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

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55 views

Equivalence of two definitions of weak solution (subtlety with null sets)

Consider $$y_t - \Delta y = f$$ $$y(0) = y_0$$ with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions: We have $y \in ...
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108 views

$L^2$ bound on solution of PDE in terms of $L^2$ norm of initial value

Let $u \in H^1((0,T)\times S)$ be the unique solution of $$u_{tt} + \Delta u =0$$ $$u|_{t=0}= u_0$$ $$u|_{t=T}=0$$ where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without ...
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50 views

sharp conditions characterizing the vanishing of scalar Jacobi fields

Let $T>0$, let some function $\kappa(t)$ smooth on $[0,T]$, and let $b$ the unique solution to the ODE $\ddot b + \kappa(t) b = 0$ with initial conditions $b(0)=0$ and $\dot b(0) = 1$. I was ...
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31 views

Recursive formula for symbol of resolvent on noncompact manifold

On a compact Riemannian manifold $(M,g)$ without boundary it was shown (by R. Seeley) how to define the complex power of an elliptic classical pseudodifferential operator $A$ of positive order $m$: ...
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53 views

persistence of regularity for nonlinear Klein-Gordon equation

I have been reading the paper on nonlinear Klein-Gordon equation(NLKG) for initial data in modulation space: For detail please see the paper "Klein-Gordon Equations on Modulation Spaces (2014)" ...
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82 views

maximum principle on compact manifolds with boundary

Let us consider the equation $Lu + f(u) = 0$ on a compact manifold $\overline{M} = M \cup \partial M$ with boundary, with Dirichlet boundary conditions. $L$ is a linear elliptic operator, and $f$ ...
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68 views

Cauchy Problem and stochastic representation for discontinuous initial data

Where can I read more about the Cauchy problem, i.e. solutions to $$ \frac{\partial u}{\partial t}+Lu=0 \text{ and } u(0,x)=f(x)$$ for some elliptic differential operator $L$ where $f$ is not ...
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82 views

$H=W$ for weighted Sobolev spaces

Meyers and Serrin's $H=W$ is well known, but how does it generalize when we add weights? Let's define $H^{m,p}(\mu_0,\dots,\mu_m)$ to be the completion of $C^\infty(\Omega)$ in the norm ...
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57 views

Existence for special Dirichlet problem

I would like to know the following: Let $M$ be a smooth surface with connected boundary. Let $f: M \rightarrow \mathbb{R}^3$ be an embedding such that the boundary $\partial M$ of $M$ is mapped onto ...
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115 views

Strichartz estimates for the wave equation

Strichartz estimates for the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$ can be stated as $$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t ...
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54 views

Isolated singularities of harmonic mappings

Can a homeomorphic harmonic mapping $f=(u,v,w):\Omega\to \Omega'$ have isolated singular points. Here $\Delta f =0$, and singular point is a point with zero Jacobian. This will extend Lewy theorem for ...
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67 views

Strong solution to parabolic equation without differentiability assumption on coefficient?

Consider on $(0,T)\times \Omega$, $\Omega$ a bounded domain $$u_t(t,x) - a(u(t,x))\Delta u(t,x) = f(t,x)$$ $$u|_{\partial\Omega} = 0$$ where $a$ is real-valued and satisfies $C_1 \leq a(r) \leq C_2$ ...
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118 views

Harmonic extension of $L^\infty$ function is in $L^\infty$?

Let $u \in H^{\frac 12}(\Omega)$ with $\int_\Omega u = 0$ and consider the solution $v \in H^1(C)$ where $C=\Omega \times (0,\infty)$ of $$-\Delta v(x,y) = 0$$ $$\partial_\nu v = 0$$ $$v(x,0) = ...
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40 views

Finite elements $W^{1,\infty}$ error estimates

Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'_h - u'\|_\infty$)? Which families of elements can be used for implementing them?
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195 views

A Lie algebra associated with a one dimensional foliation

A non vanishing vector field $X$ on a manifold is called "well behaved" if for every non vanishing smooth function $f$ we have $$C(X)\simeq C(fX)$$ This means that the centralizer Lie algebras ...
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48 views

Density of restrictions of $p$-harmonic functions on a hypersurface

Let $\omega,\Omega\subset\mathbb R^n$, $n\geq2$, be bounded smooth domains so that $\bar\omega\subset\Omega$. Let $1<p<\infty$. Define the boundary space $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...
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138 views

