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

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4
votes
1answer
215 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
122 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
98 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
65 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 ...
0
votes
0answers
68 views

Invertibility of Paneitz operator on a compact manifold without boundary

Let $(M,g)$ be a Riemannian manifold of any even dimension that we assume compact and without boundary and $P_g$ the Paneitz operator in the metric $g$. The question is the following: how to define a ...
4
votes
1answer
204 views

Bounding $\lVert{u}\rVert_{C^0([0,T];V)} \leq C\left(\lVert{u}\rVert_{L^2(0,T;V)} + \lVert{u'}\rVert_{L^2(0,T;H)}\right)$?

Define $W(X,Y) = \{ u \in L^2(0,T;X) \mid u' \in L^2(0,T;Y)\}$ where $u'$ is the usual weak derivative. Let $V \subset H \subset V^*$ be a Hilbert triple. If $u \in W(V,V)$ (so $$u \in L^2(0,T;V) ...
3
votes
1answer
158 views

parabolic PDE with almost-monotone elliptic operator, existence results?

Are there any existence results for parabolic PDE of the type $$u_t - Au = f$$ in some Gelfand triple setting ($V \subset H \subset V^*$) with $A$ an operator that it is not quite monotone but close: ...
-3
votes
1answer
159 views

$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?

Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at ...
0
votes
1answer
75 views

harmonic maps from cone to $S^2$ locally lipschitz?

Are the harmonic maps from a 2-dimensional cone to $S^2$ locally lipschitz or Holder continuous?
0
votes
0answers
105 views

Solvable PDEs and their Green's functions

I have a class of PDEs of the form $$ -\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0 $$ with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and ...
2
votes
0answers
90 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 ...
0
votes
0answers
108 views

Does the Laplace-Beltrami/surface gradient commute with orthogonal projection? (related to Galerkin method)

Let $\Gamma$ be a $C^k$ $(n-1)$-dimensional hypersurface embedded in $\mathbb{R}^n$. Let $H=L^2(\Gamma)$ and $V=H^1(\Gamma)$. Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily ...
2
votes
1answer
277 views

How fast does the Heat equation with boundary condition $\frac{\partial u}{\partial \vec{n}}=u^2$ decay?

Consider the heat equation $\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u$ in a bounded domain (say the interval [0,$\pi$]) with boundary condition $$\frac{\partial u}{\partial \vec{n}}=u^2$$ with ...
1
vote
1answer
82 views

Equivalence of distributional and viscosity solution in parabolic case?

In the answer to http://math.stackexchange.com/questions/166286/viscosity-solution-vs-weak-solution H. Ishii, "On the equivalence of two notions of weak solutions, viscosity solutions and ...
2
votes
0answers
115 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) ...
0
votes
0answers
39 views

Well-posedness of a Stefan problem with Faedo-Galerkin approach

Given a domain $\Omega$ which is divided by $\Omega_1(t)$ and $\Omega_2(t)$ and the interface $\Gamma(t)$, does anyone have a reference to where a Stefan problem of the type $$\frac{d}{dt}H(u) - ...
2
votes
0answers
107 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
votes
0answers
132 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 ...
0
votes
1answer
109 views

semiclassical principal symbol

What is the semiclassical principal symbol $\sigma_h$ of the operator $h^2\Delta-1$ (here $\Delta=-\sum_j\partial^{2}_{x_j}$)? $h^2\Delta-1$ is a second order semiclassical partial differential ...
2
votes
0answers
94 views

Almost a Green formula

Let $\Omega$ be the half-space $\mathbb{R}^{n-1}\times \{ x_n>0 \}$, let $v \in L^2(\Omega)$ and $\phi\in \mathcal{C}^{\infty}(\overline{\Omega})$ with compact support in $\overline{\Omega}$. What ...
1
vote
1answer
162 views

Finite propagation speed of second order operators

I was reading about the finite propagation speed of the wave equation on a Riemannian manifold. I was wondering if instead of the Laplace-Beltrami operator $\Delta$ we consider the equation ...
0
votes
2answers
135 views

A basic question about JL Lions' transformation of a Stefan problem

In J.L Lions' book "Quelques méthodes de résolution des problèmes aux limites non linéaires" (page 196), the author considers a two-phase problem with moving boundary separating the interface. The ...
1
vote
1answer
202 views

