# Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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### Estimate for the operator $A A_D^{-1}$

Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\mathrm{div}g(x)\nabla$
...

-1
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0
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59
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### Reversing heat transfer with respect to time

Fact: One can easily compute heat dispersion in a plane using the heat equation.
Question: Has any research been done on computing the process in the reverse time direction?
That is, given a heat map $...

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62
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### Weak derivative of $|u|^s$ $(s>1)$ [migrated]

I often come across instances in texts where people calculate the weak derivative of $|u|^s$ for $s>1$ as $s|u|^{s-1} \operatorname{sign}(u) \partial_x u$ for some $u\in W^{1,s}(\Omega)$.
However, ...

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132
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### How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?

Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, Is $\operatorname{Id}-K$ a proper map? I think maybe it has ...

3
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1
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160
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### Weighted Lebesgue space with exponential weights: smoothing effect and properties

I am researching whether there are weighted Lebesgue spaces of the type
$$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$
...

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75
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### Non-linear PDE involving norms

I'm interested in searching for a solution $\textbf{v} = [v_1,v_2,v_3]$ to the following PDE
\begin{equation}\label{Eq1}
\dfrac{\partial}{\partial t}v_i =\nu \left|\bigtriangleup \textbf{v}\right|+ \...

1
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0
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141
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### Linear third order water wave pde admitting particular gamma factor solution. How do you understand evolution on vertical strip in complex plane?

I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...

-1
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1
answer

98
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### Proving that $\max_{w \in B(z)} e^{f(w)} \leq Ce^{f(z)}$

Let $f : \mathbb R^2 \to \mathbb R $ be a smooth function statisfying
$$
0 < \alpha \leq \Delta f(w) \leq \beta < \infty, \ \ \forall w \in \mathbb R^2
$$
where $\Delta$ denotes the Laplace ...

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154
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### Bounding the vorticity equation [closed]

Assume there is a solution to the vorticity equation (in component form)
\begin{equation}\label{Eq1}
\dfrac{\partial}{\partial t}v_i + |\textbf{v}|~|\nabla u_i|\cos \alpha_i - |\textbf{u}|~|\nabla v_i|...

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53
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### Solutions for a system of PDE's

Let $ \Omega_s(x) $ solve the system
$$s \dfrac{\partial^2}{\partial s^2}\Omega_s(x)=\pm x\dfrac{\partial}{\partial x}\Omega_s(x) $$
$$2\sqrt{s}\frac{\partial}{\partial s} \sqrt{\pm\Omega_s(x)}=\sqrt{...

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72
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### Spectrum of Laplace-Beltrami operator on tensors

Let $(M, g)$ be a complete Riemannian manifold diffeomorphic to $\mathbb{R}^n$. Under appropriate geometric assumptions concerning the geometry near infinity, but without any curvature sign ...

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65
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### Closed form ODE solutions for Jacobi field/eigenfunction of Laplacian on hyperbolic space

I'm trying to compute Jacobi fields of the hyperbolic disk $\mathbb{H}^m$ considered as a minimal hypersurface in $\mathbb{H}^{m+1}$ in the half model. References to literature or solutions to the ...

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57
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### How to calculate the minimum of this functional

Let $ n:[0,2\pi]\to S^2 $ and $ m:[0,2\pi]\to S^2 $ be two smooth regular curves on $ S^2 $. Assume that $ |n(0)-n(2\pi)|=2 $ and $ |m(0)-m(2\pi)|=2 $, i.e. the start and end points of $ n,m $ are ...

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48
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### Convex combination of cyclically monotone sets

I want to show the following statement, but I am not sure how.
Proposition(?):
Let $C \in \mathbb{R}^d$ be a compact convex set, and let $u, v : C \to \mathbb{R}$ be smooth convex functions.
Suppose
$$...

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1
answer

133
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### Why $-\Delta_g u+\lambda=\lambda \frac{e^{2 u}}{\int_M e^{2 u} d \mu_g}$ type PDE is called 'mean-field equation'?

Why $$
-\Delta_g u+\lambda=\lambda \frac{e^{2 u}}{\int_M e^{2 u} d \mu_g}
$$type PDE is called 'mean-field equation'? It's closely related to moser-trudinger inequality, there are many classical ...

2
votes

1
answer

78
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### Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function

Context: I am reading a paper on Long-Time Asymptotics of the thin film equations, in which the authors consider the strong solutions of the thin film equation in 1-D and transform them using a time-...

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96
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### Regularity of elliptic equation with Neumann boundary conditions

In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...