Help in understanding “Local well-posedness for the Maxwell-Schrodinger system”

Is there someone who knows the following paper "Local well-posedness for the Maxwell-Schrodinger system" by M.Nakamura and T.Wada. I'm trying to study it but I've some doubts. In particular I'm not ...
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90 views

Showing existence of positive weak solution of a PDE by CoV

Given the following PDE $$ \begin{cases} -\Delta u+\alpha=u^q &x\in\Omega\\ u=0 &x\in\partial\Omega \end{cases} $$ where $\Omega\subset\mathbb R^3$ is open bounded with smooth boundary, ...
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64 views

Question on the local existence theory for the classical solution for the incompressible fluid dynamics equation

For the incompressible fluid dynamics system \begin{equation} \begin{split} &\nabla\cdot v=0,\\ &\dfrac{\partial v}{\partial t}+v\cdot \nabla v+\nabla p=A(S,D)v, \end{split} \end{equation} ...
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90 views

variation norm of a Fourier transform

Motivated by certain uniform estimate in oscillatory integrals, I am now trying to calculate the Fourier transform of the function ${\large e^{i|t|^{\epsilon}}/t}$ on $\mathbb{R}$, where $\epsilon\in ...
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79 views

A bilinear estimate in Lp space

Let $\varphi(D)$ be a Fourier multiplier with symbol $\varphi(\xi) = \xi/(1+|\xi|^2)$. It's easy to prove that \begin{equation} \|\varphi(D)u^2\|_{H^s(R)}\lesssim \|u\|^2_{H^s(R)} \quad (*) ...
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86 views

Free Endpoint of Minimization Problem

Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > ...
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42 views

Introduction to free boundary problems (that are not Stefan problems)

Could someone recommend some notes/papers that deal with existence/regularity of free boundary problems arising from parabolic equations (excluding Stefan type equations)? I am thinking of eg. ...
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26 views

Is it classical that the solution to an hyperbolic equation equation is Lipschitz -continuous $[0,\infty)\to L^1(\mathbb{R})$?

Recently I am reading a paper "Global solution and smoothing effect for a non-local regularization of a hyperbolic equation" published on J.E.E, 2004. In the proof, the authors write "It is classical ...
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54 views

decomposition of tempered distributions by entire analytic functions

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$ Let $j\in \mathbb N$ ...
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73 views

Measurability of solution of diffusion equation in sub sigma algebra

I want to solve the following problem: Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$. Let $a( x;. )$ and $f(x;.)$ be ...
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118 views

How does perturbation method guarantee its solution for the perturbed pde $\Delta u + \epsilon u^2 =0$

[This may not be a research-level question; if it violates any term of this website, I will delete it right away] My question is quite simple: Suppose we are given a PDE of with a boundary condition ...
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58 views

Bogomol’nyi’s Formula for the Critical Action

I'm studying Aigner's paper 'Existence of the Ginzburg-Landau Vortex Number' (2001) and I have some difficulties to prove the equality (3.1) , which is ...
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146 views

Existence of solution?

I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here. Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf Let ...
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82 views

Uniform bounds for a coupled parabolic system of PDE (linear)

Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$. Consider the system, with $u^\epsilon, v^\epsilon ...
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105 views

A linear operator equation (PDE) with non-monotone term

I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple. There is a linear operator $L:{D}(L) ...
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70 views

presence of turbulent phenomena in systems of linear pde?

Are there linear systems of PDE that are known to have solutions which exhibit turbulence, or can turbulence be firmly classified as a fundamentally non-linear phenomenon, similar to solitons or shock ...
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36 views

Does the following measurable Halmilton-Jacobian equation admit a Lipschitz solution?

