Tagged Questions

Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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2
votes
1answer
214 views

Classical theory for the incompressible Euler equation (reference request)

I have recently been interested in the incompressible Euler equation, but since I am new to the topic, I would like to inquire what are the standard sources/references (for self-study) regarding the ...
1
vote
0answers
100 views

on high order Laplacian

Roughly speaking, we have good understanding of the solution to heat equation $u_t-\Delta u=0$, on bounded or unbounded domain. For example, we know the decay rate, we know it generates analytic ...
6
votes
0answers
117 views

Reference Request: Elliptic differential operators in the Fréchet setting

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...
0
votes
1answer
53 views

properties of frequency-uniform decomposition operator $\square_{k}^{\sigma}$

Let $\rho \in S(\mathbb R^{n})= \text{Schwartz space}, \ \rho:\mathbb R^{n}\to [0,1]$ be smooth radial function verifying $\rho(\xi)=1$ for $|\xi|_{\infty}\leq \frac{1}{2}$ and $\rho(\xi)=0$ for ...
3
votes
1answer
113 views

Techniques to show existence for a PDE with dynamic boundary condition

Let $\Omega$ be a bounded domain. I am looking for techniques to show existence of solutions to dynamic boundary problems of the form $$\Delta u = 0 \quad\text{on}\quad \Omega \times (0,T)\\ ...
0
votes
0answers
56 views

Thesis of Serge Resnick - Dynamical Problems in Non-Linear Advetive Partial Differential Equations

I'm seeking the thesis Dynamical Problems in Non-Linear Advetive Partial Differential Equations by Serge Resnick. In the virtual library of The University of Chicago ...
3
votes
2answers
168 views

Existence for ODE in Banach space (accretive operators and Crandall-Liggett)

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form $$\frac{du}{dt} + Au = f$$ where $A$ is an accretive nonlinear operator under some ...
2
votes
0answers
60 views

Integrability of $D^2u$ for $\infty$-harmonic function $u$?

Consider infinity harmonic functions; that is, functions satisfying $\Delta_\infty u = 0$ with $$\Delta_\infty u = \langle Du, D^2 u \, Du \rangle = \sum_{i,j} \frac{\partial^2 u}{\partial x_i \, ...
6
votes
1answer
243 views

Are there nontrivial real functions of 2 real variables with gradient having constant euclidian norm on each level line?

Let $F$ be the class of locally Lipschitz continuous functions $z=f(x,y)$, from $\mathbb R \times\mathbb R \to\mathbb R,$ such that the euclidean norm $|\ \mathrm{grad}\ f (x,y)\ |$ of its gradient ...
5
votes
0answers
270 views

Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
0
votes
0answers
56 views

Guassian upper bound of the heat kernel implies ultracontrativity?

For spaces satisfies uniformlly local doubling and Poincare inequality (for example, Riemannian manifold with Ricci curvature bounded below RCD(K,N)). By Sturm's paper, we have bounds on the heat ...
0
votes
1answer
136 views

Fractional Laplacian on compact hypersurface/manifold via harmonic extension?

Let $M$ be a sufficiently smooth compact hypersurface of dimension $n-1$ in $\mathbb{R}^n$. In pages 10-11 of this paper, the authors define $\mathcal{M} = M \times (0,\infty)$ and consider the ...
1
vote
0answers
87 views

Boundary gradient estimate

Assume $U$ is the unit disk and $\bar U$ its closure and let $u\in C^2(U)\cap C(\bar U)$ be a real function, with $u(z)=0$ for $z\in \partial U$. If $$|\Delta u|\le A|\nabla u|^2+g(z),$$ for some ...
0
votes
0answers
56 views

Why is it impossible to reduce a linear PDE of the second order in more than two independent variables to canonical form globally

It is known that in the case of more than two independent variables, it is usually not possible (especially in the case of PDE with the variable coefficients) to reduce a linear partial differential ...
3
votes
0answers
67 views

Local energy decay for variable-speed, divergence-form wave equation in non-trapping medium without obstacles

I'm looking for a reference in the literature describing local energy decay for solutions of a smooth-coefficient, variable-speed wave equation, in divergence form, with compactly-supported initial ...
1
vote
0answers
84 views

Buseman function for Riemanniam manifolds with two ends and $Ric\ge -(n-1)$ [closed]