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71
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### Finiteness of theta vanishing in the KP direction for locally planar curves

I believe the main question is Question 2 at the end, and for experts it might be completely okay to skip directly to it (assuming I'm not saying any nonsense).
My motivation comes from pure algebraic ...

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1
answer

160
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### A variant of Hardy's inequality for "convolutions"?

Consider Hardy's inequality on $L^{2}(\mathbb{R}^3)$. This inequality states that:
$$\int_{\mathbb{R}^3} dx \, \frac{|\psi(x)|^2}{|x|^2} \le K \int_{\mathbb{R}^3} dx \, |\nabla \psi(x)|^2.$$
I want to ...

3
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0
answers

90
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### Conformal Killing vector fields on manifolds that are not asymptotically flat

Let $M = [1,\infty) \times S^2$.
Equip $M$ with the metric $g = dr^2 + r^2 (\gamma + h)$ where $\gamma$ is a metric on $S^2$ and $h$ is a $(0,2)$ tensor on $M$ that satisfies
$$h = O(1/r),\quad \...

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0
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31
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### Reference for an IBVP for the linear homogeneous 1-D Schrödinger equation

I asked the following question on MSE some time ago, but got no answer (sorry if the question is not appropriate for MO).
Consider the following initial boundary value problem for the linear ...

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answers

74
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### Can we study concavity of vorticity equation?

The vorticity equation is well known given as
\begin{equation}\label{Eq1}
\dfrac{\partial}{\partial t}\textbf{v} + (\textbf{u}\cdot \nabla )\textbf{v} - (\textbf{v}\cdot \nabla )\textbf{u} = \nu \...

1
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1
answer

169
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### Let $u$ be harmonic on domain $D\subset \mathbb R^d$, how far can we extend $u$ holomorphically?

If $u$ is harmonic then it is real analytic so then it can be extended locally holomorphically. I also know that if $u$ is harmonic on a ball in $\mathbb R^d$ we have that the radius of convergence is ...

1
vote

1
answer

88
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### Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data

I already asked the question on MSE, and have tried to figure it out myself.
But the problem seems trickier than expected, so I guess MO is a better place to ask..
For the sake of completeness, I ...

3
votes

1
answer

115
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### Nature of a certain invariant on smooth field of positive definite matrices

I initially asked on math.stackoverflow but have since come to understand this forum may be more appropriate, as this is indeed a question that arose in writing a research article.
Denote $g$ a ...

4
votes

0
answers

242
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### Does there exist research about equation like $u_{tt}=\det(D_{x}^{2}u)+\dots$?

I have asked this question on Mathematics Stack Exchange yesterday, but there still is no reply.
Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, ...

1
vote

2
answers

160
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### Exterior differential systems on compact three-manifolds and Cartan-Kähler theory

Let $M$ be a compact three-manifold. I am interested in the following equation on $M$:
$ d e^i = \sum_{j,k}^3 \Theta^i_{jk} \, e^j\wedge e^k\, , \qquad i =1,2,3$
together with the following condition:...

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359
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### Stokes equation and Helmholtz decomposition

I apologize in advance for the long question, but it involves some work I been thinking about and would like help with. The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational ...

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58
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### Are there spectral Galerkin methods for PDE of the form $\partial_tu=\nabla\cdot f(\nabla u)\nabla u$?

Question is in the title. The nonlinearity due to the term $f(\nabla u)$ makes it difficult to directly apply the spectral Galerkin method as it can be done for PDE of the form $\partial_tu=\nabla\...

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0
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50
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### Dispersive equations at low frequencies and time oscillations

It seems to me that nearly all the common linear dispersive equations have dispersion relations which vanish at the zero spatial frequency. For example:
The Schrodinger dispersion relation is $\omega(...

3
votes

1
answer

133
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### Stochastic representation of Laplace equation with Neumann boundary condition

Consider nice domain $D\subset \mathbb R^d$ and $\Delta u =0$ with $u\big|_{\partial D}=g$. It is well known that $u(x)=E^x[g(B(\tau))]$ where $\tau$ is exit time of $B$ from the domain $D$.
What if ...

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0
answers

112
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### Conformal laplacian on asymptotically flat manifolds with boundary

Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies
$$\...

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1
answer

169
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### Can we approximate a Hölder pdf by higher-order Hölder pdf's?

$\newcommand{\RR}{\mathbb R}\newcommand{\NN}{\mathbb N}$
Let $\alpha \in (0, 1)$ and $j \in \NN$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\RR}^d)$ the space of real-valued functions $f$ on $\...

