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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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1 vote
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73 views

Compute surface Sobolev norm using local coordinate

For a bounded $\Omega\subset \mathbb{R}^n$ with Lipschitz boundary, there are various definitions of fractional Sobolev spaces (a.k.a. Sobolev-Slobodeckij spaces) on $\partial \Omega$, either by using ...
1 vote
0 answers
177 views

Eigenvalues and eigenvectors of non-symmetric elliptic operators

We know that the operator $A=\Delta$ with domain $D(A)=\{u\in W^{2, 2}(\Omega): u=0 \ \ \text{on } \partial\Omega\}$ (say $\Omega$ is a bounded nice domain) has eigenvalues $\lambda_1>\lambda_2\ge \...
2 votes
0 answers
72 views

Maximal Lyapunov exponent of Schrödinger-Newton equation

I am trying to determine the sign of the maximal Lyapunov exponent of the Schrödinger-Newton equation $$ \partial_t \psi(t,\vec{x}) = i\left(a\nabla^2 + \int_{\mathbb{R}^3} \frac{|\psi(t,\vec{y})|^2}{|...
2 votes
0 answers
188 views

Elliptic regularity for a system of PDEs

I am considering a system that can be simplified to the following problem. Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$, and consider the following coupled ...
0 votes
1 answer
148 views

Proof of vanishing viscosity error rate

Consider the initial value problem associated to the parabolic equation $u^\epsilon_t + u^\epsilon_x = \epsilon u^\epsilon_{xx}$ and the corresponding hyperbolic problem $u_t + u_x = 0$. What is a ...
4 votes
0 answers
160 views

An estimate for the Benjamin-Ono equation from T. Tao's well-posedness paper

In https://arxiv.org/abs/math/0307289 (eq. (8)), for a (smooth) solution of the equation $$u_t - uu_x + Hu_{xx} = 0$$ ​ (where $H$ denotes the Hilbert transform) the following estimate is stated (...
3 votes
1 answer
553 views

Lax-Milgram and the existence of solution to parabolic equation

I think it is standard and common to use Lax-Milgram theorem to prove the existence of solution to elliptic equation. However, can we use it to establish the existence of parabolic equation? I do not ...
2 votes
0 answers
138 views

Update on Viskov's paper on random processes, Lagrange inversion, and the Heisenberg–Weyl algebra

"A Random Walk with a Skip-Free Component and the Lagrange Inversion Formula" by Viskov presents connections among Lagrange inversion and measures of random Lévy processes. The freely ...
0 votes
0 answers
60 views

How to prove this estimate involving the Stein Derivative?

Recall the Stein Derivative, $$\mathcal{D}^{s} f(x)=\left(\int \frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2 s}} d y\right)^{1 / 2}.$$ I want to show that, $$\left\|\mathcal{D}^{s}(f g)\right\|_{2} \leq\left\|f \...
8 votes
1 answer
380 views

Lavrentiev phenomenon between $C^1$ and Lipschitz

Does there exist a (onedimensional) integral functional of calculus of variations (with $f$ finite everywhere) $$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt
 $$ such that $$ \inf_{y\in Lip([a,b])}F(y)<\inf_{...
6 votes
1 answer
376 views

Lavrentiev phenomenon between $C^1$ and $C^2$

Does there exist a (onedimensional) functional that exhibits the Lavrentiev phenomenon between $C^1$ and $C^2$ that is $$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt \quad\text{or possibly}\quad F(y)=\int_a^b f(...
5 votes
1 answer
561 views

Dependence of the Hölder exponent in De Giorgi-Nash-Moser

I am curious about the Hölder exponent obtained by the De Giorgi-Nash-Moser theory, as a function of the ellipticity. More precisely: suppose $u$ satisfies weakly $$ D_i(a^{ij}D_ju)=f $$ on the $d$-...
0 votes
1 answer
73 views

Meaning of $K'[\cdot]$ when $K$ is an symmetric Onsager (matrix) operator

My question is from the section 5.2 of the monograph "Entropy Methods for Diffusive Partial Differential Equations" written by Ansgar J$\ddot{\text{u}}$ngel. To be specific, I do not know ...
1 vote
0 answers
159 views

Sobolev interpolation inequality for relatively compact subdomains

I was looking at Nicolaescu's Lectures on the Geometry of Manifolds (3rd edition). In Theorem 10.2.29 he presents (without proof) the following inequality: For $m \geq 1, p \geq 1, 0 < r \leq R$ ...
1 vote
0 answers
79 views

Reference for smoothness of Nemytskii operator on fractional Sobolev spaces

Let $\varphi:\mathbb{R}\to\mathbb{R}$ be smooth and bounded (together with all of its derivatives). Define the operator $$ \big(N_\varphi x\big)(t)=\varphi\big(x(t)\big) $$ for $x\in H^s(T^d)$, the ...
2 votes
0 answers
104 views

Vortex equation on Riemann surface and a similar equation

Let's take a Riemann surface $(X,\omega)$ and a holomorphic line bundle $L$ on it with a hermitian metric $h$ on $L$. $g$ be a real valued smooth function on $X$ and we consider the following two sets ...
1 vote
1 answer
677 views

