# 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.

2,558
questions

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83 views

### Fractional Schrödinger equation

Let $\mathcal{F}$ denotes the Fourier transform.
It is known that $\mathcal{F}(e^{-4\pi^2 i t |x|^2})(\xi)= e^{i |\xi|^2/4t}{(4\pi i t)^{-n/2}} \ (x, \xi\in \mathbb R^n).$
My question is: what is ...

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**0**answers

28 views

### Failure of entropy condition for a singular limit of higher order regularization for a conservation law

Consider a regularization of the conservation law
$$\partial_t u + \partial_x f(u) + \epsilon \partial_{x}^3 u = 0$$
How does one prove that the limit function $u$ of $\{u_\epsilon\}_\epsilon$ as $\...

**1**

vote

**0**answers

29 views

### Existence and uniqueness of entropy solutions for a scalar conservation law

Consider the conservation law
$$(\ast) \qquad u_t + \partial_x(u^\alpha) = 0$$
where $\alpha > 0$.
For what values of $\alpha$ is it known that there exists a (unique) entropy solution for the ...

**5**

votes

**0**answers

114 views

### Wave equation with porous medium term

The classical porous media equation is
$$u_t + \Delta(u^m) = 0 \quad m>1.$$
Has the (degenerate) wave equation
$$u_{tt} + \Delta(u^m) = 0$$
been subject of studies? What would the physical ...

**7**

votes

**1**answer

110 views

### BV functions and wave equation

What is the role played by BV functions in the study of (possibly nonlinear) wave equations?
I think that one would need assume small initial data in $L^1$ or $H^1$ to get a well-posedness result (...

**3**

votes

**1**answer

93 views

### $\Delta_g f = 0$ on the Riemannian Manifold $(\mathbb{R}^3 \setminus B , g)$ with conditions on the boundary and at infinity

Consider the manifold $\mathbb{R}^3 \setminus B$ where $B$ is the ball with radius 1 with riemannian metric $g$ (not necessarily the euclidean metric).
I am looking for solutions to $\Delta_g f = 0$ ...

**5**

votes

**1**answer

367 views

### Research topics in Microlocal Analysis

Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution ...

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votes

**1**answer

98 views

### Numerics for continuity equation with Sobolev vector field

Has any work been done about numerical methods for the continuity equation
$$
\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,
$$
where $...

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votes

**0**answers

74 views

### Comparing different types of a PDE solutions

A few days ago I was reading the paper:
"Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system" - Feireisl, Jin, Novotny, 2012 [Arxiv].
...

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**0**answers

67 views

### How we can do the derivative for this equation w.r.t.to time t>0

Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...

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votes

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33 views

### Difference quotients of solutions of ODE and PDE in Sobolev setting

In the post Difference quotient for solutions of ODE and Liouville equation, it was showed that if $\Phi$ is the solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\...

**7**

votes

**1**answer

220 views

### Generalized Hodge Decomposition on Manifolds with Boundary

This question is motivated by the problem of finding heat kernels to use for the renormalization of quantum field theories on manifolds with boundary.
If $(\mathscr{E}, Q)$ is an elliptic complex on ...

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votes

**1**answer

89 views

### Eigenvalue problem involving fractional laplacian

Does the problem
$$(-\Delta)^s u=\lambda u \text{ in } \mathbb R^N $$ admit a non-trivial solution when $s\in (0, 1)$ and $\lambda>0.$

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vote

**1**answer

101 views

### How to prove the following Whittaker formula

I am a theoretical physicist and
I need help in proving the alternate Whittaker formula
$W _ { k , m } ( z ) = \frac { \Gamma ( - 2 m ) } { \Gamma \left( \frac { 1 } { 2 } - m - k \right) } M _ { k , ...

**2**

votes

**1**answer

110 views

### Difference quotient for solutions of ODE and Liouville equation

Suppose that $\Phi$ is the solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
How does one prove that
$$\...

**5**

votes

**2**answers

146 views

### $W^{k,1}$ regularity for elliptic equations

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and assume $u$ is a solutions of $\nabla \cdot (a \nabla u)=f$ with $a>c>0$ in $\Omega$, where $a\in C^k(\bar{\Omega})$. Is the following ...

**5**

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**0**answers

123 views

### Initial data and heat equation

We assume all solutions to be bounded here!
Let $y_{+},y_{-} \in C_c^{\infty}$ be two positive functions.
If we then consider the heat equation
$$\partial_t u(t,x) = \Delta u(t,x)$$ for two ...

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votes

**0**answers

73 views

### Explicit formula for Neumann heat kernel

It is well-known that
$u(x,y,t)=(4\pi t)^{-n/2}(e^{-|x-y|/4t}+e^{-|x-y'|/4t})$, $x,y\in \mathbb{R}^n_+=\{x\in \mathbb{R}^n|x_n\geq 0\}$, $y'=(y_1,\dots,y_{n-1},-y_n)$, is Neumann heat kernel of $\...

