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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7
votes
3answers
328 views

How to find the associated conservation law from a given symmetry

It is a very well-known fact that any conservation law associated with some given PDE has an associated invariance (by Noether's Theorem). However, it is completely mysterious for me how to compute/...
1
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0answers
73 views

Poisson equation on exterior of a ball

Let $ B_1^c$ denote the compliment of the unit ball centered at the origin in $ R^N$ where $N \ge 3$. I am interested in $ -\Delta u(x)=f(x)$ in $ B_1^c$ with $ u=0$ on $ \partial B_1^c$. In ...
1
vote
0answers
64 views

Commutator estimates for $-(-\Delta)^s$, with $s \in (1,2)$

I'm currently trying to work with the non-local operator given by $$ (-\Delta)^{\frac{s}{2}}f(x)= c_s\text{P.V} \int_{-\infty}^\infty \frac{-f(x+y)-f(x-y)+2f(x)}{|y|^{1+s}} dy, $$ where $f :\mathbb ...
0
votes
0answers
40 views

Fokker-Planck equation with zero diffusion coefficients on the boundaries

I am currently working on a Masters project in which I will be looking at numerical methods for a specific PDE. The PDE that I will be considering is a Fokker-Planck equation on a unit hyper-cube. ...
2
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0answers
79 views

Sufficient conditions for constant solutions

Let $u : \mathbb{R}^N \to \mathbb{R}$ be a smooth negative function that satisfies $$-f(x)u(x) + B(x) = \Delta u(x),~\forall x\in \mathbb{R}^N,$$ where $B(x)$ is a smooth positive function and $f$ is ...
1
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1answer
73 views

Understanding traveling waves as critical points of the constrained energy

I am trying to understand a (straightforward I guess) statement of an old (but outstanding) paper on stability of solitary waves. Let us consider the following functionals: $$ V(u)=\dfrac{1}{2}\int_\...
2
votes
0answers
33 views

Current optimal regularity level for Benjamin-Ono equation

Ionescu and Kenig showed global wellposedness for the Benjamin-Ono equation for all (in particular, low) regularities $s\geq 0$. At $s=0$ they used modified $X^{s,b}$ spaces in order to avoid ...
1
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0answers
29 views

Examples of invariant measures for Hamiltonian PDE

If $X$ is a symplectic space and $H$ is a Hamiltonian on $X$, then we have the non-normalized Gibbs measure $e^{-\lambda H}dm$ for any $\lambda\in\mathbb R$ with $dm$ being the Haar measure on $X$, ...
0
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0answers
30 views

Simplest example of the method of normal forms?

For the KdV equation there is a method of correction terms. It was reinterpreted in a paper by Bourgain as a special case of the "method of normal forms." I am writing this question to seek a simple ...
7
votes
1answer
256 views

How to tell, roughly, which PDE's are interesting to analyse?

How can one tell which PDE's are, roughly speaking, perhaps more interesting to analyse? Physical motivation is one reason. For example, the KdV $u_t+u_{xxx} - 6uu_x=0$ for a function $u:\mathbb R\...
1
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0answers
52 views

Normalized $p(x)-\mathrm{laplacian}$ is uniformly elliptic?

The normalized $p(x)-\mathrm{laplacian}$ is defined by $$-\Delta_{p(x)}^{N} u = -\operatorname{tr}\Big( \big( I + \frac{(p(x)-2)}{|Du|^{2}}Du \otimes Du\big)D^{2}u\Big)=0 ,$$ from now on, two ...
4
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0answers
130 views

A metric $w$ on a Kahler manifold is extremal if and only if the gradient vector field of the scalar curvature is holomorphic

I am trying to understand the calculation in An introduction to Extremal kahler metrics. On the fourth line of page 55 the author calculated that $\int_{M} - 2 S R^{\bar k j} \partial_{j} \partial_{\...
3
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0answers
155 views

Can every diffeomorphism be rescaled into a volume preserving one?

This is a cross-post. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $f:D \to D$ be a diffeomorphism. Does there exist a smooth $h \in C^{\infty}(D)$ such that $h\cdot f$ is an ...
3
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0answers
96 views

Second derivative estimates

I am in big trouble since I don't see how to proceed (I don't need the exact calculation) with the following estimates. In one of his papers, Lin proves the following result: Let's consider a ...
1
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0answers
18 views

Reference request: Transverse parabolic Schauder estimates

Is there a version of the parabolic Schauder estimates for transversely parabolic linear PDE's on a manifold with a Riemannian foliation for functions that are constant on the leaves of the foliation? ...
7
votes
1answer
71 views

Sobolev topology on essentially compactly supported Sobolev-“functions”

The locally convex space of essentially compactly-supported $p$-integrable "functions" $\operatorname{L}_{\mathrm{comp}}^p(\mathbb{R}^d,\mathbb{R})$ is defined as the set $$ \bigcup_{n \in \mathbb{N}} ...
2
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0answers
73 views

