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
votes
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
2answers
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
2answers
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
0answers
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
0answers
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
1answer
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 $$x(t)=x(0)+\int_0^...
-1
votes
1answer
95 views
Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]
Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds?
$$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$
6
votes
1answer
90 views
Different ways to prove $L^p$-estimates for the heat equation
Let $p \in (1,\infty)$. We are interested in strong $L^p$-solutions to the heat equation in $\mathbb{R}^n$.
$$
\begin{cases}
\partial_t u = \Delta u + f \\
u(0) = 0.
\end{cases}
$$
It is well-...
1
vote
1answer
118 views
Band limited initial data : regularity for Navier–Stokes equation defined on a torus $\mathbb{T}^m$
Consider the Navier–Stokes equation and the Euler equation defined on a torus (periodic solutions).
Let the dimensionality of the space $\mathbb{T}^m$ be $m\ge 3$.
Link to the problem (paper "...
4
votes
1answer
138 views
Is a specific product function orthogonal to all harmonic functions
Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...
5
votes
1answer
154 views
First order PDE in complex variables?
Consider the equation
$$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}.$$
Similarly, we can look at complex variables and consider the equation and ...
0
votes
0answers
68 views
Equation $u_t - u_{tx} - u_{xx} = 0$
Consider the following heat equation with a perturbation given by a second order mixed derivative:
$$u_t - u_{tx} - u_{xx} = 0$$
Does this equation have a name? How can one prove a wellposedness ...
13
votes
2answers
804 views
Reference request: the theory of currents
I am a graduate student and want to study the theory of currents. What is a good reference for a beginner?
I should be familiar with the theory of distributions or generalized functions on $\mathbb R^...
3
votes
0answers
79 views
Determining what happens to the spectrum of Schrödinger operator as boundary condition changes
I recently came across a problem in research, and I'm asking about it here after trying in math stack exchange with no luck.
Suppose I have a metric graph $G$ (or even a closed interval, to make ...
-3
votes
1answer
142 views
Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$
The Euler equations are given as $$ \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$$
Now I want to rewrite these same equations but with a new ...
2
votes
1answer
147 views
Prove convergence of whole sequence $f_n$ of solutions to a differential problem to a limit $f$ (without uniqueness assumptions)
Let $\{f_n\}_n \subset C^\infty \cap L^2(\mathbb R^N)$ be a sequence of functions that solves a linear differential equation $F_n(f_n, \nabla f_n) = 0$. Suppose that there exists a subsequence $n_k$ ...
4
votes
0answers
99 views
Definition of homogeneous Sobolev spaces
As we know the inhomogeneous Sobolev space (we only consider $s>0$)
$${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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
3answers
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
0answers
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
$$
(u=u_\epsilon)\\
\partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\
u(0,x)=u_0(...
5
votes
1answer
151 views
Showing integrability of a locally integrable function on a bounded domain under some additional assumptions
Suppose $\Omega\subset \mathbb{R}^3$ is a smooth and bounded domain, and $f:\Omega\to[0,\infty]$ is a given function which is finite almost everywhere and satisfies
Assumption A: For all $g\in C_0^1(\...
13
votes
0answers
194 views
A Green's function for the Laplacian on k-forms
Let $X$ be a compact, oriented, Riemannian $n$-fold. Then we have a Laplacian operator $\Delta = d d^{\ast} + d^{\ast} d$ from $\Omega^k(X)$ to itself. We have the Hodge decomposition $\Omega^k(X) = \...
2
votes
0answers
102 views
Green's function for Robin boundary condition
Let $\Omega$ be a bounded subset in $\mathbb{R}^n$, $n \ge 3$, with smooth boundary $\partial \Omega$. Assume given $a^{ij} \in W^{2, p}(\Omega)$ with $a^{ij} = a^{ji}$, $f \in L^p(\Omega)$ and $g \in ...
7
votes
0answers
118 views
Gromov's compactness theorem via Sacks-Uhlenbeck and Schoen-Uhlenbeck
Gromov's compactness theorem for pseudo-holomorphic curves, in section 1.5 of "Pseudoholomorphic curves in symplectic manifolds," is very well known. I'm aware of the following two proofs:
...
4
votes
1answer
171 views
Finding super(sub)-harmonic functions for an elliptic operator
I am looking for a super(sub) harmonic function for an elliptic operator.
Let $n$ be a positive integer. We denote by $(\cdot,\cdot)$ and $|\cdot|$ the standard inner product and norm on $\mathbb{R}^n$...
3
votes
0answers
49 views
Boundedness of Calderon-Zygmund type operator
I am trying to prove the following fact.
Suppose $\varphi\in C_c^\infty(\mathbb{R}^1)$. Define
$$T(u)(x):=P.V.\int_{\mathbb{R}^1}\frac{\varphi(x)-\varphi(y)}{|x-y|^{\frac32}}u(y)dy$$
where P.V. means ...
0
votes
0answers
49 views
Integral inequality with Fractional Laplacian
Is the following inequality true
$$
\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
0answers
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 ...