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**0**answers

39 views

### Inequality for fractional power norm (sectorial operators)

How could we prove following inequality:
$\int\limits_{0}^{l} u^{3}(x) dx \leq \sqrt{l} \cdot|| u||_{\frac{1}{2}}^{3}$
where
$ || u ||_{\frac{1}{2}} = ||A^{\frac{1}{2}}(u)||_{L^{2}} + || u ||_{L^{...

**1**

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**1**answer

106 views

### Role of the divergence of the vector field in transport equations: mass concentration?

Consider the continuity equation
$$\partial_t u(t,x) + \mathrm{div}(a(t,x)u(t,x)) = 0,$$
where $u: [0,T]\times \mathbb{R}^N \to \mathbb{R}$ is the solution and $a:[0,T]\times \mathbb{R}^N \to \...

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60 views

### Role of absolute continuity of divergence of BV function in proof of renormalization property

In the paper http://cvgmt.sns.it/paper/436/, the author proves the renormalization property for the flow generated by a vector field $a(t,\cdot) \in BV(\mathbb{R}^N; \mathbb{R}^N)$.
Heuristically, ...

**5**

votes

**1**answer

142 views

### Equivalence of viscosity and weak solutions for the Poisson equation

Suppose $\Omega$ is a bounded smooth domain in $\mathbb{R}^d$.
How does one prove that weak solutions are viscosity solutions and vice versa for the problem
$$
\begin{cases}
-\Delta u = f(x) & \...

**3**

votes

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43 views

### How to prove that most of the systems of PDEs cannot be decoupled?

Let's consider the following system
\begin{align}
\frac{d}{dt}\begin{pmatrix}
u\\v
\end{pmatrix}=\begin{pmatrix}
f(x,t,u,v,u_x,v_x)\\g(x,t,u,v,u_x,v_x)
\end{pmatrix}.
\end{align}
If there locally ...

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votes

**0**answers

75 views

### Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case

By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...

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35 views

### Definition of differential operator on the boundary

Let $\Omega \subset \mathbb{R}^N$ be a bounded (smooth) domain.
Consider the following problem
$$
\begin{cases}
F(D^2u)=0 \quad &\text{ in } \Omega\\
u=0 \quad &\text{ on } \partial \Omega
\...

**1**

vote

**1**answer

116 views

### Derivative and Jacobian determinant of solution of ODE [closed]

Let $\Phi$ be the unique solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
where we have assumed $f$ smooth.
...

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votes

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41 views

### generalised Ornstein-Uhlenbeck semigroup

I have some trouble using the Hille-Yosida theorem to prove that certain operators generate contraction semigroups.
To give an example, I would like to find some classes of functions $\alpha :\mathbb{...

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vote

**0**answers

41 views

### Elliptic equation with Neumann boundary condition: RHS in $L^2$ implies solution in $L^\infty$?

Consider the homogeneous Neumann problem $$-\Delta u + ku = f$$ $$\partial_\nu u = 0$$
on a smooth, bounded domain $\Omega$.
If $f \in L^2(\Omega)$, do we obtain the regularity $u \in L^\infty(\...

**0**

votes

**1**answer

48 views

### Boundary behavior of Greens functions on smooth bounded (planar) domains

It is well known that for any smooth bounded (connected) domain $\Omega\subset\mathbb R^d$ with $d\ge2$, we can define a Green's function $G:\Omega\times\Omega\to\mathbb R$ in $\Omega$ which is smooth ...

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45 views

### How to solve the following problem of partial differential equation?

How to solve the adjoint problem for the initial-boundary value problem of the wave equation
\begin{align}
u_{tt}-u_{xx}=0,
\end{align}
The research area is $(t, x)\in [0, T]\times[0, L]=:\Omega.$
...

**2**

votes

**1**answer

73 views

### About the continuity of a function on BV

For a fixed $u \in BV(\mathbb{R}^N)$, consider the function
$h:(0,+\infty) \to BV(\mathbb{R}^N)$, given by
$h(t) = u (tx)$.
Is $h$ continuous?

**5**

votes

**2**answers

103 views

### Most general conditions for (weak or classical) solutions to Poisson's equation

I thought I knew this but have found it surprisingly difficult to find good references. I am interested in solving
$$
\left\{
\begin{align}
& \Delta \psi = - \rho & & \mbox{in } \mathbb{...

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votes

**0**answers

53 views

### Bi-linear estimate for free Schrödinger equation

Consider the homogeneous Schrödinger equation:
$$ i\partial_t u + \Delta u=0, u(x,0)= u_0(x)$$
Proposition 4.4 Let $\lambda>0$. Assume that $u, v$ are solutions to the homogeneous Schrödinger ...

**1**

vote

**1**answer

62 views

### Strong convexity of internal energy with respect to Wasserstein metric

It is well known that the internal energy (see, e.g., Definition 3.32 in and Proposition 3.33 in 1) is geodesically convex with the 2-Wasserstein distance. I was wondering under what condition, the ...

