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
2answers
186 views

Reconstructing density from integrals along specific manifolds

Let $\Phi_t : \mathbb R^n \to \mathbb R^n$ be the time-$t$-map associated to an ODE $\dot{x}=F(x)$ and let $H: \mathbb R^n \to \mathbb R$. Let $F$ and $H$ be sufficiently smooth (e.g. $C^k$ or ...
0
votes
0answers
36 views

How to use persistence of regularity to obtain global well posedness result in $H^{s}(\mathbb R), (0<s<1)$?

[This question occurs to by the answer given by W. W ;and may be trivial for MO; But I am keen to understand the process:"local existence and persistence of regularity implies global well posedness" ] ...
3
votes
1answer
86 views

Weak convergence in $W^{1,p}_0$

Note from the answerer : this question stems from this article. I ask this question in http://math.stackexchange.com/questions/1206617 I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so ...
0
votes
0answers
29 views

regularity of non-local linear elliptic equation

$\alpha\in (0,1)$, $u$ satisfies: \begin{equation*} b\cdot \nabla u(x)+\sum_{i=1}^d \int_{R} \left[u(x+se_i)-u(x)-s\mathbb{I}_{\{|s|\leq ...
1
vote
0answers
51 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 $f$ ...
-4
votes
0answers
52 views

Differentiability of a function [closed]

Can someone help me out to give a bound of the first derivative w.r.t. $x$ of the function $\min_{x,y\in\Omega}\{1,\frac{d(x,\partial\Omega)d(y,\partial\Omega)}{|x-y|^2}\}$?. ...
5
votes
2answers
142 views

Well-posedness of Fokker-Planck equation

Consider the following equation on $[0,T]\times\mathbb{R}^n$ \begin{eqnarray} &\partial_t\rho=\mathrm{div}(\rho\nabla V)+\Delta\rho\\ &\rho|_{t=0}=\rho^0, \end{eqnarray} where $V\in ...
-1
votes
0answers
30 views

Global existence of power-type nonlinear Schrodinger equations on compact manifolds [closed]

Consider the nonlinear Schrodinger equation $$i\partial_t u + \Delta u = K|u|^ru$$ on a compact manifold, may be with boundary (with Dirichlet boundary conditions). It is known that on $\mathbb{R}^n$, ...
-4
votes
0answers
35 views

Solving a tough a PDE shifting data [closed]

How would I solve this one: $u_t-\nabla^2u = f(r,\theta, t) \quad r<a, t>0$ $u(r,\theta, 0)=\phi(r,\theta) \quad r<a$ $u=h(\theta) \quad r=a$ So I guess I need to make the BC's ...
3
votes
3answers
165 views

Are linearizations of involutive PDEs locally solvable?

A possibly soft question for you guys and gals. Say a system of analytic PDEs has been completed to involution (in the sense that it's geometric symbol has a Pommaret basis, or has vanishing ...
0
votes
0answers
38 views

inflow/outflow Boundary Conditions for flow in pipe

I have a question about boundary condition of solving Navier-Stokes equation through pipe. When I simulate the flow in pipe using periodic boundary condition, it works good. But when I tried to change ...
-1
votes
1answer
113 views

Analysis of Sobolev spaces [closed]

I just wanted to know wthether the following is OK or not. Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...
2
votes
1answer
69 views

distance formula of warped products

Given a warped product, I want to compute the ditance of any two points. First get the equation for the geodesic, then compute the length of the geodesic. Consider the two-dimensional surface $$ ...
2
votes
0answers
91 views

Sobolev space for manifold with boundary

For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...
0
votes
0answers
24 views

On a remark regarding to initial-boundary elliptic estimate

In the paper "On the motion of the free surface of a liquid" see [CL], the authors proved the initial boundary elliptic estimate (proposition 5.28) $$ ||\nabla^r q||_{L^{2}(\Omega)}+||\nabla^{r-1} ...
3
votes
1answer
69 views

Blow-Up for Semi-Linear Wave Equations

I am reading C. D. Sogge's book "Lectures on Non-Linear Wave Equations". As an exercise, I attempted to fill out the details of the proof of Theorem 5.1 (Local Existence of Solutions for Semilinear ...
0
votes
0answers
74 views

if $u_\epsilon \rightarrow u$ weakly in $L^2$ then also $\partial_t u_\epsilon \rightarrow \partial_t u $ weakly in $L^2$

