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.
2,396 questions
0
votes
0answers
14 views
2d incompressible Euler equations under periodic boundary conditions
It is well known that the 2d incompressible Navier-Stokes equations under periodic boundary conditions always have global smooth solutions, given smooth initial conditions.
I tried searching for a ...
7
votes
0answers
63 views
A geometrical problem in terms of a convex function
I wish to know whether the following problem has ever been investigated.
Let $D$ be a convex domain in ${\mathbb R}^d$, with smooth boundary $\partial D$. Let $\vec V:\partial D\rightarrow{\mathbb R}^...
2
votes
1answer
81 views
Local Sobolev embedding on complete Riemannian manifold
Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose $Ric\geq(n-1)\kappa$. Let $B_p(r)$ be a geodesic open ball.
Q Can we find a constant $C=C(\kappa,r,m)$(...
0
votes
1answer
163 views
Stone–von Neumann theorem?
The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...
0
votes
0answers
16 views
Reference request: Numerical methods for Hamilton-Jacobi-Bellman equations with state constraints
I have two questions on numerical methods for solving Hamilton-Jacobi-Bellman (HJB) equations with state constraints.
Consider an optimal control problem given by
$$
v(x) = \max_{\{u(t)\}_t} \int_o^\...
1
vote
0answers
46 views
Can the rank of harmonic maps decrease far from the boundary?
This is a cross-post.
Let $\mathbb D^n$ be the closed unit disk in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a smooth immersion, and let $\omega:\mathbb D^n \to \mathbb{R}^n$ be the ...
5
votes
1answer
219 views
Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?
$\newcommand{\R}{\mathbb R}$
$\newcommand{\N}{\mathbb N}$
$\newcommand{\de}{\delta}$
$\newcommand{\sig}{\sigma}$
$\newcommand{\Average}[1]{\left\langle#1\right\rangle} $
$\newcommand{\IP}[2]{\Average{...
1
vote
1answer
95 views
Poisson equation on noncompact manifold
Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold and $C^\infty_c$-functions are dense in $L^p_k$ space.
For the equation
$$\Delta u=f,$$
...
1
vote
0answers
26 views
Quasilinear elliptic problem on fractal domain
Consider the following quasilinear elliptic equation
$$\nabla_x (A(x,u(x),\nabla_x u(x))) + f(u,x) = 0 $$
on a bounded domain $\Omega$, augmented with homogeneous Dirichlet boundary data: $$u|_{\...
2
votes
1answer
57 views
Quasilinear elliptic problem: Ellipticity-type conditions
Consider the following quasilinear elliptic equation
$$\nabla_x (A(x,u(x),\nabla_x u(x))) + f(u,x) = 0 $$
on a bounded domain $\Omega$, augmented with homogeneous Dirichlet boundary data: $$u|_{\...
0
votes
0answers
37 views
Dirichlet problem and schauder estimate for manifolds
Let $M$ be an n dimensional Riemannian manifold without boundary. Let $U\subset M$ be a bounded domain with smooth boundary. Let $f \in C^{\alpha}(U)$, $g\in C^{2,\alpha}(\partial U)$. Consider the ...
7
votes
1answer
153 views
Almost orthonormal projection and orthonormal projection in Hilbert space
Let $(e_i)_i$ be a family of vectors in a Hilbert space being almost orthonormal but not quite, i.e.
$$\langle e_i, e_j \rangle \approx \delta_{i,j} + \alpha e^{-\vert i-j \vert} $$
and $\alpha$ is ...
1
vote
1answer
345 views
A question of the Schrodinger Semigroup --By B. Simon
The question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).
On the Theorem C.1.2(subsolution estimate) of the paper, it says that: If $Hu=0$, where $H=-\...
1
vote
1answer
68 views
The difference between the nonlocal and local conditions problems
In some problems involving ordinary differential equations, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed.
In this paper: Existence and uniqueness ...
0
votes
1answer
47 views
Finding Free surface elevation in semi-infinite channel
A semi-infinite channel of finite depth is occupied by an ideal fluid layer initially at rest . the vertical finite end of the channel is fixed and only a part of the horizontal bottom , with finite ...
0
votes
1answer
82 views
Analytic approach to geodesic connectedness in Semi-Riemannian manifolds
Can you point out a reference (or references) that deal with analytical methods (rather than methods from differential geometry) for the study of geodesic connectedness on Semi-Riemannian manifolds?
