# 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.

2,558
questions

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### 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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**1**answer

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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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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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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**1**answer

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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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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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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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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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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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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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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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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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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146 views

### $W^{k,1}$ regularity for elliptic equations

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and assume $u$ is a solutions of $\nabla \cdot (a \nabla u)=f$ with $a>c>0$ in $\Omega$, where $a\in C^k(\bar{\Omega})$. Is the following ...

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**1**answer

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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**0**answers

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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**1**answer

236 views

### A condition for Laplacian

Let $u\in L^{2}(\mathbb{R}^{2}) $ with $-\Delta(u) -c (x^{2}+y^{2})u \in L^{2}(\mathbb{R}^{2})$ where $c>0$.
Is true $-\Delta u \in L^{2}(\mathbb{R}^{2})$?
Thank you in advance.

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1k views

### 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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**3**answers

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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**1**answer

378 views

### Regularity of solution to a hyperbolic pde

I have a question concerning 2nd order evolution equation of the form $u''(t)+A(t)u(t) = f(t)$ in $L^2(0,T;V^*)$, where $f\in\ L^2(0,T;H)$ holds. Under what assumptions is it possible, to guarantee a ...

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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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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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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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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: ...

**4**

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**1**answer

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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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 ...

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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 $...

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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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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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135 views

### Von Neumann stability vs eigenvalues of amplification matrix

My, limited, understanding of the stability analysis of PDEs is that broadly speaking there are two methods: von Neumann analysis which looks at the growth the error of the solution, as described here
...

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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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**1**answer

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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629 views

### Spectrum of Dirichlet Problem for Laplacian on a Parallelogram

Let $ M \subset \mathbb{R}^2 $ be parallelogram constructed by putting together two equilateral triangles (so that all sides of the parallelogram have length 1, and the internal angles are 60 and 120)....

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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 $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...

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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 ...

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942 views

### Elliptic operators corresponds to non vanishing vector fields

Added, June 19, 2019: The main motivation of this post is to associate an index to differential operator associated to a dynamical system such that the index has an interesting ...

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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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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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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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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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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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902 views

### Alternative proof of Varadhan's formula on Riemann manifolds

Consider Varadhan's famous formula for the kernel of the heat equation on a manifold:
$$ \lim_{t \rightarrow 0} t \log h(t,x,y) = - \frac{d(x,y)^2}{4} .$$
I do not have access to his 1967 two papers,...

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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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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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424 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, ...

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**1**answer

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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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 ...