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

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24 views

convection/transport with different velocities

What is the prototypical model for convective transport of a quantity whose constituents move with constant but varying velocities? In order to illustrate what a mean: Suppose that a large number of ...
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72 views

Probability that d-Brownian Motion ,d>3, avoids a set A

In other words, the probability that Brownian motion stays within $A^{c}$. So far I found that it is 1, for random cylinders and thorns (http://www.math.upenn.edu/~pemantle/papers/burdzy.pdf). What ...
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34 views

$L^\infty$ estimate for a fourth order (hyperbolic) equation

Consider the following fourth order equation $$u_{tt}+u_t= d\Delta u-\Delta^2u+f,$$ with Dirichlet or Navier boundary conditions, that is on $\partial\Omega$, we assume that ...
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34 views

reading request on linear elliptic systems of pdes, strong solutions

Does anyone have some references where I could find results on strong solutions to linear elliptic systems of pdes ? Regards
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124 views

Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$

We have a bounded domain $\Omega$ of $\mathbb{R}^n$. Let $$u \in L^2((0,T);H^1(\Omega)) \cap H^1((0,T);H^{-1}(\Omega))\cap L^\infty((0,T);L^\infty(\Omega)).$$ I want to show for $r \geq 2$ that ...
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55 views

Compatibility of initial and boundary conditions

Suppose we consider the heat equation $$\partial_t u = \Delta u, x \in \text{int}D^2, t > 0$$ where $D^2$ is the closed unit disc in $\mathbb{R}^2$, subject to Neumann type boundary conditions ...
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85 views

elliptic regularity when right hand side in weak $L^p$

I am interested in the following question (whose answer i assume is well known) but just not by me. Suppose $u,f$ are smooth functions defined on $B_1$ and $ \Delta u = f$ in $B_1$ with $u=0$ on $ ...
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126 views

$b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{-1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$

This question stems from the proof of Theorem A.1 on page 425 of this paper. Let $Q=(0,T)\times \Omega$. Suppose $b_n \rightharpoonup b$ in $L^q(Q)$ for any $q < \infty$ and $b_n \to b$ in ...
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70 views

Estimate for an integral of a function of the solution to a PDE

Let $\Omega \in \mathbb{R}^3$ be a bounded smooth domain. Assume that smooth functions $\sigma_1,\sigma_2$ satisfy $\sigma_1-\sigma_2 \in C_0^\infty(\Omega)$ and $\lambda\leq \sigma_1, \sigma_2 \leq ...
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96 views

How to choose appropriate norms in the analysis of PDEs?

In many papers on the analysis of nonlinear PDEs, the authors could always choose appropriate norms to do their estimates. Sometimes these norms look very odd, which I don't know how these authors ...
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51 views

References for LWP of a NLS Equation

I am studying the LWP of $$i \partial_t \psi + \Delta \psi = \left| \psi \right|^{p-1} \psi + \frac{1}{\left| x \right|^2} \psi$$ in $\mathbb{R}^{1+2}$ given appropriate Cauchy data. It will probably ...
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53 views

About approximate eigenvalue

I am in trouble when read the book "D.Henry, Geometric Theory of Semiliner Parabolic Equations". The question is relate to Page 104,proof Lemma 5.1.4. Suppose $X$ is a real Banach Space, $M$ is a ...
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24 views

Why Does a quadratic phase term in BNLS causes collapse?

I've heard a couple of times that in the Biharmonic Nonlinear Schrodinger Equation, $i\psi_z + \Delta ^2 \psi + |\psi | ^{2\sigma } \psi =0 $, $\psi (x, 0) = \psi _0 (x) \in H^1( \mathbb{R} ^d ) $ ...
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37 views

Consequence of elliptic estimates up to the boundary

Consider the half space $\Omega=\{x=(x_1,...,x_N)\in\mathbb{R}^N:x_N>0\}$. Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ a positive bounded solution of $$ \begin{eqnarray*} \Delta u+f(x_N,u)=0, ...
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77 views

Commuting Derivative and Convolution type integral

Suppose $\Gamma$ is a smooth curve, $f$ and its derivative belong to some $L^p(\mathbb{C})$(i.e $f\in W^{1,p}$) and kernel $K(|z-y|)\in\mathbb{C}\times\mathbb{C}$ has only singularity on ...
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34 views

How can one use stability analaysis of finite differences methods in linear Schrodinger to the NLS?