First derivative of cut off function

I am working on proving the following: Let $\rho(x)= \frac{2}{2+x^2}$, $\theta >1$ (assumed integer here) and $B \subset H^1_{ul}$,(uniformly local Sobolev space), be any subset which is bounded in ...
0 votes
0 answers
70 views

Normal vector to a level set and fractional Laplacian

Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
3 votes
0 answers
68 views

Eigenvalues of an elliptic operator on shrinking domains

This was probably done somewhere 100 times, but I can't find a reference. Assume that we have a bounded star-shaped domain $\Omega\subset \mathbb{R}^n$ with piece-wise smooth boundary and a general ...
2 votes
0 answers
80 views

Solution verification of some PDE with an additional condition

Provided a probability density $\rho:\mathbb R_+\to\mathbb R_+$ (as nice as possible), consider the PDE $$ \begin{cases} \partial_t p = \dfrac{\partial_{xx}p}{2\big(1+m(t)\big)^2} - \partial_x p, &...
2 votes
0 answers
86 views

Best constant for Sobolev-type inequality

I am currently reading a paper from Del Pino/Dolbeault about optimal constants of GNS inequalities (http://capde.cmm.uchile.cl/files/2015/06/pino2002.pdf). The author wants to prove the following GNS ...
2 votes
1 answer
215 views

Reference request for spectral theory of elliptic operators [closed]

I want to learn the spectral theory of linear elliptic operators in bounded and unbounded domains in $R^n$, in particular for Laplacian and Schrodinger operators. Please suggest me some reference. I ...
3 votes
2 answers
130 views

Link between controllability of ODEs and controllability of transport equations

What is the relationship between the controllability of the ODE $$\dot x(t) = v(x) + u(t)$$ using a control $u$ and the controllabilty of the transport equation $$\rho_t(t,x) + \mathrm{div}(v(x) \rho(...
4 votes
1 answer
395 views

Decay of solutions to the wave equation $\ddot\phi(t, x)+\frac{n p}{t}\dot\phi(t,x)-t^{-2p}\Delta\phi(t,x)=0$

For the physical motivation of this question, see my question 669101 on physics StackExchange. The question is this: Let $\hat M=\mathbb R^n$ or $\hat M =(\mathbb R/\mathbb Z)^n$ for some $n\in\...
1 vote
0 answers
120 views

Liouville theorem for an elliptic equation with gradient perturbation

How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation? Let $u \in L^2(\mathbb R^n;\mathbb R)$ be a smooth solution of $$ -\Delta u + v \cdot \nabla u = 0 ...
5 votes
0 answers
546 views

Regularity of solution to Laplacian equation with Neumann data on Lipschitz domain

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$ and let $u\in H^1(\Omega)$ be a weak solution to \begin{equation} \begin{cases} -\Delta u=0 \quad &\mbox{in $\Omega$}\\ \frac{\partial ...
26 votes
3 answers
3k views

Does Ricci flow depend continuously on the initial metric?

Consider a version of Ricci flow for which short time existence and uniqueness are known, e.g. the Ricci flow on a closed manifold. Does the solution $g_t$ for small $t$ depend continuously on the ...
1 vote
0 answers
96 views

BV estimate for conservation law $u_t +( v(x)f(u))_x=0$

Let $u_0 \in BV(\mathbb R)$ and $f:\mathbb R \to \mathbb R$ be Lipschitz. Consider the Cauchy problem $$ \begin{align*} u_t +( v(x)f(u))_x&=0\\ u(0,\cdot) &= u_0 \end{align*} $$ What is the ...
1 vote
0 answers
72 views

Elliptic systems with two dimensions

Let $ \Omega\subset\mathbb{R}^2 $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq 2 $ and $ 1\leq\...
47 votes
3 answers
3k views

Why is the Vandermonde determinant harmonic?

It can be checked that the Vandermonde determinant defined as $$V(\alpha_1, \cdots, \alpha_n) = \prod_{1 \le i < j \le n}(\alpha_i-\alpha_j) $$ is a harmonic function, that is $\Delta V = 0$ where ...
1 vote
0 answers
85 views

Boundary estimates for elliptic systems

Let $ \Omega\subset\mathbb{R}^d $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq d $ and $ 1\leq\...
1 vote
0 answers
133 views

Regularity of solutions to heat equations

Let $d$ denote a positive integer. Let $f$ be a positive function on $\mathbb{R}^d$. We also assume that $f$ is bounded above and below. That is, there exists $C>0$ such that $C^{-1}\le f(x)\le C$, ...
1 vote
0 answers
47 views

Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter

Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example $$ \left\{\begin{aligned} \Delta \Phi &=\sigma f(x, y) \...
5 votes
0 answers
304 views

Similarity in Navier-Stokes equation and convolution in finite abelian groups?

