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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1answer
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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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 $\...
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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 ...
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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
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1answer
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
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1answer
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
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1answer
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 ...
4
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1answer
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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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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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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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
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1answer
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 ...
2
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1answer
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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1answer
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
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1answer
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
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2answers
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 ...
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0answers
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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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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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^{...
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1answer
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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0answers
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, ...
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1answer
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) & \...
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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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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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0answers
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 \...
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1answer
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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0answers
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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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(\...
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1answer
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
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1answer
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
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2answers
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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0answers
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 ...
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1answer
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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0answers
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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0answers
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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0answers
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
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1answer
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
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0answers
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(\...
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2answers
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 ...
2
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1answer
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. ...
2
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1answer
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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1answer
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 } \...
2
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0answers
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
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1answer
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
1answer
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(...
2
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0answers
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
1answer
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 $W^{1,1,2}(I \times \Omega)=L^2(I; W^{1,2}(\Omega)) \cap W^{1,2}(I; W^{-1,2}(\Omega))$ ...
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0answers
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
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1answer
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+\...