# 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.

2,855
questions

**7**

votes

**3**answers

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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**0**answers

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**

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**0**answers

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

**0**answers

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**

votes

**0**answers

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**

vote

**1**answer

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

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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

**1**answer

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**

vote

**0**answers

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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**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

vote

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**0**answers

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

**0**answers

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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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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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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**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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**

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**0**answers

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

**0**answers

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**

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**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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} \...