} u_t - u_x = -\frac{1}{2} (u+v)\\ v_t + v_x = -\frac{1}{2} (u+v) \end{cases} $$ Let's write the solution as $w=(u,v)$ ...
Riku's user avatar
  • 839
2 votes
0 answers
86 views

Continuity of the entropy of the solution of a parabolic PDE at $t=0$

Consider the following initial value problem for a parabolic PDE : $$\begin{cases} \textrm{div}\big(A\,\nabla u(t,x) + b(x)\, u(t,x)\big) \,=\,\partial_t u(t,x) \quad x\in\mathbb R^d\,,\ t>0 \\[4pt]...
tituf's user avatar
  • 311
2 votes
0 answers
84 views

Method of characteristics and explicit formula for an IBVP for the transport equation

Let us assume $c:(0,1) \to (\epsilon,+\infty)$ and consider the PDE $$ \begin{cases} u_t+c(x)u_x = 0, \\ u(0,x) = g(x) \\ u(t,0) = f(t) \end{cases} \qquad (t,x) \in (0,T)\times(0,1) \label{1}\tag{$\...
Riku's user avatar
  • 839
2 votes
0 answers
229 views

Weighted Sobolev norm in terms of Spherical harmonics coefficients

Let $M = [1,\infty) \times S^2$. Consider the weighted Sobolev space $H^k_{\delta}(M)$ with the Sobolev norm: $$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV $$ ...
Laithy's user avatar
  • 969
2 votes
0 answers
65 views

Regularity of solution to first order time dependent variational problem

Consider the following first order evolution problem over some regular bounded domain $\Omega\subset\Bbb R^d$ $$\frac{\partial\phi}{\partial t}(\mathbf{x},t) +\vec V(\mathbf{x},t)\cdot \nabla\phi(\...
demlevi33's user avatar
  • 153
2 votes
0 answers
94 views

From some priori estimates can we estimate higher Sobolev norm?

Suppose $u$ is a smooth function on bounded set $\Omega$ with smooth boundary such that $$\|u\|_{W^{1,p}(\Omega)}\le C\|\phi\|_{W^{1-1/p,p}(\partial\Omega)}$$ where $u|_{\partial\Omega}=\phi$. Can we ...
Curious student's user avatar
2 votes
0 answers
147 views

Dimension of critical set of p-harmonic function

Let $\Omega\subset \mathbb{R}^n$ be a smooth domain and $u\in W^{1,p}(\Omega)$ a non-constant $p$-harmonic function, for some $1<p<n$. Question: What is the Hausdorff dimension of the critical ...
cork_twist's user avatar
2 votes
0 answers
125 views

Regularity up to boundary of a solution $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ to $\Delta^2 u = -\text{div}\, F$

Let $\Omega\subset \Bbb R^n$ be a $C^{2}$ domain (open and bounded) and let $p\in(1,\infty)$. Suppose $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ is a weak solution to the fourth-order elliptic system ...
BigbearZzz's user avatar
  • 1,245
2 votes
0 answers
49 views

Determining a space of differentiability

I have a questions and maybe you are able to assist with this? Let us consider the space $X:=\mathrm{L}^2[0,\pi]$. On $X$ we consider the family of operators $(P(t,s))_{t\geq s}$ defined by $$ P(t,s)f:...
Daniella Dannell's user avatar
2 votes
0 answers
162 views

$\int_{\mathbb{R}^{N}\setminus\Omega}\vert x-z\vert^{-N-\alpha} dz = c \ \forall x\in\partial U$ implies $dist(x,\partial\Omega)=c, x \in \partial U$?

Let $\alpha \in \mathbb R_+$, $\Omega \subset \mathbb R^N$ and $U \subset \Omega$. Is it true that if $$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-\alpha} dz = \text{constant} \quad \text{for all ...
user175203's user avatar
2 votes
0 answers
657 views

Convergence of operator in norm resolvent sense and their eigenvectors

Let $\{T_n\}_{n=1}^\infty$ and $T$ be (unbounded) self-adjoint operators and $T_n\to T$ in norm resolvent sense, that is, for some $z\in \mathbb{C} \setminus \mathbb{R}$, $\|(zI- T_n)^{-1}- (zI- T)^{-...
user270619's user avatar
2 votes
0 answers
71 views