Reference request: Boundary behavior and quantitative lower bound for the principal eigenfunction of an elliptic PDE in a ball $B(r)$

Consider the elliptic eigenvalue problem $$ \begin{cases} \int_{B(r)} A(x) \nabla u \cdot \nabla \phi \, dx &= \ \ \frac{\lambda_1}{r^2}\int_{B(r)} u \phi \, dx \\ \qquad \qquad \qquad \quad ...
1
vote
1answer
87 views

well-posedness of heat equation with Neumann BC and periodic data

On a domain $\Omega$ with $f \in L^2(0,T;H^{-1})$ such that $f(0) = f(T)$, consider $$u_t - \Delta u = f\quad\text{on $\Omega$}$$ $$\frac{\partial u}{\partial \nu} = 0\quad\text{on $\partial\Omega$}$$ ...
1
vote
0answers
101 views

Weak periodic solution of parabolic PDE

Take $$ u_t(t) + A(t)u(t) = f(t), $$ $$ u(0) = u(T), $$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
4
votes
0answers
234 views

Linearizing and solving a nonlinear PDE numerically

Im trying to solve the following (transport & diffusion) nonlinear PDE numerically (via finite volume on a cuboid region. Some Material gets cooled down, s.t. in some areas the material becomes ...
-1
votes
1answer
177 views

Why can't I get global existence to linear PDE in this way? [closed]

For any $n > 0$, standard theory implies there is a unique $u_n \in L^2(0,n;V)$ with $u_n' \in L^2(0,n;V^*)$ such that $$u_n' + Au_n = f\quad\text{as an equality in $L^2(0,n;V^*)$}$$ $$u_n(0) = ...
2
votes
1answer
182 views

Inequality in the Sobolev space $H^1$

I've found the following inequality $$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ...
1
vote
0answers
118 views

Half-wave group $e^{it\sqrt{-\Delta_g}}$ for large $t$

Consider the Laplace-Beltrami operator $\Delta_g$ on compact Riemannian manifold $(M,g)$, then $e^{it\sqrt{-\Delta_g}}f$ is the solution of the following Cauchy problem. $$ ...
2
votes
2answers
157 views

Perturbations of positive-definite self-adjoint operators

I was reading Kato's book on Perturbations of Linear Operators and have the following questions: If we have a self-adjoint operator, what kinds of perturbations (other than relatively bounded ones) ...
3
votes
1answer
162 views

(Ref req) Schrödinger heat kernel is a weak solution of parabolic Schrödinger equation

If we have nonnegative $V \in L^1_{\textrm{loc}}(\mathbb{R}^{n})$, then the operator $H = -\Delta + V$ can be defined on $L^{2}(\mathbb{R}^{n})$ via quadratic form methods. This is done by, for ...
0
votes
0answers
74 views

Dirichlet integral of harmonic functions on manifold controlled by radius?

M is an n($\ge2$)-dim Riemannian manifold with Ricci curvature bounded below. $\Omega$ is a domain in M. $$\Delta u=0, x\in \Omega;u=f ,x\in \partial \Omega$$ f is Lipschitz. Is there a constant c ...
0
votes
0answers
68 views

Harmonic/functional analysis question: Uniform bounds for $(1 - \varepsilon\Delta)^{-1}$ as $\varepsilon\to 0$?

Is there a space $X$, compactly embedded in $H^{-1}(R^2)$, such that the operators $(1 - \varepsilon \Delta)^{-1}$ are bounded from $L^1(R^2)$ into $X$, with operator norms that are in turn bounded by ...
5
votes
1answer
397 views

Do exist infinitely differentiable, compactly supported non zero solutions of the free Schrodinger equation?

I would like to get an answer for the following problem (and possibly be pointed to the relevant literature): given the one dimensional free Schrodinger equation $ i \, f_t + f_{xx}/2 = 0$ for the ...
0
votes
1answer
138 views

A Poincaré-type inequality with logarithmic function

For any function $f(x)$ we denote $\bar{f}:=\frac{1}{\Omega}\int_\Omega f(x)\,dx$. Let $\Omega\subset \mathbb{R}^n$ be a bounded smooth domain and $u(x)> 0$ be a smooth function defined on ...
1
vote
1answer
201 views

Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?

Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous? I define $H^{\frac 1 ...
1
vote
2answers
84 views

References for well-posedness of weak solutions to Stefan problem

Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)? ...
1
vote
1answer
67 views

Does replacing 'local' with 'global' in the definition of viscosity sub(super)solution change the definition?

Let $F(x,u,Du,D^2u)=0$ for $x\in\Omega$ be our equation of interest. One way of defining a viscosity subsolution is this: We say $u$ is a viscosity subsolution iff for any test function $\phi\in ...
1
vote
2answers
146 views

How to show this integral on boundary of Lipschitz domain is finite?

Sorry for asking a basic question but this did not get answered on M.SE. Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that $$\int_{\partial\Omega} ...
12
votes
0answers
6k views

Otelbayev's approach to Navier-Stokes [closed]

Recent news post that Mukhtarbai Otelbayev from Eurasian National University has shown existence of strong solutions of the Navier-Stokes equation in the article "Existence of a strong solution of ...
1
vote
1answer
110 views

What fails when we try to extend existence and unique for parabolic PDEs for 'PDEs which are 'parabolic in two components''?

What I have in mind is that on $(0,T)\times\Omega$, we have a parabolic pde operator $L$, we have unique solution to $Lu = f$ when f is Hoelder for some coefficient strictly between 0 and 1. This ...
1
vote
1answer
204 views

Harmonic function defined on a cone

It's well known that: Given a continuous function defined on the boundary of the disk, then there exists a unique harmonic function in the interior of the disk. What if we replace the disk by a cone? ...
0
votes
2answers
142 views

Defining surface integral on boundary of $C^1$-domain

Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to ...
3
votes
2answers
269 views

Dirichlet energy of a harmonic function bounded above by the energy of the boundary function?

$\Omega$ is a domain in $R^2$ with sufficient smooth boundary. Given an absolutely continuous function f difined on $S^1$($[0,2\pi]$). Then there exists an unique harmonic function u defined on ...
1
vote
1answer
126 views

Pohozaev result for equations with weights

I am interested in nonnegative solutions of $-div( e^{-\gamma(x)} \nabla u(x)) = e^{-\gamma(x)} u(x)^p$ in $\Omega$ with $ u=0$ on $ \partial \Omega$. Or instead the equation $ -\Delta u + ...
-1
votes
1answer
79 views

Question regarding to the basis of L^p space via compact self adjoint operators. ( eg: inverse of -laplacian )

Do eigenfunctions of inverse of elliptic operator (eg: Laplacian) form basis of $L^P(\Omega)$ ? For p=2 we know the answer is yes, I am looking for p>2. More generally, is it true that eigenfunctions ...
2
votes
1answer
85 views

Conditions for existence of $m$-th differentiable root of a non-negative definite matrix

In M.I. Friedlin's famous paper "On the Factorization of Non-Negative Definite Matrices", he shows that if a non-negative definite symmetric matrix $a(x)=\{a^{ij}(x)\}_{i,j=1}^n$ is in ...
4
votes
1answer
153 views

K-Theory of Algebra of Zeroth Order Pseudo differential operators

Any one knows a reference for computing K_0 of Algebra of zeroth order Pseudo's on a closed manifold in terms of explicit generators? Thanx!
1
vote
0answers
160 views

Reverse Holder Inequality and the higher integrability of the gradient of a solution to Euler's equation for a certain functional

In Giaquinta-Giusti's (1978) paper "Nonlinear Elliptic Systems with Elliptic Growth" (thm 1.1) they consider the following system: \begin{equation} \sum_{i, j=1}^{n}\sum_{\alpha, ...
1
vote
2answers
163 views

Interior Schauder estimates with weights

Suppose we have $u(x)\in H_2^{loc}(\Omega_{\rho})$, where $\Omega_{\rho}=\{x\in \mathbb{R}^n, |x|>\rho\}$, and in $\Omega_{\rho}$, $u$ satisfies the equation $$ \Delta u-V(x)u=0, $$ where $V$ is a ...