I have the following question: Let $F:\Omega\times \mathbb{R}^n\to [0,\infty)$ be a convex Finsler norm, which means that $F(x,\cdot)$ is convex with respect to the second variable. $F(\cdot,v)$ ...
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135 views

Half Laplacian; (definitions of) and regularity

I have a question regarding the half Laplacian $ (-\Delta )^\frac{1}{2}$ on some smooth bounded domain $ \Omega$ in $R^N$. I am attempting to clarify some confusion with the various definitions. ...
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93 views

Laplacian mapping on various function spaces

I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$. If $ 1 <p< ...
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152 views

Stokes operator without dirichlet boundary condition

Let $\Omega$ be a domain, then the following stokes operator is quite well known : $\mathcal{H} \rightarrow \mathcal{V}_{\sigma} $ $f \rightarrow u$ such that $ - \Delta u = f $ where $ ...
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126 views

Compact embedding

Let $\Omega$ be a domain in $\mathbb{R}^d$ (not necessarily bounded, no regularity assumption) and $K \subset \Omega$ a compact. Is it true that the embedding $H^1_0(\Omega) \rightarrow ...
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87 views

$L^p$ norm of solution to porous medium equation decreases in time: how to make formal calculation rigorous?

Let $u \in C^0([0,\infty);L^1(M)) \cap W^{1,1}_{\text{loc}}((0,\infty);L^1(M))$ with $u(t) \in H^1(M)$ for a.e. $t$ be the solution of the porous medium equation $\dot u = \Delta (u^m)$ on a compact ...
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83 views

Comparison principle using truncation for porous medium equation

For a porous medium equation (eg. $u_t - \Delta \Phi(u) = f$), is it possible to obtain a comparison principle for very weak solutions (eg. if $u_0 \geq 0$ and $f \geq 0$ then $u \geq 0$ a.e.) using ...
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172 views

Is this function space a “classical” Sobolev space?

I apologise if this is indeed classical but my functional analysis is quite rusty... My work recently led me to the norm: $(\|u\|_p)^p=\int_D (|u|^p+|\Delta u|^p)d\lambda$ where $D$ is the unit disk ...
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55 views

Well-posedness of a certain linear Cauchy-problem

I am interested in solutions to the linear Cauchy problem $$\Bigl(\frac{\partial^2}{\partial t^2} + a(t, x)\frac{\partial}{\partial t} + \sum_{j=1}^n b_j(t, x) \frac{\partial}{\partial x_j}\Bigr)u(t, ...
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101 views

A question on the maximum principle for Schrodinger operators

Let $(M^n,g)$ be a closed Riemannian manifold, and $L=-\Delta+V$ be a Schrodinger operator, $V\in C^{\infty}(M)$. In answers to the two questions (First eigenvalue of Schrödinger operator is simple 1 ...
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139 views

Estimate the smallest eigenvalue of a Schrodinger operator

There are several results on the estimate of the number of negative eigenvalues of a Schrodinger operator, see a recent paper of Grigor'yan-Nadirashvili-Sire and references therein. I wonder how to ...
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93 views

On the proof of a $W^{2,p}$ estimate - regularity on eliptic PDE

I see this proof on http://www.math.uiowa.edu/~lwang/cccalderon.pdf and I couldn't understand what he did. If $||f||_{L^p(B_{4})} = \delta$ is small and the measure $|\{ x \in B_1; ...
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86 views

Limit Toward Discontinuous Point of Dirichlet Boundary Value

The question arises from a paper on Schwarz's domain decomposition method (click here). We consider a bounded domain in $\mathbb{R}^2$ and a curve splits it into two, see the figure below. Now we ...
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90 views

Lyapunov Schmidt; basic example

I am attempting to understand the Lyapunov-Schmidt method with a simple example but I am running into trouble. Here is the example I am considering. Suppose $ v>0$ satisfies $ -\Delta v - v=0 $ ...
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116 views

Showing a normal-derivative operator is a (sort of) contraction (related to Crandall-Liggett and PDEs)

Denote by $\mathbb{E}(g)$ the solution of the PDE $$\Delta v(x,y) = 0 \quad\text{in $\Omega$}$$ $$v(x,0) = g(x) \quad\text{on $\partial\Omega$}.$$ Let $X=L^1(\partial\Omega)$. Let $h(t)$ be a ...
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39 views

Does there exist a base $\{e_j\}_{j\geq 1}$ of $H(\Omega)$ such that $\{e_j\}_{j\geq 1}$ is linearly independent in $L^2(\omega)^d$?

Does there exist a base $\{e_j\}_{j\geq 1}$ of $H(\Omega)$ such that $\{e_j\}_{j\geq 1}$ is linearly independent in $L^2(\omega)^d$? Where $\omega\subset\subset \Omega$ with $\Omega$ is a $C^2$ ...
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111 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 ...
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96 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 ...