**1**

vote

**2**answers

162 views

### Dealing with a Matrix ODE in integral form

Supose we have
$v'(t) = A(t)v(t)+b(t),$ with $v(0)=0$
where $v(t),b(t) \in \mathbb{R}^n $, $A$ is a $n\times n$ real matrix and $t \in [0,1]$
Now, if A and b are discontinuous but integrable ...

**0**

votes

**2**answers

488 views

### duality argument in PDE

Can anyone please explain the term 'duality argument', or the difference between this term and the weak formulation in PDE analysis? Or give some references?
Occasionally I see this term appears in ...

**0**

votes

**0**answers

362 views

### What is the Schauder estimate on usual Hölder space for parabolic type equations

What is the Schauder estimate on usual Holder space for parabolic type equations ?

**5**

votes

**0**answers

263 views

### Rigorous results on the method of multiple scales

The method of multiple scales (Scholarpedia) is a technique used to obtain approximate solutions to differential equations, most commonly when some of the more standard approaches to perturbation ...

**1**

vote

**0**answers

115 views

### The existence of the solution of $u_t+u_0u_x+u_{0x}u+u_{xxx}=0,u(x,0)=0$

Prove the existence of the solution of the Cauchy Problem:
$$u_t+u_0u_x+u_{0x}u+u_{xxx}=0,u(x,0)=0$$
where$u_0\in C^{\infty}(R).u_0,u_{0x}\to 0,when |x|\to \infty$
AS Robert Bryant's comment,$u=0$ is ...

**1**

vote

**0**answers

99 views

### The existence of the solution of the perturbed KdV Equation(semi-group operator)

Consider the perturbed KdV Equation$$u_t-6uu_x+u_{xxx}=\epsilon u,u(x,0)=f(x)$$where $f(x)=v(x,0)$，$v(x,t)$ is a soliton solution.$u$ satisfies the condition$u\to 0 $when $|x|\to\infty$
I want to use ...

**1**

vote

**3**answers

829 views

### book on PDE on manifolds

let $M$ be a Riemannian manifold and $\alpha$ be any some unknown form on $M$. I am interested in solutions or some references of the equation of type $(d + \delta) \alpha = 0$ where $\delta$ is the ...

**3**

votes

**2**answers

171 views

### second order operator with real coefficients and not locally solvable

This question is a follow up of the following answer I posted recently:
Counterexamples in PDE
Is there a second order partial differential operator with real coefficients which are not solvable in ...

**2**

votes

**0**answers

349 views

### Optimal transport warping implementation in Matlab

I am implementing the paper "Optimal Mass Transport for Registration and Warping", my goal being to put it online as I just cannot find any eulerian mass transportation code online and this would be ...

**1**

vote

**1**answer

164 views

### Constructing an example of Hamiltonian flow

I have this Hamiltonian flow generated by
$$
h(x, \xi ) = V(x) + \sqrt{\xi ^2 + 1}, \quad x, \xi \in \mathbb{R}^3,
$$
so the defining equations are
...

**0**

votes

**1**answer

264 views

### Solution to a second order PDE

Looking for a general solution to the following second-order PDE, where the unknown is a function $f(x_1, x_2)$ of two variables:
$
0=a^2f+a^2x_1{\partial f\over \partial x_1}+b^2x_2{\partial f\over ...

**2**

votes

**1**answer

144 views

### Differential finiteness of power series with hypergeometric coefficients

Let $f$ be a formal (or convergent near the origin, as you wish) power series with variables $x_1$, ... , $x_n$ and complex coefficients.
$$ f = \sum_I f_I \mathbf x^I $$
This power series is said to ...

**3**

votes

**1**answer

270 views

### Moving under the influence of a vector field

I have a continuously varying vector field $v(p)$ on $\mathbb{R}^2$, and a particle
at point $p$ in the plane that can move in a direction $u(p)$ as long as $u(p)$ is
turned at most $\pi/2$ left of ...

