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

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3
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233 views

Controlling the Second Eigenvalue of a Schrödinger Operator

Consider a bounded domain $\Omega$ (with smooth boundary) in some Riemannian $n$-manifold $M^n$. Let $L$ be the operator $$ L=\Delta+V $$ where $\Delta$ is the Laplace-beltrami operator on $M$ (so is ...
3
votes
0answers
163 views

Generalizations of group algebras for arbitrary manifolds?

In the analysis of partial differential equations on Euclidean spaces, one of the most useful properties of the Fourier transform (and the related integral transforms) is that they take ...
2
votes
0answers
113 views

Heat kernel and Wiener measure

A theorem by Barry Simon says that for arbitrary open sets $\Omega\subset \mathbb{R}^n$, we have $$[\exp(t\Delta_{\Omega}^D)](x,y) = \mu_{x,y,t}\lbrace \omega \text{ } \vert \text{ } \omega(s) \in ...
2
votes
0answers
119 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 $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
2
votes
0answers
45 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 ...
2
votes
0answers
62 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 ...
2
votes
0answers
134 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} ...
2
votes
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74 views

Is a certain set of periodic solutions of the 2D Navier-Stokes equations closed generically?

I would be interested to know if a certain set of periodic solutions for the two-dimensional Navier-Stokes equations is closed generically. Many similar (yet not identical) set-ups can be found in the ...
2
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106 views

geometric irregularities in pde's

The following question is intended for a person more acquainted with the works of Yves Laurent. see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French) ...
2
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104 views

How to show the identity $\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$?

I am reading this paper. Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$ On page 5 of ...
2
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125 views

Estimates on gradients of diffusion semigroups

Consider the Dirichlet or Neumann Laplacian on a manifold with boundary. Suppose we have some estimate of the form $$||e^{t\Delta} f||_{L^p} \leq C(t)||f||_{L^q}$$ for some $p, q$. For a specific ...
2
votes
0answers
66 views

Deduce global estimate from scaling-invariant local estimate

Let $(M,g)$ be a non-compact Riemannian manifold, with finite volume (or compactly exhausted, or any nice condition you would like, except for compactness). Suppose I have a tensor $T$ on $M$ of which ...
2
votes
0answers
82 views

slightly subcritical elliptic pde; the linearized equations

Let $ p_m \nearrow \frac{N+2}{N-2}$ and consider the family of elliptic problems $$-\Delta u_m(x)=u_m(x)^{p_m} \quad B \qquad \quad u_m =0 \quad \partial B,$$ where $B$ is the unit ball ...
2
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71 views

Linear heat equation with initial condition of generalized function

I am consider a very simple heat equation over the interval $[0, 1]$ with a Neumann BC and a very bad initial condition, written as: $\partial_tu(t, x) = \partial^2_xu(t, x) + a(t, x)u(t, x)$, for ...
2
votes
0answers
72 views

Extra regularity of Poisson problem having nonzero Neumann boundary condition in convex domain

Let $\Omega\subset\mathbb{R}^2$ be a convex simply connected domain having piecewise smooth boundary, $f\in L^2(\Omega)$ and $g\in H^{\frac 1 2}(\partial\Omega)$. Grisvard in [1] among others prove ...
2
votes
0answers
60 views

What's a good resource for Hormander symbols of type (1/2, 1/2)?

I'm currently working with some pseudodifferential operators of Hormander class $L^{m}_{\frac{1}{2},\frac{1}{2}}$ and unfortunately many of the usual tools break down, due to difficulties with their ...
2
votes
0answers
175 views

Alternative representations of Sobolev space

Is there a way to represent a Sobolev space as the image of a fractional integral operator over an $L^p$ Lebesgue space? Yes, as it was comment, there is an answer for that in the book "Singular ...
2
votes
0answers
124 views

How to pick out harmonics based on boundary conditions?

(..this is almost a continuation of my last question (which got closed!)...) Let me first rewrite one of the main results of this paper, http://calvino.polito.it/~camporesi/JMP94.pdf in a coordinate ...
2
votes
0answers
121 views

W^2,p regularity for solutions of elliptic equations

I'm stucked in the following (maybe classical) issue concerning the $W^{2,p}$ regularity of solutions of a second order elliptic equations like $Lu=f$ in a bounded domain (say a ball) $\Omega$. I have ...
2
votes
0answers
91 views

Why pseudoconvexity is important in Partial differential equation theory?

