# 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.

2,962
questions

**4**

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21 views

### Does the 'reproducing kernel formula' for a bounded open set $U$ define an equivalent norm on the Sobolev space $H^1_0(U)$

We refer to the 'reproducing convolution formula with a kernel' for an open bounded domain $U$ of $R^n$, $n \geq 2$ discussed in the paper of G. Talenti (Annali de Matematica, Dec 1976) on Best ...

**1**

vote

**0**answers

44 views

### Examples/applications of parabolic PDEs that are not posed on domains or manifolds

Are there any examples of parabolic PDEs
$$u' - Au = f$$
posed in a Gelfand triple setting $V \subset H \subset V^*$ with $V$ and $H$ chosen NOT as spaces of functions (or distributions) over a domain ...

**0**

votes

**0**answers

36 views

### Two types of limits of viscosity solutions

I actually posted this on math.stackexchange but it wasn't getting responses even after a bounty. I thought maybe it is too specialized so I'll post it here. I'm currently reading the user's guide to ...

**3**

votes

**0**answers

54 views

### Elliptic equations in semi-infinite strips

Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Let $g=(g_{jk})_{j,k=0}^n$ be a Riemannian metric on $\mathbb R^+\times \Omega$ with smooth bounded components. Is there a good ...

**3**

votes

**0**answers

56 views

### Green operator of elliptic differential operator and radius of convergence

Let $E \to X$ be a hermitian vector bundle over a compact Kähler manifold and let $L$ be a self-adjoint elliptic linear differential operator on $E$. Suppose that $E \to X$ and $L$ are real-analytic. ...

**2**

votes

**1**answer

83 views

### An inequality involving fractional Laplacian

I have to prove that for $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n)$, (i.e. $u$ is a Schwartz function):
$$ |(-\Delta)^su(x)|\leq c_{n,s}|x|^{-n-2s},\quad\forall x\in\mathbb{R}^n\setminus B_1(0), $$
...

**5**

votes

**1**answer

78 views

### Examples of applications of hyperbolic conservation laws

I am giving a talk in front of my applied PDE research group on hyperbolic conservation laws, the most basic form of which is the PDE $$ u_t + f(u)_x = 0 $$ where $u$ is the conserved quantity and $f$ ...

**2**

votes

**1**answer

89 views

### Positive subharmonic functions with constant integral blowing up at boundary

Say, we're given smooth functions $f_n$, $n=1,2,3,...$ defined on a smooth bounded domain $\Omega\subset\mathbb{R}^d$ satisfying
$\Delta f_n\ge 0$ (subharmonic)
$f_n\ge 0$
$\int_\Omega f_n=I>0$ ...

**2**

votes

**0**answers

31 views

### Mixed boundary value problems for Heat equation

This might be a very simple question, but basically I am looking for a good reference for studying the heat equation on Riemannian manifolds with boundary, specifically when data is put on lateral ...

**1**

vote

**0**answers

37 views

### Well-posedness for a wave equation with degenerate coefficients

Let $\Omega$ be a bounded domain with smooth boundary and consider the following initial boundary value problem:
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
t\partial_t(t\partial_t u)-\...

**4**

votes

**0**answers

80 views

### Auxiliary spaces/conditions for orbital stability of traveling waves

In the context of orbital stability, probably one of the most used theorem to show the orbital stability of traveling waves is the one from Grillakis-Shatah-Strauss "Stability theory of solitary ...

**3**

votes

**0**answers

50 views

### Strong maximum principle for a PDE with coefficient in $L^1$

Let $U$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$. Set $N = \frac{2n}{n-2}$. I am interested in the following equation:
$$
-\Delta \phi + R \phi + \phi^{N-1} = 0
$$
...

**4**

votes

**1**answer

266 views

### Gradient of a function defined on a Riemannian-manifold

If I have a smooth positive scalar function $h$ defined on a 2-dimensional manifold $M$, then $h:M\rightarrow (0, \infty)$, where the metric of $M$ is $g=\frac{dx^2+dy^2}{y^2}$.
$h$ must satisfy the ...