**3**

votes

**2**answers

261 views

### delay differential equation

I'm looking for exact solutions, if such exist, for the following non-linear delay differential equation (DDE):
$ y_x(x) = A y(x-1)^a $
where $ 0 < a < 1 $ and $ A > 0 $ are given ...

**3**

votes

**1**answer

181 views

### ODE in symmetric definite positive matrices

It is easy to solve the ODE: $\frac {dx}{dt} = a - b x^2$ with $x(0)=0$, $a>0$, and $b>0$, indeed all one has to do is write $dt = \frac {dx}{a-bx^2} = \frac 1 {2\sqrt a}(\frac {dx}{\sqrt ...

**2**

votes

**1**answer

92 views

### Are the real components of s-roots subharmonic?

Suppose $f(z)$ is an analytic function on a domain $D$ which maps negative axis to negative axis. For $s>1$ consider the function $$u(z)=\Re \sqrt[s]{f(z)}$$ with the branch cut along the negative ...

**1**

vote

**0**answers

73 views

### Semi implicit DAE integration using an implicit Runge Kutta scheme

I'm looking for some references regarding integration of DAEs in the form
$M(t) \frac{dy}{dt} + G(y(t),t) + f(t) = 0, \quad y \in R^n, M(t) \in R^{nxn}$
using a high order implicit Runge Kutta ...

**1**

vote

**0**answers

174 views

### Optimal Sobolev Inequality

Recall the Optimal Sobolev Inequality:
Let $(M^n,g)$ be a smooth,
compact Riemannian $n (\geq 3)$ manifold with $\hbox{inj}_g\geq i_0,
|Ric(g)|\leq \Lambda g$. Let ...

**1**

vote

**0**answers

158 views

### A priori estimate for Yamabe solution

We know Schoen's compactness on Yamabe problem:
Let $(M^n,g)$ be a smooth compact Riemannian manifold of dimension
$3\leq n\leq 24$ without boundary. Denote $\Phi$ to be the full set
of arbitrary ...

**2**

votes

**1**answer

170 views

### A priori estimate of elliptic complex

On a compact Riemannian maniflod $(M,g)$, for an elliptic
complex
$\mathcal{C}_0\overset{L_1}{\longrightarrow}\mathcal{C}_1\overset{L_2}{\longrightarrow}\mathcal{C}_2$ where $L_1$ and $L_2$ are ...

**1**

vote

**1**answer

177 views

### topological equivalence of ODEs

Let $n$ be a non-negative integer. $\;\;$ Let $\: f : \mathbb{R}^n \to \mathbb{R}^n \:$ and $\: g : \mathbb{R}^n \to \mathbb{R}^n \:$ be Lipschitz.
Define the relation $\stackrel{f}{\sim}$ on ...

**1**

vote

**2**answers

490 views

### The spectrum of Schrodinger Equation [closed]

Consider the Schrodinger Equation$$\psi_{xx}-(u-\lambda)\psi=0$$
with the condition
1.when $x\to|\infty|,u\to0,u_x\to0$
2.$\psi|_{x\to \infty}=0$
How to prove that spectrums are real?
...

**4**

votes

**1**answer

295 views

### Lie algebra “generated” by matrix-valued curve?

Let $A(t)$ be a $n\times n$-matrix-valued continuous (plus possibly other niceness conditions; see below) curve, with the matrix entries being complex in general. If I am not mistaken, $A(t)$ ...

**1**

vote

**1**answer

136 views

### dilation operator green function

how can i solve $ -ixDG(x,s)-iG(x,s)/2= \delta ( \frac{x}{s}-1) $
i do not know , since it is a first odrder differntial operator, the formal solution i've found would be
$ G(x,s)= \sum_{n} ...