It's well known that if M is a Riemannian manifold with $Ric \ge 0$ and contains a line $\gamma $. Set $${\gamma _ + } = \gamma | {_{[0, + \infty )}} ,{\gamma _ - } = \gamma | {_{[ - \infty ,0)}} ...
2
votes
0answers
79 views

Weighted energy estimate for the heat equation of higher order

The question is originally related to Hardy's uncertainty principle, convexity and Schrodinger evolutions. In this work the authors deduce a convex property of Schrodinger equation by doing it first ...
0
votes
1answer
118 views

Nonlocal (parabolic) PDEs in the Sobolev space setting

Can someone recommend me some literature on nonlocal parabolic problems (eg. of the form $$u_t + (-\Delta)^s u = f$$ where the nonlocal operator is the fractional Laplacian) in the setting of Sobolev ...
1
vote
0answers
102 views

Passing to the limit in a PDE (subsequence problems)

For $w \in L^2(0,T;H^1)$, consider the PDE $$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$ where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and ...
4
votes
2answers
491 views

Limit cycles as closed geodesics(geodesible flow)

The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$: \begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation} This equation defines a foliation on ...
0
votes
0answers
55 views

Interpolation with time continuity

If $u(x,t)$ is a function depends on $x\in\Omega$ and $t\in[0,T]$. The following result could be found in L.C. Evans's book "PDE". Suppose $u\in L^2(0,T;H_0^1(\Omega))$, with $u_t\in ...
4
votes
1answer
316 views

Existence of positive solutions of a linear PDE on closed manifolds

I was wondering is there a sufficient condition (or sufficient and necessary condition) for the existence of positive solutions of the following linear PDE on a closed manifold $(M, g)$, ...
2
votes
1answer
87 views

Elliptic Fourier integral operators

I know what it means for a pseudodifferential operator $A\in\Psi(\mathbb{R}^n)$ to be elliptic at a point $(x,\xi)\in T^*\mathbb{R}^n$: the principal symbol of $A$ is non-vanishing at the point. But ...
2
votes
0answers
63 views

Changing the test function space in a weak formulation of parabolic PDE

Suppose we are interested in the existence of a $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that $$(u(T),\varphi(t))_H -\int_0^T \langle \varphi'(t), u(t) \rangle_{V^*,V} + \int_0^T ...
5
votes
2answers
195 views

Probabilistic Interpretation of First Dirichlet Eigenvalue?

The first Dirichlet eigenvalue of a compact domain $\Omega\subset\mathbb{R}^n$ with smooth boundary is the smallest positive number for which there exists a non-trivial solution to $$ -\Delta\psi = ...
1
vote
1answer
237 views

A question about the first eigenvalue for two Kahler metrics

While reading the paper of Gang Tian, "Kähler-Einstein metrics with positive scalar curvature". In the proof of Theorem 1.6, he pointed that if two Kahler metrics $\omega $ and $\omega'$ satisfies ...
2
votes
2answers
381 views

Elliptic theory on compact manifolds

Maybe this is silly. On a bounded set $\Omega\subset\mathbb{R}^n$ consider the equation $$ \Delta u=f \quad\text{ in $\Omega$}$$ $$ u=0\quad\text{ on $\partial\Omega$}.$$ One has the following ...
1
vote
0answers
114 views

Bound for a certain integral expression

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...
4
votes
0answers
151 views

$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in ...
3
votes
1answer
114 views

Checking initial data in parabolic PDE with no control on time derivative

It is possible to define a weak solution of a parabolic PDE $$u_t - Au = f$$ $$u(0) = u_0$$ as $u \in L^2(0,T;H^1)$ such that $$-\int_0^T\int_\Omega u(t)\varphi'(t) + \int_0^T\int_\Omega ...
2
votes
0answers
161 views

Scattering for rapidly decaying solutions of NLS

Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions of the Nonlinear Schrödinger Equation" the following property. Given the problem \begin{equation} \left\{ \begin{array}{rl} ...
0
votes
0answers
50 views

Solvability and uniqueness of Fokker-Planck BVP

I have been searching for solvability for the following BVP, I believe that this is a Fokker Planck's equation but I can't find any comprehensible text on existence and uniqueness of the solutions of ...
2
votes
1answer
120 views

Extending a harmonic function in a ball to subharmonic in a larger ball

Consider the Laplace equation in a ball $B(r) \subset \mathbb{R}^n$ of radius $r$: $$ \begin{cases} -\Delta u &= 0, \quad \text {in} \quad B(r), \\ \ \ \ \ \ \, u&= g, \quad \text {in}\quad ...
0
votes
0answers
64 views