9
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2
answers

441
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### Wick rotation for Laplace and wave equations

I have seen Wick rotation used to describe the relationship between the heat and Schrodinger equations. That is, if $u(t,x)$ solves the heat equation then $v(t,x):=u(it,x)$ solves the Schrodinger ...

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198
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### Navier Stokes Equation

The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided as:
\begin{equation}\label{Eq1}
\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...

5
votes

1
answer

265
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### Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces

It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...

2
votes

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answers

87
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### What is known about the analytic continuation of Maz'ya's modified harmonic zeta function $\sum_{n=1}^{\infty} e^{-zH_n}$?

Question:
If we let $H_n = \sum_{k=1}^{n} \frac{1}{k}$ be the harmonic numbers then we can consider the modified zeta function
$$ f(z) = \sum_{n=1}^{\infty} e^{-zH_n } = \sum_{n=1}^{\infty} e^{-z(\ln(...

4
votes

1
answer

119
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### Reference request: Uniformly elliptic partial differential operator generates positivity preserving semigroup

I am looking for a reference of the following result:
Let $\Omega\subset \mathbb{R}^n$ be be a bounded domain with smooth boundary. Let
$$A = \sum_{i,j=1}^n \partial_i ( a_{ij} \partial_j) + \sum_{i=1}...

3
votes

0
answers

70
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### Dirichlet-to-Neumann map is analytic

Let $M^n$, $n \geq 2$, be a compact smooth manifold with boundary and let $I \ni t \mapsto g_t$ be an analytic (with respect to t) $1$-parameter family of Riemannian metrics on $M$. For each $t \in I$,...

3
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0
answers

74
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### Embedding theorems for Dini continuous functions

Are there embedding theorems for the space of Dini continuous functions on a Euclidean domain, or even just on an interval? Ideally, I am looking for something like the classical Morrey inequalities ...

0
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1
answer

107
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### Decompositions of $\partial_i$ to the radial direction and rotations in higher dimensions

We know in dimension $3$,
\begin{align}
\partial_{i}= \frac{x_i}{r} \partial_{r} - \varepsilon_{ijk} \frac{x^j}{r} \frac{R^k}{r} ,
\end{align}
where $\varepsilon_{ijk}$ are Levi-Civita symbols ...

2
votes

0
answers

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### $H^s$-mild solution for Navier–Stokes : why do we restrict attention to the function spaces "without Fourier zero mode"? (Related to Terence Tao blog)

This question has been triggered by the Definition 32 and Remark 33 in the blog of Terence Tao.
There, every function space is restricted to ones without the Fourier zeroth mode. And the Remark 33 ...

1
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0
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53
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### Solution to hyperbolic linear second order PDE

I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...

2
votes

1
answer

138
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### A question on biharmonic functions

Let $w$ be a function $\mathbb R^n\to \mathbb R$ with the following properties:
$w$ is globally $\alpha$-Hölder continuous, $\alpha \in (0,1)$;
$w$ is biharmonic on $\{w>0\}$;
$w$ is subharmonic ...

2
votes

0
answers

114
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### Uniqueness of the solution to systems of first-order linear PDEs

Context:
Let $\Omega \subset \mathbb{R}^p$ be an domain.
For functions $A_{jk}^i : \Omega \to \mathbb{R}$ and $B_k^i : \Omega \to \mathbb{R}$ with some regularity, I am interested in the following ...

4
votes

2
answers

246
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### Regularity of solution of $(-\Delta + w)f = 0$

I am studying the following Schrödinger equation:
$$(-\Delta + w)f = 0$$
which represents a quantum state with zero energy. Here $w$ and $f$ are defined on $\mathbb{R}^{3}$. For simplicity, let us ...

0
votes

0
answers

63
views

### What does the big O notation mean in this variation of Wasserstein distance?

I am reading section 13.2 in the book Optimal Mass Transport on Euclidean Spaces by Francesco Maggi.
Given $\varepsilon>0$, a vector field $\mathbf{u} \in C_c^{\infty}\left(\mathbb{R}^n ; \mathbb{...

2
votes

1
answer

115
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### A priori estimates to $u_t - \Delta u = u^2$ [closed]

My research is now considering the a priori estimates on the equation
$$
\begin{cases}u_t - \Delta u = u \min(u,c) \\
u(0,y) = u(1,y)\\
u_x(0,y) = u_x(1,y)\\
\partial_n u(x,0) = \partial_n u(x,1) = 0
\...

1
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0
answers

61
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### Time reversal of infinite-dimensional SDE

Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...

1
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0
answers

284
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### Solutions of a Gauss–Codazzi-like system of nonlinear PDEs

Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$.
$$
\begin{cases}
\tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...