**1**

vote

**0**answers

17 views

### Well-posedness of hyperbolic system with constant coefficients in finite domains

I'm studying the PDE
$$
\frac{\partial u}{\partial t} + A_x\frac{\partial u}{\partial x} + A_y\frac{\partial u}{\partial y} + A_z\frac{\partial u}{\partial z} = 0
$$
with $A_x, A_y, A_z$ being ...

**3**

votes

**1**answer

131 views

### Discovery of norm in PDE

We have seen so many norms we need for PDE. For example, for elliptic PDE, we require a continuous version of $C^k$, i.e. $C^{k,\alpha}$. Roughly speaking, under appropriate norm, we could capture the ...

**2**

votes

**0**answers

74 views

### Lions/diPerna type commutator estimates for differential operator in Fokker-Planck type equation

I have a question about a particular commutator estimate as it occurs in the study of Fokker-Planck equations with low regularity data, see e.g. [1,2].
Denote by $\rho_\varepsilon$ some usual ...

**6**

votes

**2**answers

152 views

### What part of the subject of model reduction of parametrized PDE is directly related to machine learning?

I'm planning to apply for a postdoc position where the mentor claims to be doing research in combining two very different fields: partial differential equation (PDE) and machine learning (ML), but her ...

**5**

votes

**2**answers

539 views

### Arzelà-Ascoli theorem and Hölder spaces

Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f_i\}$ be a sequence of functions bounded in the Hölder norm $C^{k,\alpha}(B)$ for a given integer $k\geq 0$ and $\alpha\in (0,1)$.
Does there exist ...

**2**

votes

**0**answers

95 views

### Does the minimal surface system in the plane have the weak unique continuation property?

Let $\Omega \subset \mathbf{R}^2$ be a domain in the plane and suppose that $u : \Omega \to \mathbf{R}^k$ is a smooth function for which the graph of $u$ is a smooth minimal surface in $\Omega \times \...

**2**

votes

**0**answers

213 views

### Singularity of L^1-solutions to elliptic PDEs on the puntured ball

Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^n$. Then it is true that if $u\in L^1(\mathbb{B})$ such that $\Delta(u)=0$ on $\mathbb{B}\backslash\{0\}$, then $\Delta(u)$, as a distribution on $\...

**2**

votes

**1**answer

104 views

### Unique solution of a 1-D ODE with a bounded positive right-hand-side

Consider the initial value problem $$\dot x(t) = F(t,x), \quad t \in (0,T)$$ with given initial datum $$x(0) = x_0 \in \mathbb R.$$ More precisely we consider the integral equation $$x(t)=x(0)+\int_0^...

**-1**

votes

**1**answer

95 views

### Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]

Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds?
$$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$

**6**

votes

**1**answer

90 views

### Different ways to prove $L^p$-estimates for the heat equation

Let $p \in (1,\infty)$. We are interested in strong $L^p$-solutions to the heat equation in $\mathbb{R}^n$.
$$
\begin{cases}
\partial_t u = \Delta u + f \\
u(0) = 0.
\end{cases}
$$
It is well-...

**1**

vote

**1**answer

118 views

### Band limited initial data : regularity for Navier–Stokes equation defined on a torus $\mathbb{T}^m$

Consider the Navier–Stokes equation and the Euler equation defined on a torus (periodic solutions).
Let the dimensionality of the space $\mathbb{T}^m$ be $m\ge 3$.
Link to the problem (paper "...

**4**

votes

**1**answer

138 views

### Is a specific product function orthogonal to all harmonic functions

Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...

**5**

votes

**1**answer

154 views

### First order PDE in complex variables?

Consider the equation
$$f'(x)+ g(x)f(x)=0$$
This equation is an ODE and has a solution $$ f(x)=C e^{ \int_1^x g(x) \ dx}.$$
Similarly, we can look at complex variables and consider the equation and ...

**0**

votes

**0**answers

68 views

### Equation $u_t - u_{tx} - u_{xx} = 0$

Consider the following heat equation with a perturbation given by a second order mixed derivative:
$$u_t - u_{tx} - u_{xx} = 0$$
Does this equation have a name? How can one prove a wellposedness ...

