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

learn more… | top users | synonyms

4
votes
1answer
125 views

Minimal surfaces + Semi-Geodesic Coordinates

Let $(M,g)$ be a three dimensional smooth Riemannian manifold and suppose that $\Gamma$ is an embedded minimal surface in $M$. Define the Fermi or semigeodesic coordinates around this surface through ...
3
votes
2answers
203 views

Isothermal-related functions in higher dimensions

I am interested in getting some geometrical or analytical perspective in studing the following complex pde. I would appreciate any help. Consider $ (M,g)$ to be a 3 dimensional Riemannian manifold ...
0
votes
2answers
177 views

Frobenius Condition for a specific first order pde

I would appreciate it if Someone would be kind enough to share some insights about the following question: Suppose $(M,g)$ is a 3 dimensional Riemannian manifold. Consider the following system of ...
0
votes
0answers
120 views

Notion of solution of pde

Let's consider the following Schrodinger equation $$iu_t+\Delta u+F(u)=0$$ in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in ...
3
votes
0answers
136 views

Strong solution to $u_t - \Delta_p u = f$

For $p > 1$, consider the equation $$\langle u_t, v \rangle + \int_\Omega |\nabla u|^{p-2}\nabla u \nabla v = \langle f, v \rangle$$ $$u(0) = u_0$$ $$u|_{\partial\Omega} =0$$ for all $v \in ...
1
vote
0answers
48 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 $j\in \mathbb N$ ...
0
votes
0answers
75 views

Matrix equation

Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that ...
1
vote
0answers
69 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 ...
4
votes
1answer
104 views

$L^p$ stability of the Beltrami equation

Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} ...
2
votes
0answers
118 views

What's the idea behind various equivalent norms on Besov spaces $B^{s}_{p,q}$?

I am trying to understand Besov spaces; and I am eager to see why the various norms are equivalent on it. Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $ \operatorname{supp} \phi ...
2
votes
1answer
271 views

Sobolev Space, “characteristic function” for the weak derivative

Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$, working in the space $H_0^1(\Omega)$ with the inner product $$(u,v)_{H_0^1} = \int_\Omega \nabla u \cdot \nabla v$$ for $u\in H_0^1$ and ...
1
vote
1answer
52 views

Condition Number and CFL Condition in Finite difference Methods

when applying a Finite Difference scheme for an IVP, two factors come to mind when considering stability: One factor would be the condition number of the approximation operator. The other factor ...
0
votes
1answer
152 views

Green's function and eigenvalues with multiplicity

Green's function of a differential operator contains a lot of information of that operator. In particular, if we have a differential operator on a compact manifold with discrete spectrum, then Green's ...
1
vote
1answer
186 views

Smooth curves in a Frechet space

Is the space $C^{\infty}([0,1];C^{\infty}(S^1))$ equal with the space $C^{\infty}([0,1]\times S^1)$ ? I am interested in characterizing the smooth curves in the space $C^{\infty}(S^1)$ where $S^1$ is ...
2
votes
1answer
122 views

A Liouville theorem involving an advection term

I am curious whether there are any Liouville theorems for the following pde: $$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } R^N $$ where $a(x)$ is a smooth bounded vector field ...
9
votes
3answers
233 views

Reference request: Systems of linear PDES with constant coefficients

I am looking for a reference for the following statement: Assume that $P_1, \dots, P_k \in \mathbb R[x_1, \dots, x_m]$ and consider a system of PDEs \begin{align} P_i(\partial / \partial x_1, \dots, ...
4
votes
1answer
139 views

Dependence of solutions on parameters in partial differential equations

In the standard homogenization problem $$-\nabla.\left(A\left(x,\frac{x}{\epsilon}\right)\nabla u^{\epsilon}(x)\right)=f\ \mbox{in } \Omega,$$ the homogenized matrix $A_0$ is given in terms of ...
1
vote
0answers
114 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 ...
1
vote
0answers
54 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 ...
4
votes
1answer
110 views

Reference request: Wasserstein metric spaces for non linear weights/mobility?

There is a very nice theory of gradient flows in metric spaces by Ambrosio, Gigli and Savaré. One particularly important application is the quadratic Wasserstein setting, where the metric space in ...
1
vote
0answers
133 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 ...
3
votes
2answers
214 views

Stochastic methods for solving very high-dimensional PDE

I am looking for stochastic methods for solving a very high-dimensional PDE (with one time dimension and very large number of spatial dimensions), which would reduce it to a lower-dimensional problem, ...
3
votes
1answer
165 views

semi-classical Green's function

I am reading Gutzwiller's papers on the relation between Hamiltonian flows and solution to Schrodinger equations. In the two papers, he gave a semi-classical approximation of the Green's function to ...
1
vote
1answer
155 views

Nonlocal Stefan problems

Has there been much work in the setting of Stefan (or general free boundary) problems with some type of nonlocality? A search on Google and MathSciNet give me only a handful of results which greatly ...
3
votes
1answer
166 views

What are the applications of Grillakis Shatah and Strauss paper?

I am studying the following paper. Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197. ...
4
votes
0answers
123 views

Pseudodifferential operators on compact manifolds with boundary

I have heard that the square root of the Dirichlet (or the Neumann) Laplacian is not a pseudodifferential operator on compact manifolds with boundary. The context in which this was said was that ...
2
votes
0answers
119 views

Better Sobolev inequality holds in this case when assuming doubling and Poincare inequality?

