# 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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### PDE involving nonlocal integral

Let $\textbf{F}$ be smooth non-zero vector field over $\mathbb{R}$. Does there exist a vector field $\textbf{v}:\mathbb{R}^3\times [0,\infty)\rightarrow \mathbb{R}^3$ such that
$$\textbf{v} - \int_{\...

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### Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces

The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is,
$[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$.
Now I ...

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### Dirichlet-to-Neumann estimate for minimal graphs

Let $\Omega \subset \mathbf{R}^n$ be a smooth, bounded domain. The Dirichlet problem for the minimal surface equation
\begin{equation}
(1 + \lvert Du \rvert^2) \Delta u - D_i u D_j u D_{ij} u = 0
\end{...

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### Why does failure of boundedness of this operator for $p<q$ implies its failure for $p>q^{\prime}$?

I am reading the paper "P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (1981), pp. 47-55" where $L^p-L^p$ boundedness of the operator
$...

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### A strong maximum principle for varifolds of arbitrary codimension

Let $M$ be an $n$-dimensional Riemannian manifold and $N$ a hypersurface in $M$. Let $p \in N$ and $\kappa_1 \leq \cdots \leq \kappa_{n-1}$ be the principal curvatures of $N$ at $p$ with respect to a ...

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### Effective way for studying PDEs

I am new to this stack, and thought my question belongs here.
I am a first-year graduate student currently taking my second course on PDEs (basically covering Evans ch. 5 and onwards). I am planning ...

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### Maximal domain of an unbounded linear operator in a weighted Hilbert-space

Let's consider the following (unbounded) linear operator. (So called transport operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...

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### Are Neumann Laplacian eigenfunctions in $C(\overline{\Omega})$?

Consider that $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ (in the distributional sense) such that for some $\lambda>0$ we have that:
$$\begin{cases} \Delta u(x)=\lambda u(x), & x\in\Omega\...

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### Express $Q_0 u + Q_1 \Delta u + Q_2 \Delta^2 u + Q_3 \Delta^3 u=0$ as a conservation law for $u(\vec x, t) : \mathbb R \times \mathbb R \to \mathbb R$

In the study of certain PDEs, it is beneficial to write them as a conservation law so that the energy of the system may be defined. More facts such as causality can be proven by considering surface ...

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### PDE involving curl

Let $G:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be smooth vector field over $\mathbb{R}^3$. For which vector fields $F:\mathbb{R}^3\rightarrow\mathbb{R}^3$ does the PDE
$$\dfrac{\partial}{\partial t}\...

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### Looking for English version of a paper of Jean Ginibre

I am in serious need of an English translation for the following paper:
Séminaire Bourbaki by Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires
périodiques en variables d’espace, d'après ...

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### Can I characterize functions (in 2D), which will have compactly supported/support contained Poisson solution

I have the problem of solving Poisson equation in 2D
$$\Delta u = f$$
Let's say for a moment I want to solve it on $\mathbb{R}^2$, for $f(x,y), x\in \mathbb{R}, y\in \mathbb{R}$.
I know however that ...

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### Gradient of the internal energy defined on the Wasserstein space $\mathcal P_2 (\mathbb R^n)$

I'm reading the derivation of gradient of the internal energy functional defined on the Wasserstein space $\mathcal P_2 (\mathbb R^n)$ from Section 2 of Lecture 18 from the book Lectures on Optimal ...

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### Maximal regularity heat equation

Considering the heat equation on the flat torus $\mathbf{T}^d$, we have the maximal regularity estimate
\begin{align*}
\forall \varphi\in\mathscr{D}(\mathbf{R}\times\mathbf{T}^d),\quad \|\Delta \...

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### Is the $n/2$-th heat kernel coefficient topological?

I have asked the same question on math.SE, without much success so I'm trying my luck here too.
Let $M$ be an $n$-dimensional manifold, with $n$ even and consider the heat kernel of the Laplacian on $...

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### What does the continuity equation $s=-\nabla \cdot(\rho\, \nabla u)$ mean?

I'm reading about gradient flows in Wasserstein space in this note.
Let $\mathcal{P}_2 (\mathbf{R}^d)$ be the set of probability measures with finite second moment. In [Ott01] the tangent space $T_\...

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+50

### Weighted Poincare inequality for $p$ harmonic functions

Suppose $u$ is $p$-harmonic, i.e., it solve $-\operatorname{div} |\nabla u|^{p-2} \,\nabla u = 0$ where $1<p<\infty$. Then is the following inequality true?
$$
\int_{S_1} (u-k)^2|\nabla u|^{p-2}...

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### Generalized functions in infinite dimensions

What theories are there for generalized functions (distributions) in infinite dimensions?
In particular, suppose your "infinite dimensional manifold" is $\mathfrak{M}:=C^\infty(S^1)$. Is ...