**13**

votes

**2**answers

804 views

### Reference request: the theory of currents

I am a graduate student and want to study the theory of currents. What is a good reference for a beginner?
I should be familiar with the theory of distributions or generalized functions on $\mathbb R^...

**3**

votes

**0**answers

79 views

### Determining what happens to the spectrum of Schrödinger operator as boundary condition changes

I recently came across a problem in research, and I'm asking about it here after trying in math stack exchange with no luck.
Suppose I have a metric graph $G$ (or even a closed interval, to make ...

**-3**

votes

**1**answer

142 views

### Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$

The Euler equations are given as $$ \pmb{u}_t +\pmb{u}\cdot D\pmb{u} = Dp$$ $$div\mbox{ }\pmb{u} = 0$$
Where $$u = [u_1,u_2,\ldots u_n]^T$$
Now I want to rewrite these same equations but with a new ...

**2**

votes

**1**answer

147 views

### Prove convergence of whole sequence $f_n$ of solutions to a differential problem to a limit $f$ (without uniqueness assumptions)

Let $\{f_n\}_n \subset C^\infty \cap L^2(\mathbb R^N)$ be a sequence of functions that solves a linear differential equation $F_n(f_n, \nabla f_n) = 0$. Suppose that there exists a subsequence $n_k$ ...

**4**

votes

**0**answers

99 views

### Definition of homogeneous Sobolev spaces

As we know the inhomogeneous Sobolev space (we only consider $s>0$)
$${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} ...

**1**

vote

**0**answers

76 views

### A Riemann Hilbert problem on the unit square

Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$.
Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...

**2**

votes

**0**answers

113 views

### Mixed partial derivatives of planar functions converging to delta distribution

Given a sequence $(f_k)_{k\in\mathbb{N}}\subset C^2(\mathbb{R}^2)$ of strictly positive functions $f_k\equiv f_k(x,y)$ with $\|f_k(x,\cdot)\|_{L^1}=1$ for all $x\in\mathbb{R}$ and such that for each $...

**0**

votes

**0**answers

45 views

### How to prove this integral inequality in a 2-D region?

Let $\Omega$ be a 2D region. Now we have a partial differential equation system describing the characteristics of the region:
\begin{align*} \nabla \cdot (h_0^3 P_0 \nabla P_0) &= 0 \\ \nabla \...

**4**

votes

**1**answer

183 views

### Some question about the spectral function of Laplace operator on $\mathbb{R}^n$

I am studying the paper of Seeley, R., A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of (R^3), Adv. Math. 29, 244-269 (1978). ZBL0382.35043. There are some ...

**2**

votes

**1**answer

181 views

### Elliptic operators and Leibniz rule

Let $M$ be a manifold. Does it necessarily admit an elliptic operator on $C^{\infty}(M)$ which satisfy Leibniz rule?
Let $M$ be a symplectic manifold with the standard Poisson structure on $C^{\...

**2**

votes

**1**answer

146 views

### Reconstructing the metric on $CP^2$ with special one forms

I know that $(z_1,z_2)$ are the affine\inhomogeneous coordinates on the complex projective space $CP^2$. Now I have four one forms $(Y_1,Y_2, Y_3, Y_4)$. I want to rewrite the Fubini Study metric on $...

**4**

votes

**1**answer

84 views

### Boundedness of Riesz potential on Hardy space

I encounter the following claim in one paper:
If $(-\Delta)^{\frac14}u\in L^{2,\infty}(\mathbb{R})$, then $u\in BMO(\mathbb{R})$. Equivalently in its dual version, if $u\in \mathcal{H}^1(\mathbb{R})$,...

**3**

votes

**0**answers

70 views

### Estimate in vanishing viscosity for the difference $\Vert u^\epsilon - u^\eta \Vert_{L^2(\mathbb R^N)} $

Consider the following advection-diffusion equation
$$
\begin{cases}
u^\epsilon_t + f(u^\epsilon)_x = \epsilon \Delta u^\epsilon\\
u^\epsilon(0,\cdot) = u_0,
\end{cases}
$$
How can one prove an ...

