**1**

vote

**0**answers

77 views

### A Lie algebra assiciated 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 ...

**1**

vote

**3**answers

223 views

### What are some good sanity checks for simulating BNLS?

After doing some googling, I failed to find any explicit solution for the Biharmonic Nonlinear Schrodinger Equation, which states:
$$
i\psi (x,t) _t - \Delta ^2 \psi (x,t) + |\psi (x,t) | ^{2 \sigma} ...

**1**

vote

**0**answers

15 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?

**11**

votes

**2**answers

436 views

### Surjectivity of curl

Let: $\mathbb R^3\ni x\mapsto v(x)\in\mathbb R^3$ be a vector field with null divergence belonging to the Schwartz class such that
$$
\int_{\mathbb R^3} v(x) dx=0.
$$
Is it true that there exists a ...

**0**

votes

**1**answer

77 views

### Uniform $L^p-L^{p'}$ bound of a Fourier multiplier

Let $(\tau,\xi)\in\mathbb{R}\times \mathbb{R}^n$, and consider the function
$$
m_{\epsilon}(\tau,\xi)=\frac{1}{\tau+|\xi|^4+\epsilon|\xi|^2+i}
$$
in $\mathbb{R}^{n+1}$. My first question is that does ...

**3**

votes

**2**answers

143 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

**1**answer

142 views

### What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases?

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an orientation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data ...

**2**

votes

**0**answers

31 views

### Space of p-harmonic functions

Let $\Omega \subset\mathbb{R}^d$, $d \geq 2$, be a sufficiently nice set to make the following question meaningful.
I am interested in the space of p-harmonic functions on $\Omega$; that is, the ...

**6**

votes

**0**answers

117 views

### Have heat kernels for generalized Laplacians on non-compact manifolds been constructed?

Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be ...

**0**

votes

**0**answers

87 views

### Solve PDE system almost equal to the Ernst equation of general relativity

I am trying to find a solution to the following elliptic quasilinear system of PDEs:$$\Delta G+\nabla G\cdot\nabla H=0$$ $$2\Delta H-\nabla H\cdot\nabla H-e^{2H}(\nabla G\cdot\nabla G)=0$$with the ...

**0**

votes

**0**answers

35 views

### Linear elliptic estimates

i am interested in solutions of the following
$$-\Delta \phi =f \; \; \; A_\lambda, \qquad \phi=0 \; \; \partial A_\lambda,$$ where $ A_\lambda=\{ x \in R^N: \lambda <|x|<1 \}$ with $ ...

**1**

vote

**0**answers

18 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 $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; ...

**1**

vote

**0**answers

112 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 ...

**1**

vote

**1**answer

128 views

### 2D semilinear elliptic PDE

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. ...

**0**

votes

**1**answer

94 views

### Question on viscosity solution through stochastic differential equations

I have learned that for the equation $\partial_tu+a(u)\partial_xu=0$, the entropy solution could be obtained as the limit of the equation $\partial_tu+a(u)\partial_xu=\epsilon u_{xx}$ with ...

**0**

votes

**0**answers

42 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, ...

**0**

votes

**0**answers

33 views

### Dependence of weak solution of an equation on a parameter

For each $p \in [a,b]$, let $X_p$ be a Hilbert space with $Y_p \subset X_p$ a subspace and we are given a bilinear form $a_p(\cdot,\cdot):X_p \times X_p \to \mathbb{R}$.
Given $u_p$ with $p \mapsto ...

**2**

votes

**0**answers

57 views

### Analogous to a PDE but where independent variable is a function

Consider, as an example for my question, a density function $u(\boldsymbol{x},t)$ on a vector field $\boldsymbol{x}$ at some time $t$. The flow velocity vector of the density is given by ...

**12**

votes

**1**answer

287 views

### Does the Legendre-Hadamard condition imply a generalized Gårding inequality?

For simplicity, we restrict to constant coefficients. Let $A^{ij}_{ab} \in \mathbb{R}$, $1 \le i, j \le n$ and $1 \le a, b\le m$, satisfy the Legendre-Hadamard condition:
$$
...