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### Reference and hint for L^p estimates of the gradient of solutions to parabolic equation in divergence form

Considering a weak solution $u\in L^2(0,1;H^1(B_1))$ with $\partial_t u \in L^2(0,1;H^{-1}(B_1))$ to
$$\partial_t u-\operatorname{div}(A(x,t)\nabla u)=f+\operatorname{div}(F) \hspace0.5cm \text{in} \...

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### A model of pillows

(The same system with slightly different questions has been asked in MSE.)
Let $\Omega\subset \mathbb{R}^2$ be some simply connected planar domain. We seek for a mapping $\mathbf{r}:\Omega\rightarrow \...

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### How to prove the convergence of the functions with the norm above the critical point of Kondrakov?

I have a sequence of smooth functions with norm 1 in $L_q$ space. I need to prove this sequence strongly converges to some function. But I lack the compactness theorem since, $q$ is above the critical ...

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### Optimal assumption on H^2 regularity

In many text book (Evans, Gilbarg-Trudinger for example) there is a classical result of interior regularity for weak solutions to a elliptic divergence problem $\rm{div}(A(x)u)=f$ in $\Omega\subset\...

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### Pressure integrated by parts in finite element method

Most FEM texts or tutorials apply FEMs on diffusion equations where the 2nd spatial derivative is integrated by parts during weak formulation. For convection diffusion equations, there is a first ...

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### $L^p-L^q$ boundedness of this simple singular oscillatory integral operator

Let $0<\alpha<1$ and define
$$Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy.$$
The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of
$Hf(x):=\int \frac{...

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### $L^2$-extension of solutions of elliptic equations

I'm dealing with a problem characterizing $L^2$-extension of solutions of elliptic equations. The model is as follows. Suppose $L$ is an elliptic operator over $\mathbb{R}^n$, and we assume the vector ...

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### A non-linear PDE $v^2v_t=v_{xx}v-v_{x}^2$

PS : Indeed, there is a typo in my equation. Thanks to Zachary's observation.
Consider a PDE for $v: [0,1]^2\to (-\infty,0]$ satisfying
$$v_t(t,x) = \frac{v_{xx}(t,x)v(t,x)-v_{x}(t,x)^2}{v(t,x)^2},\...

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### Linear elliptic equation

Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...

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### Weak solution of elliptic differential equation of divergence type

Assume that $u\in W^{1,2}(B^n,R)$ is a weak solution to the elliptic pde of type $$\sum_{i,j=1}^n\partial_j \left(a_{ij}(x) \partial_i u(x)\right)=f\in L^p(B^n),$$ where $n/2<p<n$, and $A=(a_{ij}...

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### Approximation of viscosity subsolution

Let $u: \Omega \to \mathbf{R}$ be a $C^{0,\alpha}$ function, with $\alpha \in (0,1]$, defined on a bounded, open domain $\Omega$. Suppose that $u$ is a viscosity subsolution of the equation $\Delta U =...

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### Hyperbolic system of PDEs with elliptic-like boundary contions

Let $\Omega_1$ and $\Omega_2$ be (simply connected) domains on $\mathbb{R}^2$, with coordinates $(x,y)$ and $(X,Y)$ respectively. Given a (smooth) function $Z(X,Y)$ such that $Z\left(\partial \Omega_2 ...

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### How to implement Newton iteration in Navier–Stokes equation

Suppose the nonlinear Navier–Stokes equation
$$
\begin{cases}
(\mathbf{u} \cdot \nabla) \mathbf{u}-\nabla \cdot \mathbb{T}(\mathbf{u}, p)=\mathbf{f} & \text { in } \Omega \\
\nabla \cdot \mathbf{u}...

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### The attractive 'force' between phase interfaces in the Allen-Cahn model

The heuristic explanation of the behavior of phase transition in the Allen–Cahn model describes two 'forces' at play:
the curvature of the phase interfaces—they each 'want to' minimize length;
and an ...

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### Are two notions of generalized solution of Monge-Ampere equation equivalent?

Let $\Omega\subset \mathbb{C}^n$ be an open subset. Let $u\colon \Omega\to \mathbb{R}$ be a continuous plurisubharmonic (psh) function. The theorem of Chern-Levine-Nirenberg defines a non-negative ...

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### What is an analogous version of the Ornstein–Uhlenbeck process on Riemannian manifolds?

Recall that the Ornstein–Uhlenbeck (OU) process in $\mathbb{R}^d$ is defined by the following SDE,
$$
d Z_t=\frac{-1}{2} Z_t d t+d W_t, \quad t \geqslant 0
$$
where $\left(W_t\right)_{t \geqslant 0}$ ...