Let $G$ be a finite abelian group, $X = (x_g)_{g \in G}$ be a vector of variables. Set for $g \in G$: $$\tau_g(X) := \frac{1}{|G|} \sum_{\rho \text{ irred. }} \chi_{\rho}(-g) \exp(\sum_{s \in G} \chi_{...
6 votes
0 answers
211 views

Regularity of $|u|^{\alpha}$ when $u$ is Schwartz

Let $0<\alpha<1$. Let $D_x^{\alpha}$ denote the Fourier multiplier given by $\xi\to |\xi|^{\alpha}$. Suppose $u:\mathbb{R}^d\to\mathbb{C}$ is Schwartz (or even just smooth with compact support). ...
5 votes
3 answers
2k views

Book about fluid dynamics

Next Monday, I'll have an interview at Siemens for an internship where I have to know about fluid dynamics/computational fluid dynamics. I'm not a physicist, so does somebody have a suggestion for a ...
1 vote
0 answers
55 views

What is the the "method of ascending" in the study of elliptic systems in dimension two?

I have read a paper of Z. Shen [1]. In the paper the author mentioned we can deal with two-dimensional elliptic systems by adding a dummy variable (the method of ascending) and use the results on the ...
3 votes
1 answer
585 views

Does the following version of the Coifman–Meyer Theorem exist?

Recall the Coifman–Meyer Theorem as stated in Grafakos and Oh - The Kato-Ponce Inequality. Theorem: Let $m \in L^{\infty}\left(\mathbf{R}^{2 n}\right)$ be smooth away from the origin and satisfy $$ \...
6 votes
3 answers
541 views

About the Hausdorff dimension of removable singularities of PDE

There are some interesting phenomenons about removable singularities (or extension problems). In the theory of functions of several complex variables, we know the classical Hartogs theorem: Let $f$ ...
4 votes
1 answer
1k views

Existence of solutions to first-order PDE involving convolution

Let $f(x,\alpha)$ be a smooth function of compact support in $x$. Now, let its $\alpha$-dependence be determined by the following first-order equation, \begin{align} \frac{\partial}{\partial \alpha} ...
7 votes
3 answers
2k views

Collections of examples and counterexamples in (real, complex, functional) analysis, ODEs and PDEs

What books collect examples and counterexamples (or also "solved exercises", for some suitable definition of "exercise") in real analysis, complex analysis, functional analysis, ODEs, PDEs? The ...
1 vote
1 answer
474 views

Laplace equation with integral source terms

I am not specifically asking for a solution, but any reference on any method i could read about would be a big help. This I clarify as I am aware of the fact that MathOverflow only deals with research ...
3 votes
3 answers
383 views

Density of a functional space

Let $D$ be a bounded domain of $\mathbb{R}^n$ with smooth boundary $\partial D$. Is the following subspace dense in space $L^2(D)\times L^2(\partial D)$: $$\{(f,f\rvert_{\partial D}) : f\in C^\infty(\...
1 vote
0 answers
126 views

Estimates for the Benjamin-Ono equation

Consider the Cauchy problem for the Benjamin-Ono equation $$u_t + \frac{1}{2}(u^2)_x + \alpha \mathcal H(u_{xx}) - \beta u_{xx} = 0, \qquad t>0, \ x \in \mathbb R,$$ where $\mathcal H$ is the ...
2 votes
0 answers
73 views

Question about Gidas-Ni-Nirenberg result

Background: So I know that the Euler Lagrange equation associated with the Sobolev inequality takes the following form, $$-\Delta u = u^p$$ where $p=2^*-1$ and here we assume that $u>0$ on $\mathbb{...
2 votes
0 answers
60 views

Decay of solution for linear system with damping

Let us consider the following linear system with damping: $$ \begin{cases} u_t - u_x = -\frac{1}{2} (u+v)\\ v_t + v_x = -\frac{1}{2} (u+v) \end{cases} $$ Let's write the solution as $w=(u,v)$ ...
2 votes
1 answer
101 views

What are partial differential equations with fast reaction terms?

I know $u_t(t,x)=\Delta u^m(t,x),\;\; (t,x)\in (0,\infty)\times \mathbb{R}$ is the fast-diffusion equation when $m\in (0,1).$ But how are PDEs with fast reaction terms defined in general? I also wish ...
1 vote
0 answers
172 views

barrier functions ; harmonic functions; gradient estimates (textbook reference)

Suppose $\Omega$ is a bounded domain in Euclidean space and you can take it as convex as you wish. Is there a good place to look for a bunch of barrier type functions. In particular I am looking for ...
2 votes
1 answer
432 views

Second-order elliptic regularity with rough coefficients

Let $\Omega \subseteq \mathbb R^n$ be a bounded open set with smooth boundary. Let $k\geq 1$, $\alpha\in(0,1)$, $a_{ij},b_i,f \in C^{k,\alpha}(\Omega)$ for $i,j=1,...,n$, and define the operator $$L = ...
22 votes
3 answers
2k views

History of fundamental solutions

I have a few questions on the history of PDE. Who first wrote down the formula for the solution of the Cauchy problem for the heat equation involving the heat kernel? I have seen it called Poisson's ...

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