Closure of partial differential operators on $L^2(\Omega)$

Let $\Omega\subset\mathbb{R}^2$ open set. Consider an $\textit{uniformly elliptic}$ second order differential operator on $L^2(\Omega,\mathbb{C})$ $$ H=\sum_{|\alpha|\le 2}C_\alpha\partial^\alpha $$ ...
2
votes
4answers
172 views

Showing coercivity condition for an energy functional

Consider the energy functional $e(\cdot)$ \begin{align*} e(f,Q)&=\int_a^b \bigg\{f^4\bigg[1+\|\frac{d}{dr}Q\|^2+f^2\dot f^2\bigg]\bigg\} \,dr, \end{align*} over the space of \begin{equation*} ...
3
votes
0answers
149 views

Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$

Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required. Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b < ...
1
vote
0answers
116 views

A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form $$u_t(t) - ...
0
votes
0answers
60 views

When is Dirichlet solution from disk to Riemannian manifold Holder continuous near the boundary?

$D$ is the two-dimensional unit disk. $X$ is a compact Riemannian manifold. Let$\phi \in W^{1,2}(D,X)$ and define $$ W^{1,2}_{\phi}=\{v \in W^{1,2}(D,X):Trace(v)=Trace(\phi)\} $$ Let $$ ...
0
votes
1answer
170 views

Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banach algebra $A(\mathbb T)$ ?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
0
votes
1answer
86 views

uniqueness for Poisson equation in R^d with mildly regular data

I'm interested in Poisson's equation $-\Delta u=f$ set in the whole space $R^d$ (let's say $d\geq 3$ for simplicity) when $f$ has very little integrability, specifically $f\in L^{1+\varepsilon}$ for ...
7
votes
3answers
301 views

Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$

Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which ...
11
votes
3answers
741 views

Sobolev spaces and geometry

This is a very naive question, is there a way to geometrically understand Sobolev spaces without going through analysis and PDE's? To my knowledge, Sobolev spaces where created precisely to study ...
1
vote
0answers
96 views

How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?

(May be this is very basic question for MO) (For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
0
votes
1answer
111 views

Integrability of the Poisson integral

Maybe this is rather obvious, but I'm stuck. Let's consider the Laplace equation in the upper half plane with boundary condition $g$, $i.e.$ $$ \Delta u(x,y)=0, u(x,0)=g(x). $$ Then the solution is ...
1
vote
1answer
222 views

Are smooth functions dense in the space $\{u \in H^1(Q) \text{ with } \Delta_\Gamma u \in L^2(Q)\}$?

Define $$Q = \bigcup_{t \in (0,T)}\Gamma \times \{t\}$$ where $\Gamma$ is a compact (without boundary) hypersurface. Assume whatever smoothness is required. Define $L^2(Q) := L^2(0,T;L^2(\Gamma))$ ...
1
vote
0answers
79 views

Regularity of weak solutions for a quasilinear problem

Theorem 6.2.7 in the book Nonlinear Analisis of Gasisnski and Papageorgiou states: If $u\in W^{1,p}(\Omega)\cap L^\infty(\Omega)$ and $\Delta_pu \in L^\infty(\Omega)$ then we have $u\in ...
5
votes
1answer
220 views

Green's function for *GJMS* operator

Consider a Riemannian manifold $(M^n, g)$ of dimension $n$ with a metric $g$. We assume $M$ to be closed (compact without boundary). Let's not assume any hypothesis on the Yamabe invariant of the ...
3
votes
1answer
126 views

A question about solutions to Floer's equation which are asymptotic to a stationary point

Let $M$ be a compact symplectic manifold and $H$ a time independent Hamiltonian on $M$. Let $\alpha$ be a solution to Floer's equation $$ u(t,s): S^1 \times \mathbb{R} \to M$$ $$(du+X_H\otimes ...
0
votes
0answers
101 views

Localized convolution operators

Take our favorite differential operator $\Delta$ on $\mathbb{R}^3$. Module a constant, we can represent its inverse (in some sense) by a convolution operator, namely $$\Delta^{-1}f=f*\frac{1}{|x|}.$$ ...
0
votes
0answers
72 views

Mean value formula for high order elliptic operators

Weyl's lemma says any distributionally harmonic function $u\in L^1_{\text{loc}}$ is harmonic. One way to prove it is as follows. Take radially symmetric compactly supported and smooth approximation ...