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**0**answers

52 views

### How to prove that every weak solution is classic for a simple parabolic equation

Consider equation
$$
Lu=u_t-a(x)u_{xx}=0, \tag 1
$$
where coefficient $a$ is bounded and $a(x)\ge \delta>0$ for all $x\in \mathbb R$.
If $a$ is not smooth the conjugate operator $L^*$ can not be ...

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53 views

### Regularity estimates of the Fokker-Planck equation on the torus

The following comes from Pages 37-41 of the following paper:
https://arxiv.org/abs/1509.02505
Let $\mathbb{T}^d$ be a $d$-dimensional torus and let $C^{n+ \alpha}$ be the $n$th order $\alpha$-...

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72 views

### A reference for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$

Let $\Omega$ be an open domain with nice boundary and $u\in W^{1,p}(\Omega)$. I believe that $|u|^p\in W^{1,1}$ with
$$
\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u
$$
but couldn't find a good ...

**6**

votes

**1**answer

236 views

### Can the number of solutions to a system of PDEs be bounded using the characteristic variety?

I've recently come across a system of PDEs which I'd like to understand better. The particular system I'm interested in locally solves for a 2-dimensional Riemannian metric as the Hessian of a ...

**3**

votes

**0**answers

54 views

### Estimate on the $C^{1,\alpha}(\bar \Omega)$ norm of solution of linear elliptic Neumann problem

Let $\Omega$ be a bounded smooth domain with
$$-\Delta u + ku = f$$
$$\partial_\nu u|_{\partial\Omega} = 0$$
where $k > 0$ is a constant and $f \in L^\infty(\Omega)$. It follows that $u \in H^2(\...

**8**

votes

**2**answers

356 views

### Hyperbolic PDE in mathematics

Hyperbolic PDE (like the wave equation) are roughly speaking, PDE that satisfy the “finite propagation speed of information” property. They are ubiquitous in mathematical physics (essentially, most ...

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**1**answer

198 views

### How to use van der Corput's lemma to get the following estimates?

Recently, I'am reading Tao's article Spherically Averaged Endpoint Strichartz Estimates for the Two-Dimensional Schrodinger equation, in which there are some problems that I can't solve by myself.
...

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votes

**1**answer

124 views

### Comparing solutions of PDE problem with different initial conditions

My question(s) is about what happens with the solution of the problem if we change initial conditions.
Let's say we have a PDE problem:
$$
(1) \hspace{0.5cm} u_t+f(u)_x=0
$$
$$
(2) \hspace{0.5cm} ...

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votes

**1**answer

83 views

### Estimates for Green function for fractional Laplacian

Can the Green function for the fractional Laplacian operator be estimated from above and below.
$$ \left\{\begin{aligned}
(-\Delta_x)^{s} G(x, y)+ G(x, y)&= \delta_{y}(x) &&\text{in } \...

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**0**answers

48 views

### Second derivative of a functional defined by an integral

I was reading the following example from the book Methods in Nonlinear Analysis (Zhang, Springer) on page 10: First, everything was fine:
Example 2. Let $X = C^1(\overline \Omega, \mathbb R^N)$, $Y =...

**2**

votes

**1**answer

61 views

### Boundary regularity type results of fractional laplacian

Let $u$ be a solution of $-\Delta u= f \text{ in } \Omega$ with $u=0 \text{ on } \partial \Omega.$ If $f\in L^2(\Omega)$, then $u\in H^2(\Omega)\cap H^{1}_{0}(\Omega)$provided $\Omega$ is a smooth ...

**1**

vote

**1**answer

119 views

### Approximation of functions by tensor products

Given a function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$, can we find a sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(...

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votes

**0**answers

86 views

### Approximation of functions in $L^p(R^d;L^\infty)$

Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that
$$...

**3**

votes

**1**answer

74 views

### Question on relation between a parabolic sobolev space and a sobolev bochner space

For parabolic sobolev spaces I follow the following definition:
According to this definition, we have that $W^{1,1,2}(I \times \Omega)=L^2(I; W^{1,2}(\Omega)) \cap W^{1,2}(I; W^{-1,2}(\Omega))$
...

**1**

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**0**answers

115 views

### 6 linear PDE for only 3 unknowns?

Let $x \in (0,L)$, $t \in (0,T)$, and let $u_0 = u_0(x) \in \mathbb{R}^3$, $g=g(t) \in \mathbb{R}^3$, $P = P(x,t) \in \mathbb{R}^3$ and $Q = Q(x,t) \in \mathbb{R}^3$ be continuously differentiable ...

**1**

vote

**1**answer

105 views

### First order partial differential equation [closed]

I know there is a solution to this pde
$$\partial_{t} f(t,x)= \partial_{x}(v(x)f(t,x))$$
$$ f(0,x)=g(x)$$
( Where $v$ and $g$ are known functions)
which is given by
$$ f(t,x)=\frac{1}{v(x)} h(t+\...