I am a new researcher in mathematics and I work on convexity. Are convexity and pseudoconvexity related topics and in which respect to PDE theory ? One of the important results in PDE theory is the ...
2
votes
0answers
146 views

A contradiction to do with continuity? (involves chain rule)

Suppose for each $t$, $S(t) \subset \mathbb{R}^n$ is a domain (hypersurface). We have a diffeomorphism $D^0_t:S(0) \to S(t)$ for each $t$ such that it solves the ODE $$\frac{d}{dt}D^0_t(\cdot) = ...
2
votes
0answers
81 views

Elliptic equations with divergence-free drift terms

Given $\ \mathbf{u}\cdot \nabla c=\Delta c-a_{1}c+\rho \text{ on }\Omega $ with a $\Omega \subset %TCIMACRO{\U{211d} } %BeginExpansion \mathbb{R} %EndExpansion ^{2}$ bounded, $div$$(\mathbf{u})=0$, ...
2
votes
0answers
72 views

Helmhotz decomposition and Regularity in Stokes equation

It is known that every function $f\in L^{q}(\Omega )^{n}$ can be uniquely decomposed as \begin{eqnarray*} \ f=f_{0}+\nabla Q, \text{ (Helmhotz decomposition)} \ \end{eqnarray*} with $f_{0}\in ...
2
votes
0answers
58 views

Inclusions between $L^p$ continuous functions and Triebel-Lizorkin spaces

Working in $\mathbb{R}^{d}$, consider on the one hand the space of continuous $L^{p}$ functions (let's use $V$ to denote this space), and on the other the family $\{ F_ {\alpha}^{p, q} \}_{\alpha, q}$ ...
2
votes
0answers
109 views

Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
2
votes
0answers
107 views

linear operator associated with semilinear elliptic pde

I am reading a paper where at some point they analyse the following linear operator: $$L_\lambda(\phi)= - \Delta \phi - C_\lambda(x) \phi$$ where $ C_\lambda(x)>0$ (smooth) in $ \Omega$ a bounded ...
2
votes
0answers
91 views

Regularity of solution of nonlinear equation

Hi! Let $L$ be a linear elliptic operator of order $4$ with smooth and bounded coefficients on the ball $B_1$ of $R^{2n}$ and let $N\in C_{loc}^{0,\alpha}(R^{3})$. Let $f\in C^{0,\alpha}(B_1)$ ...
2
votes
0answers
266 views

Nonlinear PDE and Green functions

This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like $$ \partial^2\phi+V(\phi)=\delta^D(x). $$ I do not know if a real ...
2
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0answers
97 views

Examples of non-uniqueness in reaction-diffusion equations

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...
2
votes
0answers
96 views

Existence of solutions to a reaction-diffusion problem.

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...
2
votes
0answers
157 views

Similarity solutions of the imaginary time Benjamin--Ono equation

This problem arose in the course of a theoretical physics project. We seek (complex) solutions of the imaginary time Benjamin--Ono equation $$u_t-iu u_x-iu_{H,xx}=0$$ where $u_H(x,t)$ denotes the ...
2
votes
0answers
144 views

Laplacian type operator on compact Lie group

Consider the operator $S = \sum_{i,j} X_{ij}^2$ on $L^2(SO(n+1))$, where $X_{ij}$ generates the rotation of the sphere $S^n$ in the $ij$-plane keeping the $(n - 2)$ complementary directions fixed. ...
2
votes
0answers
82 views

P-laplacian equation

Hi guys, in what sense the p-Laplacian is degenerate for p greater than 2 and singular for p smaller than 2 ? Thank you!
2
votes
0answers
124 views

Extension divergence-free, curl-converging vector field

Hi. Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...
2
votes
0answers
180 views

Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...
2
votes
0answers
93 views