**5**

votes

**2**answers

525 views

### Conservative differential equations “in the wild”

Dear MO world,
I'm teaching an undergraduate course on "fun with chaos". As part of a test (on bifurcations in differential equations), I asked students to sketch phase portraits for a family of (2d) ...

**0**

votes

**1**answer

440 views

### first-order linear differential equation with boundary conditions

let be the differential equation
$ -ixDf(x)-if(x)/2= E_{n}f(x) $
with the boundary conditions $ f(x)=f(p^{k}x) $ for 'p' prime and $k=...,-2,-1,0,1,2,...$
is this possible to solve this eigenvalue ...

**2**

votes

**3**answers

298 views

### Criteria for Involutive Subbundles

Preliminaries: Let $M$ be a smooth manifold with tangent bundle $TM$. A vector subbundle
$VM$ of $TM$ is called involutive if the section space $\Gamma(VM)$ of $VM$ is closed under
the Lie bracket ...

**3**

votes

**1**answer

307 views

### Bedford-Taylor theory

The Dirichlet problem for the Complex Monge-Ampere equation on a bounded pseudoconvex domain in $\mathbb{C}^n$ was studied in Bedford-Taylor's seminal paper wherein they defined $(dd^{c} u)^n$ for ...

**1**

vote

**1**answer

354 views

### How to deduce the existence of stationary points from fixed points of evolution maps?

This is probably a very elementary question. Nevertheless I decided to post it on MO. Consider a smooth manifold $M$ and a smooth complete vector field $v:M\rightarrow TM$. Consider an autonomous ...

**1**

vote

**0**answers

149 views

### On explicit eigenfunctions

Given an algebraic surface $S$ defined by an algebraic equation such
as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest
eigenvalue $\mu_{3}$ for the differential equation $\Delta ...

**3**

votes

**3**answers

865 views

### The multiplicity of the max eigenvalue in matrix multiplication

Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq ...

**5**

votes

**1**answer

353 views

### laplace equation on manifolds with boundary

in aubin's book on page 104 theorem 4.7 there is the theorem: Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold. There exists a weak solution $\varphi \in H_{1}$ of $\Delta \varphi = f $ if ...

**3**

votes

**2**answers

368 views

### Do these kernel functions satisfy the semi-group property?

Dear Friends,
Define the kernel functions for $a\ge 1$,
$$
G_a(t,x) := \frac{C_a t}{t^{1+1/a}+|x|^{1+a}}, \qquad \forall t>0,\: x\in R\;,
$$
where the constant $C_a$ is some normalization ...

**0**

votes

**1**answer

381 views

### Continuous variation from solution of easy problem to solution of hard problem

I asked this question a week ago over on math.stackexchange and got no reply, so I am asking here with slightly different wording. I am trying to prove that there exists a solution to a problem. I ...

**-1**

votes

**2**answers

321 views

### Inverse Problem for jet equations

The following is a well known fact and due to the functorial properties of the jet functor:
Suppose you have two smooth manifolds $M$ and $N$ and maps $f:M \rightarrow N$ as well as
$g: M \rightarrow ...

**0**

votes

**0**answers

132 views

### monge ampere equation along totally real submanifolds

hi,
are there some references when solving the complex monge ampere equation along totally real submanifolds of some compact (with boundary or without) complex manifold. i know that there are a lot ...

**1**

vote

**1**answer

703 views

### Solution of Heat equation with Neumann BC in an arbitrary domain

Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition and initial condition $u(x,0)=u^0(x)$ in a bounded domain $\Omega$ with smooth boundary.
Is this true:
Any solution ...

**2**

votes

**1**answer

157 views

### differential structures on unit-root Frobenius modules.

Let $\mathcal{E}$ be the ring of series $f(T)=\sum_{n \in \mathbb{Z}} a_nT^n, \; a_n \in \mathbb{Q}_p$ such that the $\{a_n\}$ are bounded and $a_{-n}\rightarrow 0$ as $n \rightarrow +\infty$. This is ...