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### A possible characterization of subharmonic functions

Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function.
If $u$ is subharmonic then for any point $x\in \Omega$ and any $C^2$-...

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### A few questions on the paper named "On a Sobolev inequality with remainder terms"

I was reading a paper of Lu-Wei named "On a Sobolev inequality with remainder terms" (link at AMS site).
and I have quite a few questions regarding that.
(1) In page 78 they got a system of ...

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### Gradient descent relaxation dynamics of a Euler-Lagrange equation

I want to minimize the functional
$$
F=\int{L(u)}dx,
$$
where $L= u_x^2-u^2$ is the Lagrangian function of the functional. Even if its Euler-Lagrange equation is easily found and solved, I want to try ...

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### Is the Fourier multiplier $\mathcal F(G(-\hbar^2 \Delta)\psi)(p) = G(|p|^2)\hat \psi(p)$ justified for any real function $G$?

I am confused about a claim asserted in the paper "Higher Order Schrodinger Equations" published in IOP Science. The authors claim that a Fourier multiplier identity
$$
\mathcal F(G(-\hbar^2 ...

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### Existence of solution to linear inhomogeneous first order PDEs systems

Maybe I am asking a triviality. If that is the case, please let me know and I will close the question. I have searched a lot but I didn't find an adequate and forceful response.
For $i=1,\ldots, r$, ...

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### Find $f:\mathbb R^3 \to \mathbb R$ s.t. $(f - 1)\Delta f + f^2 = 0$?

I wish to solve the following nonlinear PDE that I derived in statistical physics. (I was curious if I could include higher order terms into a model for heat transfer described in a homework problem.) ...

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### Estimate of a product of functions where $1/p + 1/q >1$

I have the following problem. I need to estimate the following quantity:
$$|\nabla|^{-1}A u \nabla u \label{1}\tag{$*$}$$
I know that $A\in L^6(B_R)$, $u \in L^{14/5}\cap L^2(B_R)$ and $\nabla u \in L^...

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### Pohozaev identity for linear equations

For $-\Delta u =0$, the Pohozaev identity on say $B_1$ says
$$ \int_{S_1} |u_T|^2 \,d\sigma = \int_{S_1} |u_N|^2 \,d\sigma + (n-2) \int_{B_1} |\nabla u|^2 \ dx$$
Here $u_T$ are the tangential ...

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### Uncertainty principles and Anderson localization principles

The motivation of the question comes from the paper "Some harmonic analysis questions suggested by Anderson-Bernoulli models. Geom. Funct. Anal. 8 (1998), no. 5, 932–964" by Shubin, Vakilian ...

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### Two doubts in the paper of Brezis Merle in blow up analysis of the equation $-\Delta u=Ve^u$

I was going through the paper of Brezis and Merle [1]. In theorem 3 step 2 it's written that the boundedness of $f_n=V_ne^{u_n}$ in $L_{loc}^p(\Omega)$ implies $\mu\in L^1(\Omega)\cap L_{loc}^p(\...

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### Role of verification theorems in stochastic optimal control?

I am currently working on the optimal control of certain classes of stochastic processes and I have difficulties understanding the roles of verification theorems.
My problem is the following: I am not ...

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### Generalizing a formula with distributions — Distributional Radon transform

I will try to describe the problem, it will necessarily be incomplete, so please if you have questions or remarks to make it more clear do not hesitate to leave them in comments.
The problem
Let $a$ ...

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### Using a maximum principle to deduce regularity

Suppose $\Omega \subset \mathbb{R}$ is an bounded domain and that $u \in C(0,T; H^{2}) \cap L^{2}(0,T; H^{3})$ where $T >0$.
Consider the PDE on $\Omega \times [0,T]$
$$ \partial_{t}u = a_{1}(x,t) \...

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### Solving PDE $u_t+f(u)u_x=0$ using its physical interpretation

Let's be this PDE:
$\begin{cases}u_t+f(u)u_x=0\\
u(x,0)=\varphi(x) \end{cases}$
and $f\in 1-1$.
I have these thoughts:
We can imagine $x'x$ having sticky particles. As we know $\frac{dx}{dt}=f(u)$. ...

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### Existence theorem of weak solutions of $u_t+f(u)u_x=0$

Consider this PDE:
$\begin{cases}u_t+f(u)u_x=0\\ u(x,0)=\varphi(x)\end{cases}$
Has this PDE weak solutions whatever is $f$ or $\varphi$? I want to find an existence theorem and bibliography about that?...

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### Parabolic PDE: Zero now means zero anytime before

Studying some mathematical models I came across a simple-looking question that I do not know how to handle.
If we have the following problem:
$$\begin{cases} \dfrac{\partial Z}{\partial t}-\Delta Z=aZ-...