**0**

votes

**0**answers

47 views

### Linking theorem, elliptic pde

I am trying to solve some linear system of the form
$$ \Delta \phi +p v(r)^{p-1} \psi = -f, \qquad \Delta \psi + qu(r)^{q-1} \phi =g $$ in $ \mathbb{R}^N$ where $ f,g$ are given and $ \phi,\psi$ are ...

**1**

vote

**0**answers

38 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}
...

**1**

vote

**0**answers

83 views

### Uniqueness of the solution of a nonlinear PDE [closed]

Given the nonlinear PDE
$$
\partial^2\phi+\phi^3=0
$$
I consider the characteristics $\xi=\kappa\cdot x$. Then I look for a solution in the form
$$
\phi(x)=a\cdot\chi(\xi).
$$
Then, provided ...

**0**

votes

**1**answer

69 views

### Do constrained random walks converge weakly to the Wiener measure on the space of constrained paths (that corresponds to the heat equation)?

Let $U$ be an open subset of $\mathbb{R}^n$ such that
$\partial \overline{U}$, the boundary of $\overline{U}$ is ''nice''
(for simplicity you can assume piecewise smooth). I also want to allow the ...

**3**

votes

**2**answers

206 views

### Heat equation and evolution of number of critical points

Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...

**0**

votes

**2**answers

105 views

### Friedrichs/Poincare inequality on $S_n \times (0,\infty)$?

Should I expect the following Friedrichs/Poincare inequality to hold for $u \in C^\infty(S_n \times (0,\infty))$ with $u(x,0) = 0$:
$$\int_{S_n \times (0,\infty)}|u|^2 \leq C\int_{S_n \times ...

**0**

votes

**0**answers

31 views

### Regularity of solutions of strongly elliptic system: how smooth must the boundary be?

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with the boundary of class $C^{1,1}$.
Let $A$ be a selfadjoint operator acting in $L^2(\Omega;\mathbb{C}^n)$.
The operator $A$ is given by the ...

**5**

votes

**1**answer

948 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, ...

**4**

votes

**0**answers

257 views

### A vector space associated with a vector field on a symplectic manifold

Let $(M,\omega)$ be a $2n$ dimensional symplectic manifold and $X$ is a smooth vector field on $M$. Consider the following subvector space of $\chi^{\infty}(M)$: $$S(X)=\{Y\in ...

**4**

votes

**1**answer

520 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:
...

**2**

votes

**0**answers

65 views

### long time existence of a nonlinear parabolic equation

I am thinking about a geometric problem which boils down to the following parabolic equation:
Suppose $u=u(r,t)$, $r$ is defined on $[0,1]$ and $t>0$
$$\begin{cases}\displaystyle \frac{\partial ...

**0**

votes

**0**answers

58 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 ...

**6**

votes

**1**answer

352 views

### What is the idea behind interpolation spaces?

I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific:
Definition. Let $H$ be an ...

**3**

votes

**1**answer

274 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. $\nabla\cdot q\equiv 0$ ...

**5**

votes

**1**answer

399 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 ...

**1**

vote

**0**answers

153 views

### A integral equation with Discrete to result by inverse problem

Problem
I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here, (I know the ...

**0**

votes

**1**answer

146 views

### Cauchy problem for an overdetermined system of PDE

This question is strictly related to this one. Let us consider the differential system with constant coefficients
$$\left(\begin{array}{ccc}
B_{11} & B_{12} & 0\\
...

**5**

votes

**1**answer

132 views

### Elliptic operator on non compact manifolds with ends of the type $\Omega\times (r,\infty)\times\mathbb{R}$

A smooth manifold $M$ is a manifold with a cylindrical end if there exists a compact subset $K\subset M$ such that $M\backslash K$ is diffeomorphic to $\Omega\times (r,\infty)$ where $\Omega$ is a ...