Classical singular integral operator

I am working on a problem involved the Biot-Savart law in fluid dynamics. I found a theorem of singular integral which is intimately related to my research. Assume that $K(x)$ is a Calderon-Zygmund ...
2
votes
1answer
72 views

Convergence of semi convex functions

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
2
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0answers
40 views

Second derivative estimates for a subsolution of linear elliptic equation

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
2
votes
0answers
127 views

Integral estimate for the solution of the heat equation

Let $u$ be a solution of $\partial_t u - \Delta u =f$ with initial data $u(0,x) = 0$ on $\mathbb R^N$. How do one prove the following inequality? $$ \int_0^T \int_{\mathbb R^N} \phi(f)(- \Delta) u(...
2
votes
0answers
35 views

When does the PDE $F\cdot \nabla u+Gu+H=0$ admit a global solution $u:\mathbb R^3\to\mathbb R$?

Let $F:\mathbb R^3\to \mathbb R^3$ and $G,H:\mathbb R^3\to\mathbb R$ be some given smooth maps. In order for the PDE $F\cdot \nabla u+Gu+H=0$ to admit a global solution $u:\mathbb R^3\to\mathbb R$, ...
1
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0answers
51 views

Regularity theory for parabolic PDEs in fractional Sobolev spaces

I am trying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya and the ...
5
votes
2answers
212 views

Definition of a system being hyperbolic

Consider the $n \times n$ system $$u_t + A(u)u_x = F(u).$$ If the eigenvalues of $A$ are all real and distinct the system is called strictly hyperbolic. What is the relationship between this ...
2
votes
1answer
61 views

Hyperbolic system with no zero eigenvalue

In the $n \times n$ hyperbolic system $$u_t + A(u)u_x = F(u)$$ what's the name of the assumption that $A$ has no zero eigenvalues? Note that if the eigenvalues are all real and distinct the system ...
0
votes
1answer
71 views

Scaling argument for the heat equation in a bounded domain [closed]

We want to study the long time behavior of the heat equation $u_t - u_{xx} = 0$ in the domain $[-1,1]$. Now consider the rescaling $u^{\epsilon} = u(x/\epsilon, t/\epsilon^2)$. Then $u^\epsilon$ ...
6
votes
0answers
321 views

Lax pair of an integrable non-linear PDE

The following is a fourth-order non-linear PDE that passes the Painleve integrability test $$\left(1+x^{2}\right)^{2}u_{xxxx} + 8x\left(1+x^{2}\right)u_{xxx} + 4\left(1+3x^{2}\right)u_{xx}+ t\left(...
2
votes
0answers
34 views

Stability of weak solutions to quasilinear parabolic equations

Consider the quasilinear operator $A(x,t,\nabla u)$ satisfying $$A(x,t,\nabla u).\nabla u \geq C_0 |\nabla u|^p$$ and $$|A(x,t,\nabla u)| \leq C_1 |\nabla u|^{p-1}$$ where $1<p<\infty$. Note ...
1
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0answers
63 views

Intuition from Hopf lemma (boundary point lemma )

Consider the classical boundary point lemma: Let $L$ be an elliptic operator. Boundary Point Lemma Suppose $\Omega$ has the interior sphere property and that $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ ...
1
vote
1answer
106 views

Can a real (1,1) form $\phi$ be represented by $u\sqrt{-1}\partial\bar{\partial}u$ on Kahler manifold?

Let M be a 2-dimension (complex dimension) K\"{a}ler maifold and $\phi$ be a real $(1,1)$ form. Is it possible that there exists a function $u$ such that $\phi=u\sqrt{-1}\partial\bar{\partial}u$
1
vote
1answer
99 views

Understanding a family of Sobolev-type inequalities

I am reading Aspects of Sobolev-Type Inequalities by professor Laurent Saloff-Coste, where I found a claim on page 66 claiming the following: Denote the following inequality as $S_{r,s}^{\theta}$: $\...
0
votes
0answers
38 views

Does the function with the αth-weak partial derivative has the βth-weak partial derivative with β≤α?

The definition of weak derivative in the book Partial Differential Equations by Evans is stated as follows: Suppose $u,v \in L_{loc}^1(U)$, and $\alpha$ is a multiindex. We say that $v$ is the $\...
3
votes
1answer
197 views

Aleksandrov maximum principle for semi-convex function

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
0
votes
0answers
37 views

Weak formulation of PDE with weighted inner product

In Boyd's book on spectral methods (available here: https://depts.washington.edu/ph506/Boyd.pdf), I stumbled in section 3.5 "Weak & Strong Forms of Differential Equations: the Use-fulness of ...
2
votes
0answers
47 views

wave equation with non-smooth coefficients

Let us consider the equation $$ \sum_{j,k=0}^n \partial_j ( a^{jk}(t,x)\partial_k u) =F, \quad \text{on $(-\infty,T)\times \mathbb{R}^n$}$$ subject to $u|_{t<0}=0$. Here, $F \in L_{\text{comp}}^2((...
1
vote
0answers
36 views

Decomposition of the space of Radon measures with respect fractional harmonic capacity?

