**4**

votes

**1**answer

338 views

### Beginners Guide to Cartan for Beginners [on hold]

I am working through parts of Cartan For Beginners by Ivey and Landsberg. Thankfully some exercises have solutions, but, we would benefit from some additional guidance.
Question: I am seeking ...

**2**

votes

**0**answers

23 views

### Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)

Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse.
Does there ...

**3**

votes

**2**answers

263 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 ...

**2**

votes

**0**answers

41 views

### Application of Egorov's Theorem for Pseudodifferential Operators

Let $\;P_{0} \in OPS^{m}_{1,0}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})\;$, $\;A \in OPS^{1}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})$ and $S(t)$ the solution operator of the scalar hyperbolic ...

**2**

votes

**1**answer

80 views

### System of linear first order PDE with constant coefficients

recently in my researches I've come across the following operator
$$L\left(\begin{array}{c}
a_1\\
\vdots\\
a_n
\end{array}\right)=M_1\left(\begin{array}{c}
...

**4**

votes

**1**answer

512 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:
...

**3**

votes

**1**answer

256 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

373 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

**0**answers

87 views

### Continuously dependent on parameters [closed]

How do we check whether the solution is continuouly dependent on parameters?
Let $\Omega$ be a domain with smooth boundary. Say $f$ and $h$ are smooth. Assume that for each $\theta\in (0, 1]$, the ...

**1**

vote

**1**answer

290 views

### Strichartz estimates of damped wave equation

If $w(t,x)$ is a solution of wave equation
$$
w_{tt}-\triangle w = 0, w(0)=w_0, w_t(0)=w_1,
$$
then $w$ satisfies the following Strichartz esitmates
$$
\|w\|_{L^q_tL^r_x} \lesssim \|w_0\|_{H^1} + ...

**0**

votes

**0**answers

32 views

### Existence to an advection equation with constraints

Let $f: \mathbb R^n \to \mathbb R^n$ be a sufficiently smooth vector field and let $h: \mathbb R^n \to \mathbb R$ a scalar field. We consider the following advection equation
...

**1**

vote

**1**answer

175 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 ...

**1**

vote

**1**answer

137 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 ...

**2**

votes

**1**answer

110 views

### Well-posedness of heat equation with distributional right hand side

The question is about well-posedness of heat equation
$$
\frac{\partial\Theta}{\partial t}=\alpha^2\Delta\Theta+p(t)\delta(x-u(t))\delta(y-v(t)),~~ (x,y,t)\in\Omega\times[0,T],
$$
subjected to ...

**9**

votes

**1**answer

198 views

### applications of C$^*$-algebras in the field of PDEs

I know only a little bit about C$^*$-algebras and I want a to know if you know a nice apllication or the influence of them in the field of partial differential equations (it is better that it is ...

**1**

vote

**1**answer

42 views

### Point moving inside smooth domain?

Let $U \subset \mathbb{R}^2$ be a domain im $\mathbb{R}^3$ with smooth boundary. Let a point move inside $\mathbb{R}^3$ along the smooth curve $x(t)$. We denote by $\mbox{dist}(x(t), \partial U)$ the ...

**4**

votes

**1**answer

178 views

### Uhlenbeck's theorem novelty

This link provides a short introduction to the contributions of Uhlenbeck about regular gauge fixing. However, I feel quite puzzled about it and I do not understand the real novelty apported by this ...

**2**

votes

**1**answer

87 views

### Is the on-diagonal heat kernel “local” with respect to the metric?

Question
Let $X$ be a manifold, and $\mu_A$, $\mu_B$ two Riemannian metric on it which agree on an open subset $U\subset X$, i.e. $\mu_{A\,|U} = \mu_{B\,|U}$. Let $K_A(t;z,w)$ resp. $K_B(t;z,w)$ be ...

**0**

votes

**2**answers

127 views

### Frobenius condition

Suppose X and Y are two unit length vector fields on a Riemannian manifold which are orthogonal at each point. Is it true that the lie bracket of X, Y belongs to the span of the vector fields at each ...

**9**

votes

**1**answer

165 views

### Reference Request: Elliptic differential operators in the Fréchet setting

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...

**0**

votes

**0**answers

57 views

### How to prove it is uniformly bounded?

Let $\Omega$ be a bounded domain with smooth boundary. Say $\theta\in(0, 1]$. Let $u(x, \theta)$ be a solution to the problem $\Delta u-\theta u=g(x)$ subject to Neumann boundary condition. Suppose ...

**5**

votes

**2**answers

216 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 ...

**0**

votes

**1**answer

129 views

### A distributional normal derivative for functions in $H^1(\Omega)$

Let $\Omega$ be a smooth bounded domain with $\partial\Omega = \Gamma$. I have read this.
For all $u \in H^1(\Omega)$ such that $-\Delta u = g \in L^2(\Omega)$ in distribution, we can define the ...

**4**

votes

**1**answer

151 views

### Extension of solutions of PDEs with constant coefficients

Let $\mathcal L$ be a differential operator with constant coefficients and $\mathcal{L} f=0$ for some $f\in C^{\infty}(\mathbb{R}^n).$
Under what conditions on $\mathcal {L}$ the function $f$ extends ...