**10**

votes

**9**answers

3k views

### Textbooks for PDE between Strauss and Folland

Walter A. Strauss's Partial Differential Equations: An Introduction is a classic PDE textbook for the undergraduate students. While Folland's Introduction to Partial Differential Equations, is a nice ...

**2**

votes

**1**answer

106 views

### Local Biot-Savart law in $B(x_o,r) \subset \mathbb R^2$

Let $u: \mathbb R^2 \to \mathbb R^2$ and let $\omega = \text{curl } u$ be the 2D vorticity of $u$, where $u, \omega \in L^2(\mathbb R^2)$ and $\nabla \cdot u = 0$. The classical Biot-Savart law states ...

**4**

votes

**1**answer

590 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 ...

**4**

votes

**0**answers

57 views

### A “gradient” weak Harnack inequality for quasilinear elliptic equations

Suppose we are in the following loosely described setting:
we have a non-negative supersolution $h$ of the following elliptic equation:
\begin{equation}
\Delta h + \|\nabla h\|^2 + f(x) \geq 0
...

**1**

vote

**1**answer

146 views

### The class of bounded uniformly continuous functions in viscosity solution theory for Hamilton-Jacobi equations

Dumb question: Usually in viscosity solution theory for Hamilton Jacobi equations (with convex, coercive Hamiltonians), solutions are said to be in the class $BUC(\mathbb{R}^n)$ or ...

**4**

votes

**0**answers

85 views

### Continuity of the curve-shortening flow with respect to the curve

The curve-shortening flow is an evolution equation for a smooth closed curve $\alpha$ inmersed in a Riemannian surface $M$. The version where $M$ is the Euclidean plane is illustrated for example in ...

**0**

votes

**1**answer

112 views

### Implicit function theorem for elliptic partial differential equations

Consider the elliptic equation $-\Delta u=\alpha f(u)$ in $R^n$ and assume that for some $\alpha^* \in R$ it has a bounded smooth solution $u^*$ in $R^n$ ($f$ is a nice smooth function). Under what ...

**2**

votes

**0**answers

89 views

### Caffarelli-Silvestre extension definition of fractional Laplace-Beltrami on hypersurface

Let $\Gamma \subset \mathbb{R}^{n+1}$ be a $n$-dimensional (closed) $C^k$-hypersurface. Consider the problem
$$\nabla_\Gamma \cdot (y^{1-2s}\nabla_\Gamma u)(x,y) =0 \quad \text{for $(x,y) \in \Gamma ...

**1**

vote

**0**answers

72 views

### Frobenius-Perron operator leaving a function sub space invariant

I consider a linear subspace of $L^1(\mathbb R^n \to \mathbb R)$,
$$ \mathcal S := \{ h: L^1(\mathbb R^n \to \mathbb R) \; | \; \int_{H^{-1}(\{y\})} h(x) \, dS = 0 \text{ for all } y \in \mathbb R\} ...

**4**

votes

**1**answer

198 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 ...

**5**

votes

**2**answers

240 views

### A question on density of Lipschitz functions in weighted Sobolev spaces

Recall that for a domain $\Omega\subset \mathbb{R}^n$, the weighted Sobolev space $W^{1,n}(\Omega,\mu)$ is defined as $f\in L^n(\Omega,\mu)$ and the weak derivative $Df\in L^n(\Omega,\mu)$.
Let now ...

**10**

votes

**2**answers

2k views

### Compact Embeddings of Sobolev Spaces: A Counterexample Showing The Rellich-Kondrachov Theorem Is Sharp

Let $U$ be an open bounded subset of $\mathbb{R}^n$ with $C^{1}$ boundary. Let $1 \leq p < n$ and $p^{\ast} = pn/(n-p)$. Then the Sobolev space $W^{1,p}(U)$ is contained $L^{p^{\ast}}(U)$ and ...

**1**

vote

**1**answer

92 views

### Method of characteristics of a system of first order pdes

I asked the question on math.stackexchange.com, but didn't get any reply. So, I asked it again here. Any suggestion or hint is welcome, and thank you for your attention.
Consider the system of first ...