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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27 views

Extension of probability space problem: Hilbert space valued process V.S. random field

Maybe the question should be "Understanding the measurability: Hilbert space valued process V.S. random field" Consider the SPDE $${\rm d}u+\cdots{\rm d}t=\sigma(t,u){\rm d}W.$$ Consider the ...
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27 views

Neumann equation on manifold with edge or corner

Let $(M,g)$ be a compact Riemannian manifold with boudnary and corner, i.e. locally mdoelled in $[0,\infty)^1\times \mathbb R^{n-1}$ or $[0,\infty)^k\times \mathbb R^{n-k}$, where $n=\dim(M)$. ...
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55 views

What is the geometric or dynamic meaning of a global attractor with an infinite fractal dimension?

In Efendiev-Ôtani's article: Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium (Ann.I. H. Poincaré, AN 28,2011), is obtained that the fractal dimension ...
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1answer
82 views

Minimal assumptions such that the solution of Poisson equation is $C^2$

Take a weak solution $u$ of the Poisson equation on $\mathbb{R}^d$ $$ \Delta u = f $$ By standard elliptic regularity theory we have (for some $\alpha\in (0,1]$) $f\in C^{0, \alpha}_{\text{loc}}(\...
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2answers
260 views

Domain of definition of Laplace Operator on $L^2$

I'm trying to combine two ways of looking at the Laplacian $\Delta$ on $\mathbb R^n$ (and on other domains). Firstly, it is well known that this operator is essentially self-adjoint on $C_c^\infty(\...
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38 views

Strichartz estimates for the inhomogeneous wave equation

In the Blair, Smith and Sogge's paper Strichartz estimates for the wave equation on manifolds with boundary, the authors study integrability estimates for solution of the following problem: \begin{...
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2answers
80 views

Question on Parabolic PDEs: Improvement of the bound ${\vert \vert v(\cdot,t)-v_0(\cdot) \vert \vert}_{L^p}\le \hat C t^{1/q}$

Let $M$ be a $C^3-$compact manifold and $v \in W^{2,1}_p(M\times [0,T])$ ($1\le p<\infty$) be the solution of: $\begin{cases} \partial_t v-\Delta_{M} v=f(v), \quad M\times [0,T]\\ v(x,0)=v_0,...
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34 views

Bilinear Strichartz estimates for the Schrodinger equation

Let $d\geq 2$, $I\subset\mathbb{R}$ a time interval and $t_0\in I$. Let $u,v$ be solutions to the free Schrodinger equation on $\mathbb{R}^d$, i.e. $(i\partial_t+\Delta)u=(i\partial_t+\Delta)v=0.$ Let ...
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85 views

Optimal Sobolev regularity for $(-\Delta)^{-s}$ on domains

In the paper Chen, Huyuan; Véron, Laurent, Semilinear fractional elliptic equations involving measures, (J. Differ. Equations 257, No. 5, 1457-1486 (2014). ZBL1290.35305) in the proof of Proposition ...
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2answers
118 views

Rate of convergence of mollifiers // Sobolev norms

Following up to the question raised here, I am searching for a reference (or a simple argument) to establish (in the whole space) the following (suggested) equivalence : Given $N_1$ and $N_2$ two (...
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1answer
69 views

Literature about solitons and Hirota derivatives

This summer I'm going to learn a mini-course about soliton theory ("Soliton equations and symmetric functions" in LHSM (Russian summer school in mathematics). The web-page of this course is https://...
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27 views

Sufficient conditions for taking limits in stochastic partial differential problems

Let's say we have a (parabolic) Cauchy problem: $$ (1) \hspace{0.5cm} u_t (x,t)+A(u) \cdot u_x (x,t))=\nu \cdot u_{xx} (x,t) + \epsilon \cdot f(u) \cdot W, $$ $$(2) \hspace{0.5cm} u(x,0)=u_0(x), $$ ...
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1answer
57 views

Endpoint in commutator estimate

Let $p\in(1,\infty)$ and $J^s=(1-\Delta)^{s / 2}$ with $s>0$. Then we have the following commutator estimate by C. E. Kenig, G.Ponce and L. Vega (1991 JAMS), \begin{equation} \left\|J^{s}(f g)-f J^{...
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63 views

Partial regularity of harmonic maps into spheres

Let $u : B_1(0) \subset \mathbb{R}^n \to S^k$ be an energy minimizing (minimizing in $H^1(B_1(0); S^k)$) harmonic map. I am trying to understand the theory of partial regularity, where the main claim ...
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169 views

Elliptic Regularity with Gibbs Measure Satisfying Bakry-Emery Condition

Consider $\mathbb{R}^d$ with Gibbs measure $d\mu=Z^{-1}\exp(-V(x))dx$, where the potential $V(x)$ is strongly convex ($\nabla^2 V(x) \ge \lambda Id $). We can assume the regularity of $V$ is as good ...
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1answer
144 views

On the limiting behaviour of Sobolev space functions

Let $k$ be an integer such that $k>n/2$, and let $H^k(\mathbb{R}^n)$ denote the usual Sobolev Hilbert space. Let $f,g\in H^k(\mathbb{R}^n)$. Is it true that $\displaystyle \lim_{R\rightarrow \...
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Is there a connection between representation theory and PDEs?