**1**

vote

**0**answers

73 views

### Free Endpoint of Minimization Problem

Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > ...

**2**

votes

**2**answers

77 views

### Estimates on a heat process with fixed boundary data and zero initial conditions

Consider the following heat process:
For a given (say, smooth) domain $\Omega$ on a closed manifold $M$ we construct $p(t,x):\mathbb R_+ \times \bar\Omega \rightarrow [0,1]$, so that
$$
\partial_t ...

**0**

votes

**0**answers

36 views

### Regularity of solutions to $u' + Au = f$ for nonlinear monotone operator $A$

Consider the equation
$$u' + Au = f$$
$$u|_{\partial \Omega} = 0$$
$$u(0) = u_0$$
where $A:L^p(0,T;W^{1,p}_0) \to L^q(0,T;W^{-1,q})$ is some monotone nonlinear operator (with additional assumptions). ...

**0**

votes

**1**answer

92 views

### Prove a function, defined by integration of a harmonic function, is log-convex [closed]

Let $u$ be a harmonic function and we define
$$ q(r)=\int_{\partial B(0,r)}u^2(x)\,dx $$
The question is about to prove that $q(r)$ is log-convex, i.e., I want to show $\log q(r)$ is convex function ...

**0**

votes

**1**answer

88 views

### Complex transport equation

Consider an n dimensional Riemannian manifold with boundary.
Let $\Phi$ be a complex valued smooth function defined in M. Does there exist a NONE VANISHING complex valued function $u$ that solves the ...

**2**

votes

**1**answer

47 views

### Dichotomy for global existence or blow up for solutions of evolution problems

Consider the problem (Nonlinear Schrödinger equation)
\begin{equation}
\left\{
\begin{array}{rl}
iu_t + \Delta u\mp u|u|^{\alpha}=0\\
u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\
...

**6**

votes

**1**answer

96 views

### Is there a maximum principle for stress in continuum mechanics?

I'm working with the equilibrium equations in linear elasticity, which I have not worked with in the past. My engineering colleagues seem to "know" that the maximum Von Mises stress occurs on the ...

**1**

vote

**0**answers

28 views

### ABP estimates for semiconvex functions

Referring to the classical ABP Estimate (Gilbarg-Trudinger Lemma 9.2) I am looking for if such an estimate can be generalized to semiconvex functions. In an article of Trudinger (Comparison Principles ...

**1**

vote

**1**answer

109 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 ...

**1**

vote

**0**answers

31 views

### Introduction to free boundary problems (that are not Stefan problems)

Could someone recommend some notes/papers that deal with existence/regularity of free boundary problems arising from parabolic equations (excluding Stefan type equations)?
I am thinking of eg. ...

**1**

vote

**0**answers

21 views

### Is it classical that the solution to an hyperbolic equation equation is Lipschitz -continuous $[0,\infty)\to L^1(\mathbb{R})$?

Recently I am reading a paper "Global solution and smoothing effect for a non-local regularization of a hyperbolic equation" published on J.E.E, 2004. In the proof, the authors write "It is classical ...

**3**

votes

**1**answer

98 views

### Surfaces with specific types of second fundamental form

Given a three dimensional Riemannian manifold $(M,g)$ and a surface $\Sigma \subset M$ can one categorize surfaces where the second fundamental form of $\Sigma$ is a scalar multiple of the induced ...

**5**

votes

**2**answers

134 views

### Compactly supported functions and Sobolev spaces on manifolds

It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ ...

**4**

votes

**1**answer

104 views

### PDEs on torus $\mathbb T$

(Hope this question is o.k. for MO)
I have been learning PDE(non linear dispersive equations) techniques, mainly using harmonic analysis(kind of Strichartz estimates, estimates for unimodular ...

**0**

votes

**2**answers

171 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 ...

**17**

votes

**6**answers

3k views

### Square roots of the Laplace operator

In several places in the literature (e.g. this paper of Caffarelli and Silvestre), I've seen an integral formula for fractional Laplacians. I'd like to understand it. In this question, I'll stick to ...

**4**

votes

**1**answer

539 views

### Questions on the proof of the Serrin condition for the regularity of Navier-Stokes equations and related issues for the incompressible Euler equation

Edit: The question has been substantially modified from the original one. The original question (see below) concerned with rigorously justifying the proof of the Serrin condition. These questions have ...

**1**

vote

**3**answers

192 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

**2**answers

68 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
...

**5**

votes

**2**answers

215 views

### Poincare-like inequality on compact Riemannian manifolds

I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact Riemannian manifold without boundary. The inequality I am looking for is the equivalent of
$$ ...

**3**

votes

**2**answers

190 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 ...

**3**

votes

**2**answers

106 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 ...

**4**

votes

**1**answer

108 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 ...

**0**

votes

**1**answer

117 views

### Equivalence of two definitions of Sobolev spaces

Good morning,
I am looking for a reference about the following fact that seems to be folklore. Define the Sobolev (Beppo Levi?) space
$$
D^{1,p}(\mathbb{R}^N) = \left\{ u \in L^{p^*}(\mathbb{R}^N) ...

**0**

votes

**0**answers

113 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

**0**answers

86 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 ...