**2**

votes

**2**answers

395 views

### calabi conjecture on compact manifolds

hi,
is the calabi conjencture formulated for compact manifolds with boundary ? or only for those without boundary ? excuse me if the question is too trivial but in my literature it isn't mentioned ...

**0**

votes

**0**answers

154 views

### Inertia/Gravity in Distance Geometry

The Cayley-Menger Determinant, D(N), slickly calculates the N-dimensional simplex
volume of any N+1 points. One constraint in our 3D world is that D(4)=0.
Give each point a mass (Mi) and dynamic ...

**1**

vote

**0**answers

166 views

### Is this Stefan-type problem an open problem?

I am looking for a weak-formulation that would give me an existence and uniqueness of a solution of a Stefan-type problem. It is basically a 2-phase Stefan problem in 2D except that the free-boundary ...

**0**

votes

**0**answers

244 views

### $\partial \bar{\partial}$ on a complex manifold

hallo,
i have the following question: let $M$ be a complex $n-$dimensional manifold and $R \subset M$ be a totally real, compact, $n-$dimensional (real) manifold. let $\alpha$ be a smooth nonnegative ...

**12**

votes

**1**answer

517 views

### A polynomial recurrence involving partial derivatives

Define recursively polynomials $f_n(a,b)$ by
$$ f_0(a,b)=1,\ \ f_n(0,b)=0\ \mathrm{for}\ n>0 $$
$$ \frac{\partial}{\partial a}f_n(a,b) = f_{n-1}(b-a,1-a). $$
For instance,
$$ ...

**1**

vote

**1**answer

305 views

### Reparametrizing characteristic curves for PDE's

I'm looking for solutions for a PDE that looks like this
$$
\nabla u(\vec x) \cdot f(\vec x) = k.
$$
For some clarification, $u$ is a log-probability. And this arises from a Fokker-Plank-like ...

**0**

votes

**1**answer

286 views

### Differences between the Poisson's and elliptic Monge-Ampere equations?

I am working with a numerical application where some known techniques solve $\Delta u=\rho$ for $\rho(x,y)\geq0$ defined on a planar rectangle (given as the density of a large pointset). Since the ...

**4**

votes

**3**answers

493 views

### question about the recursion equation: $x_{n+1}x_{n−1}=x_n^2(1−4x_n)$

Hi all,
I am trying to slove the recursion equation: $x_{n+1}x_{n−1}=x_n^2(1−4x_n)$ in the form of $x_n=x_n(x_1,x_2)$ or $x_n=x_n(c_1,c_2)$, and finally get the limit of the ratio: ...

**0**

votes

**1**answer

225 views

### Nonlinear PDE ${u_{tt}}^2u_{ttxx} = 1$

I have been trying to solve this equation during fortnight
$$
{u_{tt}}^2u_{ttxx} = 1.
$$
But I still here. The only thing is change of variables $u_{tt}(t,x) = y(t,x) $ and solved the ODE $y'' = ...

**0**

votes

**2**answers

1k views

### Looking for the solution of first order non-linear differential equation ($y ′+y^{2}=f(x)$) without knowing a particular solution.

I have been working on Riccati Equation. I have tried many different methods to find a closed form for the solution of first order non-linear differential equation ($y'+y^{2}=f(x)$) without knowing a ...

**0**

votes

**0**answers

232 views

### PDE on the line

In my work I faced a system of PDE of the form
\begin{align}
\varphi^\prime_v(t,v)|_{v=X_t} = \sigma(t,X_t),\newline
\varphi^\prime_t(t,v)|_{v=X_t} = b(t,X_t),
\end{align}
where ...

**2**

votes

**0**answers

218 views

### Periodic orbits of Hamiltonian systems

Consider a second order Hamiltonian system of the type
$$
\ddot{x}+V'(x)=0, \quad x \in \mathbb{R}^N.
$$
Under very `natural assumptions' it is possible to prove the existence of a non constant ...