As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination ...
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3answers
113 views

Does the Leibniz (product) rule hold for the spectral fractional Laplacian?

Does the Leibniz (product) rule hold in some sense for the spectral fractional Laplacian (at least in 1 dimension)?
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1answer
89 views

Interpolation Inequality's Proof

Let $\Omega \subseteq R^{n}$ bounded domain. I need to prove that for $u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$: \begin{equation} \|\nabla u\|_{L^{2}(\Omega)}^{2}\leq \|u\|_{L^{2}(\Omega)}\|\Delta u\...
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40 views

Does an integration by parts formula hold for the spectral fractional Laplacian in 1-d?

Is there an integration by parts formula for the spectral fractional Laplacian in a bounded interval $[a,b] \subset \mathbb R$?
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21 views

Smooth compactly supported function with good scaling with respect to the fractional Laplacian

Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
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1answer
83 views

Is a bounded sequence of $H^1(\Omega)$ tight?

Assume $\Omega$ is a bounded subset of $\Bbb R^d$ and $ (u_n)_n$ is a bounded sequence of the Sobolev space $H^1(\Omega)$. Question: Can we say that $ (u_n)_n$ is tight in $L^2(\Omega)$ namely: ...
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1answer
111 views

Eigenvalues and Domain of the Laplace-Beltrami Operator

Assume $(M,g)$ is a compact Riemannian manifold without boundary, where $g$ is the Riemannian metric. Let $L:=-\Delta$ be the Laplace-Beltrami operator on $M$ defined by $\Delta \cdot = \text{div}(\...
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61 views

Constant in the estimate on the Green's function of the Laplacian

Given the Laplacian associated to a Riemannian manifold $(M^n, g)$, there is a Green's function $G(p,q): M \times M \to \mathbb{R}$ that satisfies an inequality of the form $$|G(p,q)| \leq Ad(p,q)^{2-...
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36 views

Schrödinger equation: well-posedness with Hartree potential and Yukawa potential

Consider the Schrödinger equation of Hartree type (HT): $$i\partial_tu +\Delta u + (V\ast |u|^2)u=0, u(x,0)=u_0$$ with $(x,t)\in \mathbb R^d \times \mathbb R.$ where $V$ is some potential. (1) when $...
1
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1answer
54 views

$C^{\alpha}$ estimates for spectral fractional Laplacian

Consider the following problem with $\Omega_{\epsilon}$ is bounded for each fixed $\epsilon>0$ and $s\in (0, 1)$. Let $u_{\epsilon}$ be a classical solution of \begin{equation} \ \ \left\{\begin{...
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1answer
66 views

Cylindrical coordinates in axis symmetric flow

I am getting stuck in a detail in a paper. It's about the axi symmetric Navier Stokes equations $$u_t - \nu\,\Delta u + u\cdot \nabla u + \nabla p=0$$ We consider in cylindrical coordinates $u=(u^r, u^...
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1answer
102 views

Dirichlet problem for manifold, how to prove $W^{1,2}_0(\Omega)$ solution is $C^{2,\alpha}(\bar{\Omega})$?

Let $M$ be an n dimensional Riemannian manifold without boundary. Let $\Omega \subset M$ be a bounded domain with smooth boundary. Let $f \in C^{\alpha}(\bar{\Omega})$, Consider the Dirichlet problem. ...
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49 views

A uniform upper bound for Fredholm index of Laplace quasi-operators on a compact parallelizable manifold

Assume that $M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...
3
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1answer
306 views

PDE system problem to find the metric

I have a big problem to solve this system: $\Delta f-hf^2=0$ $p|\nabla f|^2+hf^3=0$ where $h$ and $p$ are constants (with $h \neq0$ and $p \neq0$ and $\neq -1$), $f$ is a scalar function defined on ...
4
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1answer
77 views

Characterization of continuous weakly closed 1-forms

Recall that a differential $k$-form $\alpha$ on a smooth manifold $M$ is called weakly closed if $$\int_M \alpha \wedge d\beta = 0,$$ for all smooth forms $\beta$ of degree $n-k-1$, where $n = \dim ...
3
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275 views

