**2**

votes

**0**answers

19 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 ...

**5**

votes

**1**answer

809 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, ...

**1**

vote

**0**answers

110 views

### Runge-Kutta with all nodes at n+1 or zero weights otherwise [migrated]

So, lets say for the family of the explicit Runge-Kutta methods:
$$y_{n+1} = y_n + \sum_{i=1}^s b_i k_i$$
where,
$$k_1 = hf(t_n, y_n)$$
$$k_2 = hf(t_n+c_2h, y_n+a_{21}k_1)$$
$$\vdots$$
$$k_s = ...

**0**

votes

**0**answers

66 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*}
...

**1**

vote

**0**answers

24 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 $V_{1}$ ...

**3**

votes

**1**answer

109 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 ...

**0**

votes

**2**answers

71 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

**1**answer

247 views

### Uniqueness theorems related to Hardy Uncertainty Principle

Uncertainty Principles state that a function and its Fourier transform cannot be simultaneously sharply localised. A well known result due to G.H.Hardy says that
if $f(x)=O(e^{-\alpha^2|x|^2})$, ...

**4**

votes

**1**answer

498 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:
...

**1**

vote

**0**answers

40 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 ...

**3**

votes

**1**answer

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

**4**

votes

**0**answers

171 views

### Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...

**4**

votes

**1**answer

328 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

**2**answers

255 views

### Textbook for Partial Differential Equations with a viewpoint towards Geometry

I don't know whether I should ask this question here or not but I asked this question on MSE but didn't get any answer so I am posting it here.
Though similar questions have been asked at ...

**4**

votes

**0**answers

52 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

**1**answer

38 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
...

**4**

votes

**2**answers

475 views

### Limit cycles as closed geodesics(geodesible flow)

The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$:
\begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation}
This equation defines a foliation on ...

**2**

votes

**0**answers

58 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

**1**answer

268 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} + ...

**1**

vote

**1**answer

122 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

**1**answer

98 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 ...

**1**

vote

**1**answer

117 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 ...

**1**

vote

**0**answers

29 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 $\mathbb{R}^{2}$.
According to the comment of Joonas on this ...

**2**

votes

**2**answers

234 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

**0**answers

86 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 ...

**5**

votes

**2**answers

160 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

**0**answers

64 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

**0**answers

97 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 ...

**0**

votes

**0**answers

152 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 ...

**3**

votes

**1**answer

85 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 ...

**4**

votes

**1**answer

249 views

### $C_0$-semigroups applications

My graduation thesis was about stability theorems for $C_0$-semigroups (see the Wikipedia article for the definitions: http://en.wikipedia.org/wiki/C0-semigroup). I would like to know if there is ...

**2**

votes

**1**answer

54 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{ ...

**4**

votes

**1**answer

207 views

### Are Besov spaces $B^{s}_{p,q}$ invariant under Fourier transform?

(This may be very easy question for MO; as I am just trying to understand Besov spaces)
Let $\phi \in C^{\infty}(\mathbb R^{n})$ with
$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: ...

**3**

votes

**1**answer

487 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

164 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} ...

**3**

votes

**0**answers

76 views

### Pointwise a.e. formulation of parabolic PDE, what if null set depends on test function?

Let $u \in L^2(0,T;V)$ with $u_t \in L^2(0,T;V^*)$ be a solution of
$$\langle u_t(t), v \rangle + a(t;u(t), v) = \langle f(t), v \rangle$$
where $f \in L^2(0,T;V^*)$ and we have the usual assumptions ...

**0**

votes

**2**answers

45 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
...

**2**

votes

**2**answers

87 views

### First order pde with characteristics [closed]

Consider a first order pde of the type $$u_y+b(x)u_x=0$$ and suppose that the coefficient $b$ is not necessairly continuous (for instance with a jump in some point).
Is it still possible to apply in ...

**4**

votes

**1**answer

522 views

### eigenvalues of a Möbius strip

Consider the Möbius strip as the unit square with two opposite sides identified (with opposite directions). Consider the eigenvalue equation $\Delta u = \lambda u$ with boundary condition $u=0$. ...

**7**

votes

**1**answer

548 views

### Regularity of the Maxwell equations

As is well-known, the Maxwell equations can be phrased vectorially as,
\begin{align}
\nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\
\nabla \cdot \mathbf ...

**6**

votes

**0**answers

108 views

### Local solvability of nonlinear elliptic boundary value problems

Malgrange proves the following statement regarding local solvability of (determined or underdetermined) nonlinear elliptic systems:
Let $F_i(x,D^\alpha u)=0$ be a nonlinear elliptic system of order ...

**1**

vote

**1**answer

75 views

### $L^\infty_\mathrm{loc}$ assumption in global existence for Boltzmann equation

In short:
In P. Gérard's paper on the existence of global solutions to the Boltzmann equation from 1988 (or equivalently Cercignani's book), why are the stated assumptions (especially $A_n \in ...

**0**

votes

**0**answers

62 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 ...

**0**

votes

**0**answers

41 views

### Why Does a quadratic phase term in BNLS causes collapse?

I've heard a couple of times that in the Biharmonic Nonlinear Schrodinger Equation,
$i\psi_z + \Delta ^2 \psi + |\psi | ^{2\sigma } \psi =0 $, $\psi (x, 0) = \psi _0 (x) \in H^2( \mathbb{R} ^d ) $
...

**5**

votes

**3**answers

559 views

### Reference on Semigroup Theory and Parabolics PDE'S

Recently started to study Semigroup Theory. My background is equivalent to the first three chapters of the Jack Hale's book, Asymptotic Behavior of Dissipative Systems.
Looking for a reference to an ...

**18**

votes

**1**answer

753 views

### Why is there a connection between enumerative geometry and nonlinear waves?

I'm not 100% sure that this question is appropriate for this site. If it's not, please tell me and I'll delete it.
Recently I encountered in a class the fact that there is a generating function of ...

**5**

votes

**0**answers

86 views

### Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$:
$$
\begin{cases}
\partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\
u(0,x)=u_0(x).
\end{cases}
$$
...

**4**

votes

**1**answer

156 views

### $C^0$ estimate for solutions of elliptic PDE with Neumann BC

I am interested in a reference for (or counterexample to) a particular
$C^0$ estimate for solutions of the Laplace equation with Neumann
boundary conditions. More precisely, let $(M,g)$ be a ...

**2**

votes

**0**answers

104 views

### numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$
In this linear PDE:
\begin{cases}
B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...

**2**

votes

**0**answers

74 views

### Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE
\begin{align}
\partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\
\rho(t ...