I am looking for a citation of the above claim, I am not sure under what conditions on the sequence $u_\epsilon$ exactly this applies. $u_\epsilon$ is a sequence of functions that depend on $x,t$ ...
5
votes
1answer
1k views

Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M

Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, ...
0
votes
0answers
80 views

Schauder estimate on a bounded domain

We know if $u:\mathbb{R}^{n}\to \mathbb{R}$ is a smooth function with compact support, then we can show, via integration by part, that $$ ||\Delta u||=||{\nabla}^2u|| $$ where $||\cdot||$ is the ...
1
vote
1answer
100 views

global well posedness of cubic NLS in for initial data in $H^{s}(\mathbb R), 0<s<1$

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = ...
4
votes
1answer
534 views

Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello! Let's say I have $(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$ with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$). Now separate the Amplitude and Phase of the solution: ...
0
votes
0answers
36 views

explicit solution to linear PDE — boundary value problem [migrated]

I am looking for an explicit, closed form expression for the solution to a boundary value problem given by the linear PDE: $$ u_{xx}+au_y-bu+c=0 $$ with the BCs $ u_x(-g,y)=ku $, $u_x(g,y)=-ku$, $ ...
3
votes
1answer
302 views

First integrals of a 3D incompressible flow

Let $\Omega$ be an unbounded periodic smooth domain of $\mathbb{R}^3$. We are Given an incompressible vector field $q:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ (i.e. $\nabla\cdot q\equiv 0$ ...
1
vote
0answers
58 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 ...
5
votes
3answers
325 views

Structure of sign changes under the heat flow

Let $f$ be a smooth function on $R^2$, and define $N_f$ to be the set of points $p$ such that the nodal set of $f$ ($\{x\in R^2: f(x)=0\}$) divided every neighborhood of $p$ into four regions. Indeed, ...
7
votes
1answer
448 views

Is there a generalization of Sobolev spaces for certain locally compact groups?

I'm interested in knowing how far and how general the theory of Sobolev spaces has been developed. Classically, $H^k(U)$ for $U$ a subset of $R^n$ is given by derivatives up to order $k$ being square ...
3
votes
1answer
193 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 ...
1
vote
0answers
66 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 ...
5
votes
1answer
660 views

A question on the proof of the Serrin condition for the regularity of Navier-Stokes equations

Edit: This question has been substantially modified on January 12th, 2015. I have been studying Michael Struwe's paper "On Partial Results for the Navier-Stokes Equations", Comm. Pure Appl. Math 41 ...
0
votes
1answer
92 views

Which is the smallest space $X\subset L^{2}$ where the conservation law holds in the norm of $X$?

We formally write the solution of nonlinear Schrödinger equation (NLS) as follows: $$u(t)= U(t-t_{0}) u_{0}- i \int_{t_{0}}^{t} U(t-\tau) (|u|^{2}u(\tau)) d\tau;$$ where $U(t)= e^{it\Delta} $(free ...
2
votes
0answers
64 views

How analyze the following fully nonlinear equation

Now I want to consider the following pde $u_t(x,t)=\sigma(x,t)(1+|D_xu(x)|^2)^{1/2}$, with initial condition $u(x,0)=g(x)$ which is analytic, and on domain $D\times \mathbf{R}^{+}$, $D\subset ...
3
votes
1answer
534 views

Short time existence on Hyperbolic Ricci flow in non-compact case

We know Laplace equation (elliptic equations) $ Δ u = 0$ Heat equation (parabolic equations) $u_t − Δu = 0$ Wave equation (hyperbolic equations) $u_{tt} − Δu = 0$ we have - Hyperbolic geometric ...
1
vote
0answers
43 views

Harmonic functions subject to orthogonality condition

Suppose $(\bar{\Omega},g)$ is a Riemannian manifold with the metric being smooth where $\bar\Omega \subset \mathbb{R^3}$ is a compact domain with smooth boundary. Let $\Omega \subset \bar\Omega$. ...
4
votes
0answers
49 views

How do solutions of a PDE depend on parameters?

Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $\sigma_1,\sigma_2:\Omega\to(c^{-1},c)$ measurable (for some constant $1<c<\infty$). Let $f\in ...
0
votes
0answers
50 views

global bi harmonic functions in a riemannian manifold

Any help will be appreciated thanks! Consider $(\mathbb{R^n},g)$ to be a Riemannian manifold. For simplicity we can assume the manifold to be asymptotically euclidean outside a compact domain ...
0
votes
0answers
51 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 ...
3
votes
1answer
125 views

Gross's log Sobolev inequality proof with variational calculus?

For $f\in C^{1}(\mathbb{R}^{n})$, Gross's logarithmic Sobolev inequality says that $$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\int f^{2}\,d\mu\right)\leq \frac{2}{c}\int |\nabla ...
0
votes
1answer
150 views

Sobolev's lemma on manifolds

Let $M$ be a n-dimensional closed submanifold in $\mathbb{R}^m.$ I was looking for a version of Sobolev's lemma saying that for $f \in {W}^{k,2}$ we find a representative of $f \in C^{r}$ satisfying ...
4
votes
1answer
268 views

Schrodinger equation with magnetic vector potential

In many papers dealing with the Schrodinger equation with magnetic potential $$u_t=i(\nabla+iA(t,x))^2u$$ the authors say that this equation can be studied with Kato's methods for abstract evolution ...
2
votes
3answers
459 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 ...
3
votes
1answer
95 views

Extension of Sobolev Functions

Let $\,D\subseteq\mathbb{R}^{n-1}$ be a convex bounded domain. Let$A:D\to(0,\infty)$ be a Lipschitz continuous function. Let $\,\Omega\,$ be bounded domain in $\,\mathbb{R}^{n}\,$ of the form ...
2
votes
1answer
168 views

Exact reference for Liouville theorem

It seems hard for me to find that the solution of the following equation $$ \Delta u+e^u=0 $$ defined on a simply-connected domain $D\subset R^2$ must be of form $$ ...
0
votes
1answer
82 views

Nodal sets under the heat flow

Let $u(t,X)$ be a smooth solution of the heat equation on $R^2$ $u_t=\Delta u,$ where $(t,X)\in R \times R^2$. Suppose $\lim_{t \rightarrow 0} u(t,x,y)=x^2-y^2$. Can we prove that the nodal set of ...
1
vote
0answers
33 views

$H^1$ convergence of eigenfunctions of Schrödinger operators [migrated]

Consider the Schrödinger-Operator with Potential $V\in L^\infty(\Omega)$ with Dirichlet boundary conditions $$ H^D=-\Delta + V $$ and let $u_{i,n}\in H_0^1(\Omega)$ be the first, nonnegative ...
4
votes
1answer
124 views

Function and its Gradient with Prescribed Norms

I'm not sure if the following question is too elementary for Mathoverflow. I'm sorry if it is the case. Question: Let $n\in\mathbb{N}$ and let $1\leqslant p<\infty$. Let $\alpha,\beta>0$. ...
1
vote
0answers
45 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 ...
0
votes
1answer
130 views

$u_n$ bounded in $L^\infty(0,T;H) \cap L^2(0,T;V)$ implies $u_n \to u$ strongly in $L^2(0,T;H)$?

Let $V \subset H$ be a dense and compact embedding. Let $$\lVert u_n\rVert_{L^\infty(0,T;H)} + \lVert u_n \rVert_{L^2(0,T;V)} < C$$ where $C$ is independent of $n$. It follows that eg. $u_n ...
2
votes
0answers
80 views

Uniqueness of solution of elliptic equation with exponential nonlinearity

Consider the following equation $$\Delta v + p(r)e^v = 0$$ on $\mathbb{R}^n$ where $p(r)$ is a polynomial in $r = |(x_1,..., x_n)|$. I want to understand when equations like these have unique ...
19
votes
4answers
2k views

Classification of PDE

Recently I have been attending a course on PDE's. I was totally ignorant of the subject and wasn't that motivated to be honest. But I was intrigued and felt I had to take the course seriously both for ...
1
vote
0answers
100 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 ...