Specifically, I've seen a lot of analysis of grid stability for solving Linear Schrodinger with Forward Euler, Backward Euler and Crank-Nicolson. However, most of the usages I've seen for the same ...
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53 views

Time decay for Hartree equation with Coulomb potential

Are there any time-decay results for the solution of the Hartree equation \begin{equation}\frac{1}{i}\partial_t\phi-\Delta\phi=-(|x|^{-1}\ast|\phi|^2)\phi,\quad x\in\mathbb{R}^3\end{equation} in ...
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32 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 ...
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47 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 ...
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55 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 ...
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51 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 ...
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53 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 ...
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48 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 ...
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61 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 $L^2(\Omega,\mathbb{C})$ $$ H=\sum_{|\alpha|\le 2}C_\alpha\partial^\alpha $$ ...
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59 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 $$ ...
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95 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 $$\Delta^{-1}f=f*\frac{1}{|x|}.$$ ...
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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 ...
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67 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 ...
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104 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 ...
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103 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 ...
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38 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) - ...
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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 ...
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66 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 ...
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65 views

$|\nabla f|^2 \in W^{1,2}(M)$ for harmonic function f on manifolds with two nonparabolic ends?

We know that if a complete noncompact manifold M has two nonparabolic ends, then we can construct a nonconstant bounded harmonic function with finite Dirichlet integral defined on the whole M. ...
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222 views

Analytic solution of Poisson's equation

Consider Poisson's equation $\nabla^2 u = 1$ on a square of side-length 1 centered at the origin. Cut out a circle of radius 1/3 at the center of this square. Impose a von Neumann boundary condition ...
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137 views

Cheeger-Gromov-Taylor theory on manifolds with boundary

I was reading the paper "Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds" by Cheeger, Gromov and Taylor and I am ...
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150 views

Can a function be constructed from the direction of its gradient?

Let $\Omega$ be a bounded region in $R^n$ and $J\in (L^2(\Omega))^n$ with $|J| \leq 1$ a.e. in $\Omega$. Under what conditions the equation $Du=J|Du|$, $u|_{\partial \Omega}=f$ has a solution in a ...
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137 views

If $\phi_n$ is a sequence of mollifier converging to the identity, does $\inf f\ast \phi_n \to \inf f$?

Let $\phi_n$ be a sequence of mollifier converging to the identity $$ \phi_n(x) \to \delta_{0}(x), \text{pointwise}, $$ with $\delta_{0}(\cdot)$ the delta function at zero, and $\phi_n \in ...
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69 views

PDE with periodic solution with Galerkin method

Can anyone give me a reference to where I cacn find existence of solutions to parabolic PDE (eg. a standard linear PDE) with initial and end conditions $u(0) = u(T) = u_0$ so that periodic solutions ...
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53 views

partial maximum principle for elliptic differential operators

Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $P$ be a self-adjoint, elliptic differential operator defined on $C^\infty(M)$ with smooth coefficients. Suppose as well that the lowest ...
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72 views

Parabolic PDE; uniform bound on approximations $u'_n$ in $L^2(0,T;V^*)$ without using orthogonal basis?

Let $V \subset H \subset V^*$ be a Gelfand triple, all Hilbert and separable spaces. I consider the PDE with weak form: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that $$\langle u'(t), ...
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88 views

Reference Search for a Functional Minimization Problem

Let $u(x) \ge 0$ be a non-negative, piecewise-differentiable function on the real line. Moreover, let $u(x)$ be integrable with fixed positive mass, that is $$M \equiv\int_{x=-\infty}^\infty u(x) ~ ...
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131 views

Gradient estimates for subsolutions of elliptic equations

Let $M$ be a Riemannian manifold. Assume $u \in C^\infty(M)$ such that $u>0$ and $\Delta u + \lambda u = 0,$ where $\lambda \geq 0$. There is a poinwise estimate for $|\nabla u|$ in Peter Li's ...
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74 views

$\mathcal{D}(0,T;V)$ is dense in $W(0,T)$

Where can I find a proof that $\mathcal{D}(0,T;V)$ (the space of $V$-valued compactly supported functions on $[0,T]$) is dense in the space $W(0,T)$, where $$W(0,T) := \{ u \in L^2(0,T;V) : u' \in ...
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116 views

Harnack's Inequality and (hypo)elliptic PDE

Background: I am aware of the Harnack's Inequality for linear elliptic equations. My questions are: (a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...
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68 views

Cauchy Data Problems.

Let $\ y \dfrac {\partial u}{\partial x} - \ x\dfrac {\partial u}{\partial y}=0$. The characteristic curve obtains the function $y^2+x^2=\eta^2 $, ie a circle of radius $\eta$ and centre $(0,0)$. I ...
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163 views

A question about definition of weak derivative / integration by parts formula

We say $u \in L^2(0,T;V)$ has weak derivative $u' \in L^2(0,T;V')$ if $$\int_0^T \langle u',v\rangle_{V',V} = -\int_0^T (u,v')_{H}$$ holds for all $v \in C_0^\infty(0,T; V)$. This relation can be ...
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132 views

Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
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215 views

Eigenfunctions of the laplace on the 2-sphere with conformal metric induced by schwarzschild

Hi, it's well known that the coordinates $x_1$, $x_2$, $x_3$ are the first three eigenfunctions with positiv eigenvalues ($=2/r^2$) of the negative laplace-beltrami operator ${-}\triangle$ on the ...
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199 views

“Integration by parts” formula for functionals

We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;V)$ with $u',v' \in L^2(0,T;V')$ then $$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$ where the ...