Best constant of Gagliardo-Nirenberg inequality in exterier domain

In $\mathbb{R}^N$, we know that the best constant of Gagliardo-Nirenberg is characterized by the solution $Q$ of $-\Delta u+u=|u|^2u$ with minimal mass. One have $||u||_4^4\leq C||u||_2||\nabla ...
2
votes
0answers
120 views

Exponential decay for the gradient of a solution

Dear all, I would like to prove the exponential decay of the derivatives of a solution to the following equation in $\mathbb{R}^N$: $$ \sqrt{-\Delta+m^2} u +u= f(u), $$ where I can assume that $m \neq ...
2
votes
0answers
202 views

How to apply Lagrange Multipliers to BCs of Time Dependent problems using finite elements?

I am trying to implement a finite element scheme using the method of lines (finite difference in time and finite element in space) and enforcing boundary conditions using Lagrange Multipliers. This ...
2
votes
0answers
196 views

Recovering full regularity by energy method in the heat equation

Consider the heat equation $$ u_t = u_{xx} + f, $$ on the circle, and for a finite time interval. From Duhamel's principle one can deduce that $u\in L^\infty H^2$ if for instance $f\in L^\infty H^s$ ...
2
votes
0answers
140 views

A limit involving a regularizing kernel

I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# ...
2
votes
0answers
79 views

Asymptotics of quasilinear elliptic equations with Dirac right hand side

On a small open neighborhood $U$ of $0 \in \mathbb{R}^n$, consider the quasilinear (possibly monotone) elliptic, scalar PDE in divergence form $$\nabla \cdot(a(x,u,\nabla u)\nabla u) = \alpha ...
2
votes
0answers
152 views

Integrability of ground state solution for elliptic equation

For the solution of semi-linear elliptic equation, for example I'm considering the 2D cubic nonlinear Schroedinger equation, the correspongding elliptic equation is $\Delta u+u^3=u$, with $u>0$. By ...
2
votes
0answers
247 views

Variational Formulation of Boundary Value Problems With Unknown on the boundary

Suppose that we have a linear operator equation on $\Omega$ with Lipschitz boundary $\partial \Omega$, \begin{eqnarray} Lu &=& \frac{\partial u}{\partial t}, u(x,0) &=& u_0 \; \; ...
2
votes
0answers
178 views

Core of divergence form operator with unbounded coefficient

Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of $$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$. I also assume that ...
2
votes
0answers
436 views

Open sets and Poincaré's inequality

In many references, Poincaré inequality is presented in the following way : Let $\Omega\subset \mathbb R^d$ an open bounded set. We can find a constant $C$ which depend of $\Omega$ such that for ...
2
votes
0answers
216 views

Equation of motion of a charged string in a twisted torus.

The flat torus background: Say we want to study the sigma model of strings (closed strings $S^1= \mathbb{R}/\mathbb{Z}$) on a flat torus (for example $T^3=\mathbb{R}^3/\mathbb{Z}^3$ with a flat ...
2
votes
0answers
429 views

Comparision of cubic hermite finite element and cubic B-spline finite element (in condition nunmbers of stiffness matrix, or sth else)

Background Consider the one dimensional second order elliptic PDE, $$ \left\{\!\! \begin{aligned} & -(a(x)u'(x))'+b(x)u(x)=f(x)\qquad x\in[0,1]\\ & u(0)=u(1)=0 \end{aligned} ...
2
votes
0answers
249 views

A free boundary problem by finite difference method

I wanna discretize the following free boundary problem Find $u$ and $\Omega$ such that $\Delta u=1-\delta_0$ in $\Omega$ with the conditions $u=|\nabla u|=0$ on $\partial \Omega$. I apply finite ...
2
votes
0answers
254 views

General solutions for HJB equations in a special case.

I am reading the book of Wendell Flemming in control theorem to learn the HJB equation Here is the setting that interests me: Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ functions such that ...
2
votes
0answers
277 views

Vanishing solution of the Poisson equation at infinity

Hi, I am interested in finding some vanish bahavior at infinity of the solutions of this kind of equations: $-\Delta\phi+a(x)\phi=b(x)$ where $a(x), b(x)\in L^{p}$ with $1\leq p\leq 3$. Besides ...