It is well know that there is a generalization of Lebesgue decomposition theorem in the following way: Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
2
votes
1answer
52 views

Estimates on divergence-type operator for the matrix

Is there any result (Schauder-like estimates, $L^2$ estimates or similar) to equations of the form $$ {\rm div}(Av)=f $$ where $A$ is the "unknown" (i.e. I would like estimates on $A$ depending on $f$,...
0
votes
1answer
64 views

Existence of subsequences convergence with weak topology

Let $\left\{ {{\varphi _n}} \right\}$ is the sequence bounded in ${L^\infty }\left( {0,\infty ;H_0^1\left( {0,1} \right)} \right)$. Is there exists $\varphi \in {L^\infty }\left( {0,\infty ;H_0^1\...
8
votes
1answer
175 views

Positive solutions for semilinear parabolic equations

Let $X$ be a Banach lattice. Consider the system $$y'(t)=Ay(t)+f(t,y(t)) \qquad \text{in } (0,T) , \qquad y(0)=y_0, \qquad (*)$$ where $T>0$, $A$ generates an analytic positive semigroup $S(t)$ on $...
2
votes
1answer
199 views

Laplace equation on the disk with Robin boundary condition

Consider the following two dimensional Laplace equation on the unit disk $D$ with homogeneous Robin boundary condition: $$\Delta u = 0, ~~\frac{\partial u}{\partial n} = b(x) u(x)~~ \forall x \in \...
3
votes
0answers
79 views

Compactness of multiplication operators

Let $n \ge 3$ and $0 \le V\in L^p(R^n)$ for some $p \ge n/2$. Then the multiplication operator $$Tu=V^{1/2}u$$ is compact from $H^1(R^n)$ to $L^2(R^n)$. If $p>n/2$, this follows from the ...
3
votes
1answer
91 views

Smoothing-Strichartz estimates for the heat-Schrodinger evolution

Consider the heat-Schrodinger evolution $e^{(1+i)t\Delta}$, $t\geq 0$. For simplicity, suppose to work in dimension three. Due to the Strichartz estimates for the Schrodinger equation, and the ...
2
votes
0answers
93 views

What is the motivation to define measure valued solutions to a PDE model?

Consider the model $$\partial _{t}\mu + \partial_{x}(b(t,\mu)\mu)+c(t,\mu) \mu=0,~~~\mu \in \mathcal{M}^{+}(\mathbb{R}^{+}), t \in [0,T], x \in \mathbb{R}^{+} $$ $$ \mu(0)=\mu_{0} $$ where $ \mu (t)$...
1
vote
0answers
118 views

Rigorous error estimate for semi-discrete heat equation in bounded domain

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^N$ and $u_h$ be a solution of $$ \begin{cases} \partial_t u_h -\Delta_h u_h = f(x) & \text{ in } \Omega_h\\ u_h=0 &\text{ in } \...
1
vote
0answers
109 views

A question about Stroock's notes on the Weyl lemma

On p.4 of these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the ...
1
vote
0answers
61 views

Fixing constants of a series solution of a fourth-order PDE

The following is the PDE I want to solve, $$\left(1+x^{2}\right)^{2}y_{xxxx}+8x\left(1+x^{2}\right)y_{xxx} + 4\left(1+3x^{2}\right)y_{xx} + K\left[2x yy_{xx}+\left(1+x^{2}\right)\left(yy_{xxx} + y_{x}...
2
votes
0answers
149 views

Eigenvalue and eigenfunction convergence

Consider a bounded Euclidean domain $\Omega \subset \mathbb{R}^n$ (for simplicity, let's say, $\Omega$ has smooth boundary and is simply connected). Let $p \in \Omega$ be a point, and call $\Omega_n = ...
4
votes
1answer
105 views

Interpolation for Sobolev spaces

How one can identify the following (complex) interpolation space $$E_\theta :=[L^2(\Omega), H^2(\Omega)\cap H_0^1(\Omega)]_\theta,$$ where $\Omega$ is a regular domaine. After research, it seems that ...
2
votes
0answers
56 views

Proving that $(f,g)$ are Cauchy data for the Schrödinger equation iff $(f,g)$ satisfies an equation

I have to prove that if $f\in H^{1/2}(\partial\Omega)$ then $(f,g)$ are Cauchy data for the Schrödinger equation if and only if $$g=\gamma^{-1/2} \Lambda_{\gamma}(\gamma^{-1/2} f)+1/2 \gamma^{-1}\...
1
vote
0answers
23 views

Reference request- Shock development problem for the compressible Euler equation in 1D

I was wondering if there is any good reference discussing the shock development problem for Euler in 1D? Something in the spirit of Christodoulou's work on the same for higher dimension. I am ...
2
votes
0answers
53 views

Properties of solution to Burger's equation using Cole-Hopf transformation

I am currently looking at a $1$D-Burger's equation defined by \begin{equation} \label{ex burgers} \left\{ \begin{array}{ll} {} & \frac{\partial V_m}{\partial t} (t,x) = \frac{\sigma^2}{2} \...

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