Strict Riesz's rearrangement inequality when function is not nonnegative

The strict Riesz rearrangement inequality (Lieb- and Loss's book Analysis, Section 3, Theorem 3.9 ,page 93) says that if the functions $f,g,h,$ are all nonnegative and $g$ is strictly symmetric ...
W.J.'s user avatar
  • 379
2 votes
0 answers
162 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 ...
kumquat's user avatar
  • 185
2 votes
0 answers
175 views

Boundary terms in integration by parts for the fractional Laplacian

Let $u,v \in C^\infty(\Omega)$ and assume that $v$ is compactly supported inside a domain $\Omega$. Is it true that $$ \int_\Omega v (-\Delta)^su \, d x = \int_\Omega (-\Delta)^{s/2}v(-\Delta)^{s/2}u \...
user173196's user avatar
2 votes
0 answers
163 views

Bochner's formula for fractional Laplacian

Is there an analogue of the classical Bochner formula $\frac{1}{2} \Delta |\nabla u|^2 = |\nabla^2 u|^2$ for harmonic functions that holds for $s$-harmonic functions?
Zac's user avatar
  • 161
2 votes
0 answers
93 views

How to use Fredholm alternative to check that there are only finite eigenvalues of $H$ on the imaginary axis?

On $\mathbb{R}^3$, we consider the operator \begin{equation} \mathcal{H}= \left( \begin{matrix} -\Delta +1 -2 \phi^2 & -\phi^2 \\ \phi^2 & \Delta -1 +2 \phi^2 \end{matrix} \right) , D(...
Tao's user avatar
  • 429
2 votes
0 answers
101 views

Strongly continuous semigroups on weighted $\ell^1$ space

Let $x=(x_i)$ be a sequence in $\ell^1$ such that all $x_i>0.$ Let $T(t):\ell^1 \rightarrow \ell^1$ be a strongly continuous semigroup of, i.e. $t \mapsto T(t)y$ is continuous for every $y \in \ell^...
Sascha's user avatar
  • 536
2 votes
0 answers
52 views

Reference Request: Dirichlet operators with singular coefficients

Let $d\geq 2$, $\delta \in (0,1)$ and let $\mathcal{L}_{d,\delta}$ be the second order differential operator defined by \begin{align*} \mathcal{L}_{d,\delta}(f)(x) = \Delta(f)(x)-\delta \|x\|^{\delta-...
user69642's user avatar
  • 778
2 votes
0 answers
78 views

How to compute the functional derivative of the following functional encountered in electrical impedance tomography?

Note: I have raised this question in Mathematical stack exchange but it received no attention. That is why I proceed to here to ask this question. Please tell me if this is not appropriate, thank you....
Ken Hung's user avatar
  • 161
2 votes
0 answers
76 views

wave equation with L^2 boundary data via spectral decomposition

It is classical that if $\Omega \subset \mathbb R^n$ is a bounded domain with smooth boundary, then the equation \begin{equation}\label{pf2} \begin{aligned} \begin{cases} \partial^2_{t}u- \Delta u=0\,\...
Ali's user avatar
  • 4,153
2 votes
0 answers
163 views

Hilbert transform on weighted Sobolev spaces

Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping ...
Ali's user avatar
  • 4,153
2 votes
0 answers
164 views

What are (the different aspects of) harmonic analysis good for?

Let $G$ be a locally compact group. To the best of my understanding, harmonic analysis has three legs that all work perfectly in the case that $G$ is in addition compact and abelian, but have ...
Andrew NC's user avatar
  • 2,071
2 votes
0 answers
382 views

Poincaré inequality holds on Riemannian manifolds (min max principle)

In YuChang Xia's book "Eigenvalues on Riemannian Manifolds" Page 4 equation (1.16)Poincaré inequality: I want to know which manifold(s)/function(s) can make the inequality hold. What if we ...
Grantsome's user avatar
2 votes
0 answers
113 views

$W^{1,p}-$regularity on the boundary for solution of Laplace equation with Robin boundary condition

I came across with the attached paper and here is the part that I try to understand. If the non-tangential maximal function of $\nabla u$, i.e $(\nabla u)^*$, belongs in $L^p(\partial \Omega)$, then ...
kaithkolesidou's user avatar

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