On Solving a Fourth-Order Non-Linear PDE

I am presently working on a problem in fluid dynamics where our group is investigating the behavior of temperature and velocity at the leading edge of a flat plate when fluid flows past it. The ...
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79 views

p-Laplacian is intrinsically uniformly continuous

Definition- The operator $G$ is intrinsically uniformly continuous(IUC) with respect to $x$ in $TM-\{0\}\times Sym TM$ if there exists a modulus of continuity $w_{G}:[0,\infty)\to [0,\infty)$ with $w_{...
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1answer
202 views

Heat equation on torus

Consider heat equation on torus: $$\partial_tu(x,t) + (- \Delta)^{\alpha/2} u(x,t)=0, u (x, 0)=u_0(x)$$ where $(x, t) \in \mathbb T^d \times \mathbb R, \alpha>0$ Formally, we may write the ...
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309 views

On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold

Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field ...
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11 views

Stabilization of non-autonomuous 1-d wavs equation

I want to ask two questions about the stabilization of the equation $$\eqalign{ & {y_{tt}} = k(t,x){y_{xx}}+a(t,x){y_t}+ b(t,x){y_x}+ c(t,x){y_x} +d(t,x)y \ \ (t,x) \in {\text{ }}(0,\infty ) ...
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2answers
161 views

Diffeomorphisms as solutions to second-order ODEs

Fix $f: \mathbb{R}\times \mathbb{R}^n \mapsto \mathbb{R}^n$ and $v_0:\mathbb{R}^n \mapsto \mathbb{R}^n$. Let $X_t$ be the solution to the second-order ODE $$\frac{d^2}{dt^2}X_t = f(t,X_t), \quad X_0 ...
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113 views

Traces of manifold-valued Sobolev maps

Let $(M^m,g)$ be a compact Riemannian manifold with smooth nonempty boundary, and $N^n\subseteq \Bbb R^d$ a boundaryless isometrically embedded Riemannian manifold. For $1\le p<\infty$ we define as ...
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25 views

Underspecified Riccatti-type ODE

I came across the following Ricatti-type ODE in my reading $$ \begin{aligned} \partial_t \psi(t,x) &= \Psi(\psi(t,x)),\\ \psi(0,x)&=x,\\ \Psi(x)&\triangleq \partial_t\psi(t,x)|_{t=0^+}. \...
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58 views

A kind of heat equation on a foliated 3D manifold whose leaves are invariant under the flow of a vector field

Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let $\Delta_{\...
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1answer
94 views

Expression of solution of fractional Laplacian

Assume $s\in (0, 1).$ Suppose $u$ is a classical solution of the following boundary value problem $$ \begin{cases} (-\Delta )^{s} u=0 \text{ in } A;\\ u= g \text{ in } B, \\u= h \text{ in } C, \end{...
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0answers
31 views

Integrability condition on function determining PDE domain

I'm currently looking through the following paper which examines some dynamics of the Airy$_2$ process: https://arxiv.org/pdf/1106.2717.pdf On page 2, there appears a PDE of the form $\partial_t u +...
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54 views

Regularity of a shrunken domain

I am encountering a geometrical question that intuitively seems obvious but I have a lack of argument to prove or disprove it in a rigorous manner. Let $\Omega\subset\Bbb R^d$ be an open bounded (...
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73 views

Name for a Particular (Parabolic) PDE

This is a cross post from MSE. The original question can be found here:https://math.stackexchange.com/questions/3248114/name-for-a-particular-parabolic-pde Consider the following initial value ...
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2answers
152 views

Banach space-valued test functions in the definition of a weak solution of a PDE problem

In the literature about PDEs it is easy to find books that talk about weak solutions of a partial differential equations. A short reminder of the usual definition is given bellow. More information ...
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41 views

Wellposedness of semilinear wave equation with discontinuous source

Where can I find existence and uniqueness results for semilinear wave equations with discontinuous, i.e. $$\partial^2_{tt} u - \Delta u = f(u), \quad t >0, \ x \in \Omega$$ where $f$ is ...
2
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1answer
132 views

Metric on the phase space

I am studying PDEs whose symbols satisfy \begin{equation} |\partial^\alpha_\xi\partial^\beta_xp(x,\xi)| \lesssim M(x,\xi)\Psi(x,\xi)^{-|\alpha|}\Phi(x,\xi)^{-|\beta|} \end{equation} for all multi-...
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3answers
200 views

One dimensional heat equation with boundary conditions

Consider the heat equation $$u_t = u_{xx}$$ for $t \ge 0$, $0 \le x \le L$, given boundary conditions $$u(0,t) = u(L,t) = f(t)$$ and an initial condition $$u(x,0) = g(x)$$ for some continuous ...
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0answers
56 views

Energy-minimizing set of discrete points in a bounded domain

Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain. Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize $$ \sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|} $$ over all ...
1
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1answer
172 views

Linearization of a PDE

I have been struggling with some linearization argument of the following paper: "M. Weinstein: Modulational stability of ground states of NLS". In order to give a bit of context to my question, let us ...