Let $X$ be a Polish space and let $m$ be a locally finite Borel measure on $X$. Let $\epsilon$ be a strongly local, regular Dirichlet form on $L^2(X,m)$ with Domain $V :=\{f\in ...
2
votes
0answers
111 views

Reference request: functional analysis results used in Taubes paper (1980)

I'm studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I'm looking for a reference of three following theorems: Let $f(x)$ be a convex funtional ...
1
vote
1answer
90 views

Wave equation with linear coefficients

The following pde came up in a physics problem: $$ (Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y), $$ A,B,C,D are fixed ...
3
votes
0answers
85 views

Reference for existence results for 2D forced viscous Burgers equation

I am looking for results concering the following parabolic PDE $$u\cdot\nabla u + \Delta u = F(x),$$ where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or ...
0
votes
0answers
125 views

Calculation in From Seiberg Witten to pseudo-holomorphic curve

I am reading the Taubes's paper: From From Seiberg Witten to pseudo-holomorphic curve. I don't know how to get the result (2.17) \begin{eqnarray*} ...
2
votes
0answers
57 views

Properties of solutions of Parabolic type equations

Assume $u\in C([0,1],L^{2})$ satisfies the following Schrodinger equation $$ \partial_t u=i(\Delta u+Vu), \text{in} ~\mathbb{R}^n\times[0,1],\\ u(0)=u_{0}. $$ with $V=V_1(x)+V_2(x,t)$, where $V_{1}$ ...
-1
votes
2answers
154 views

Motivation for weak solution of a PDE (initial condition)

The following question came to me when reading the famous paper of ALT and LUCKHAUS: "Quasilinear elliptic-parabolic differential equations" When looking at a (nonlinear degenerate) PDE like $$ ...
3
votes
1answer
158 views

The complex heat kernel on a Riemann manifold

There is a vast literature available for the heat kernel. Nevertheless, I haven't been able to find almost anything useful about the kernel of the equation $\frac{1}{\mathbb{i}} \frac{\partial ...
4
votes
0answers
90 views

Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts: Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean ...
0
votes
1answer
71 views

A hyperbolic partial differential equation (wave-like) with variable-dependent coefficient and possibly singular in one variable

First, I beg your pardon since the title of the question is a bit confusing I guess. I'm working on a physical equation of the wave-like form. Explicitly, it reads ...
2
votes
0answers
67 views

Geodesics on a perturbed submanifold of $\mathbb{R}^m$ [closed]

Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the ...
1
vote
0answers
77 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 ...
1
vote
1answer
138 views

Estimates on evolution operator

Let's consider the following evolution operator in $\mathbb{R}^3$ $$S(t)=e^{(i+\delta)t\Delta }$$ How to get the following estimate $$\Vert S(t)f\Vert_2\leq C_\varepsilon t^{-\frac{1}{4}}\Vert ...
3
votes
1answer
189 views

a road from virial identity to Strichartz estimates for wave/ Schrodinger eqs?

Is there any way how virial identity implies Strichartz estimates ( or some smoothing properties) for solutions to a) wave equation b) Schrodinger equation ( say in 3d)? To keep things clear I am ...
0
votes
0answers
107 views

Log of heat kernel for positive time

A well-known theorem by Varadhan relates the logarithm of the heat kernel on a manifold and the geodesic distance function. In particular, if $d(x,y)$ is geodesic distance from $x$ to $y$ and ...
2
votes
0answers
80 views

Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE

A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which ...
2
votes
2answers
294 views

Monge-Ampere type PDE

NB: I have edited this question to clarify what the OP is asking – Robert Bryant Problem: Find a holomorphic function $f$ where where $f(x+iy) = u(x,y) + i\,v(x,y)$, such that the graph $\Gamma_u = ...
0
votes
0answers
87 views

Elliptic PDE-Fredholm PDE(Is there a contradictory situation)

Let $E$ be a smooth vector bundle on a closed manifold $M$. Assume that $D:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a diff. operator which is a fredhoolm operator, in the algebraic ...
2
votes
0answers
115 views

Is Laplacian a surjective operator?

For a closed manifold the laplacian is almost surjective operator since the index of $\Delta$ is zero and there is no a non constant harmonic function. So the codimension of the image ...
3
votes
1answer
91 views

$C_0$ semigroups on parameterized Banach spaces or moving domains

Is there any literature corresponding to one or two-parameter semigroups such that eg. $T(t) \in \mathcal{L}(X(t))$ or $T(s,t) \in \mathcal{L}(X(t),X(s))$ for parameterized Banach spaces $X(t)$??? I ...
2
votes
1answer
63 views

The asymptotic distribution of a subset of Bessel function zeroes

For a research problem I am working on in PDE, I need to obtain asymptotics for the counting function of $$\{0<\alpha <\lambda: \exists n\in \mathbb{N} \textrm{ such that }J_n(\alpha)=0 \textrm{ ...
0
votes
0answers
188 views

Heuristic probabilistic argument for the Navier-Stokes existence and smoothness conjecture

The Collatz Conjecture is a famous conjecture that has never been proven; nevertheless, there exists a simple heuristic probabilistic argument which supports its truth - in Wikipedia's words, "If one ...
1
vote
1answer
294 views

Elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...
1
vote
2answers
80 views

Solvability of quasilinear elliptic equations on closed manifolds

Is there any reference about solvability theory of quasilinear elliptic equations on closed manifolds? In particular, I am looking for solvability condition for function $f$ of following equation ...