**2**

votes

**3**answers

220 views

### How to show continuity and monotonicity of solutions to this parametrized equation?

Let $1 \le p <2$ be a parameter. Consider the equation
$$
\frac{2^{p/2} (1-\sqrt{s})^p-1}{\sqrt{s}}=-2^{p/2-1}p(1-\sqrt{s})^{p-1}. \tag{1}
$$
I am rather certain that for each $1 \le p <2$, ...

**1**

vote

**0**answers

66 views

### Perturbation in the equation $u_t=\epsilon Pu$, where $P$ is an elliptic partial differential operator

Let $Pu=\sum_{ij} \partial_j(a_{ij}(x) \partial_{i} u)$ be an elliptic operator. Consider the equation
$$
(u=u_\epsilon)\\
\partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\
u(0,x)=u_0(...

**5**

votes

**1**answer

151 views

### Showing integrability of a locally integrable function on a bounded domain under some additional assumptions

Suppose $\Omega\subset \mathbb{R}^3$ is a smooth and bounded domain, and $f:\Omega\to[0,\infty]$ is a given function which is finite almost everywhere and satisfies
Assumption A: For all $g\in C_0^1(\...

**13**

votes

**0**answers

194 views

### A Green's function for the Laplacian on k-forms

Let $X$ be a compact, oriented, Riemannian $n$-fold. Then we have a Laplacian operator $\Delta = d d^{\ast} + d^{\ast} d$ from $\Omega^k(X)$ to itself. We have the Hodge decomposition $\Omega^k(X) = \...

**2**

votes

**0**answers

102 views

### Green's function for Robin boundary condition

Let $\Omega$ be a bounded subset in $\mathbb{R}^n$, $n \ge 3$, with smooth boundary $\partial \Omega$. Assume given $a^{ij} \in W^{2, p}(\Omega)$ with $a^{ij} = a^{ji}$, $f \in L^p(\Omega)$ and $g \in ...

**7**

votes

**0**answers

118 views

### Gromov's compactness theorem via Sacks-Uhlenbeck and Schoen-Uhlenbeck

Gromov's compactness theorem for pseudo-holomorphic curves, in section 1.5 of "Pseudoholomorphic curves in symplectic manifolds," is very well known. I'm aware of the following two proofs:
...

**4**

votes

**1**answer

171 views

### Finding super(sub)-harmonic functions for an elliptic operator

I am looking for a super(sub) harmonic function for an elliptic operator.
Let $n$ be a positive integer. We denote by $(\cdot,\cdot)$ and $|\cdot|$ the standard inner product and norm on $\mathbb{R}^n$...

**3**

votes

**0**answers

49 views

### Boundedness of Calderon-Zygmund type operator

I am trying to prove the following fact.
Suppose $\varphi\in C_c^\infty(\mathbb{R}^1)$. Define
$$T(u)(x):=P.V.\int_{\mathbb{R}^1}\frac{\varphi(x)-\varphi(y)}{|x-y|^{\frac32}}u(y)dy$$
where P.V. means ...

**0**

votes

**0**answers

49 views

### Integral inequality with Fractional Laplacian

Is the following inequality true
$$
\int_{B_1(0)} f(x) (-\Delta)^\alpha u(x) dx - \frac{1}{|B_1(0)|}\int_{B_1(0)}(-\Delta)^\alpha u(x) dx \cdot \int_{B_1(0)}f(x) dx \ge 0
$$
for a strictly convex $f:\...

**2**

votes

**0**answers

35 views

### Fractional Laplacian and convolution $(-\Delta)^\alpha (u \ast \eta_\epsilon) = (-\Delta)^\alpha u \ast \eta_\epsilon$?

For $u \in L^\infty(\mathbb R)$ and $\eta_\epsilon$ mollifier, it is well-known that for the (distributional) derivative it holds that $(u \ast \eta_\epsilon)' = u'\ast \eta_\epsilon$.
Is it also true ...