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0answers
33 views
Anisotropic elliptic problems
Where can I find a reference on the following anisotropic problem?
$$(\bullet) \qquad \begin{align*} \nabla_x (A(x) \nabla_x u(x,y)) + \Delta_y(K(y) \ast u(x,y))\end{align*}=0, $$
where $(x,y) \in \...
1
vote
1answer
57 views
Elliptic interface problem without conditions on the interface
Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$.
For a model case, consider a ball split in a smaller ball and an anulus.
Consider the following elliptic ...
1
vote
0answers
36 views
Local Cancellation in Real Hardy Space
I want to show the following asymtotic estimate in Hardy space over $\mathbb{R}^n$: Let $a\in \mathbb{R}^n$. I want to show the function
$$
f(x)=\mathbb{1}_{B(0,1)}-\mathbb{1}_{B(a,1)}
$$
is asymtotic ...
0
votes
0answers
67 views
Biharmonic equation
Let us consider for $0<\alpha\leq V(x)\leq \beta$ and $0\leq K(x)<\gamma$ the equation
\begin{equation}\label{\star}
\Delta^2u+V(x)u=g(x, u)+K(x)u,
\end{equation}
where $|g(x,s)|\leq \varepsilon|...
3
votes
0answers
43 views
Inverse Laplacian and convolution in Albeverio's “Solvable Models in quantum mechanics”
I asked this question on math.stackexchange.com two weeks ago but got no answers so far and I got no clues from literature, so maybe someone here knows a reference. I hope it is ok to ask this ...
0
votes
0answers
97 views
Green function of a Laplace operator in an annulus
How to find the Green function of Dirichlet laplacian in an annulus?
0
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0answers
39 views
Existence and uniqueness for semilinear parabolic problem using fixed point approach
Where can I find a proof of existence and uniqueness of solutions for a semilinear parabolic problem
$$u_t -\Delta u +f(t,x,u,\nabla u) =0$$
which is based on a fixed point approach?
2
votes
1answer
64 views
Well-posedness of wave equations with time-dependent coefficient
Let us consider the following wave equation
\begin{array}{rrr}
y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in}
& (0,T)\times (0,1), \\
y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \\
y(0,x)...
5
votes
1answer
159 views
Obstruction to Navier-Stokes blowup with cylindrical symmetry
Is there a known obstruction to cylindrically symmetric solutions (with swirl) of incompressible 3D Navier-Stokes blowing up in finite time ?
EDIT: in the whole space $\mathbb R^3$, I forgot to say.
...
1
vote
0answers
31 views
Parabolic regularity for the 2D Navier-Stokes equations in a bounded domain
Suppose we consider 2D Navier-Stokes equations in a bounded domain $\Omega \subseteq \mathbb R^2$, together with suitable boundary conditions so that we can consider the vorticity equation: $$\omega_t ...
0
votes
0answers
41 views
Elliptic Dirichlet problems with measure boundary data
Can you point out any references on the Dirichlet problem for divergence-type elliptic operators with a Radon measure as boundary data?
2
votes
1answer
80 views
Boundary condition for elliptic problems and domain decomposition
This question is motivated by one that has been previously asked on this website: Elliptic problem on a domain split in two subdomains
Consider an open domain $U$ split in two non-overlapping ...
3
votes
1answer
181 views
Elliptic problem on a domain split in two subdomains
Consider the following elliptic problem in a split domain:
$$ (\ast) \quad\begin{cases} -\Delta u=f_1 \quad &\text{ in } U_1\\
-\Delta u =f_2 & \text{ in }
U_2\\
u=g & \text{ on } \...
3
votes
1answer
170 views
Heuristics for boundary Harnack inequality
What is the heuristic idea of the proof of the boundary Harnack inequality presented in the appendix of Caffarelli's 1998 lectures on the obstacle problem (page 38 here)?
2
votes
1answer
122 views
Bounded solution for parabolic equation
Let $\Omega_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation
$$u_t=\Delta u + a(x)u, \;\; (t,x)\...
3
votes
0answers
48 views
Green's function of time-dependent Stokes equation
It is well known that the Green's function of a standard parabolic equation in a bounded domain, say
\begin{align}
\partial_tG(x,y,t)-\Delta_x G(x,y,t)&=0
&&\mbox{for}\,\,\,(x,t)\in \...
1
vote
0answers
46 views
Reference request: Proof of Lemma 3.3 in Linear and Quasi-linear Equations of Parabolic Type
The following lemma in Linear and Quasi-linear Equations of Parabolic Type by Nina Uraltseva, Olga Ladyzhenskaya, and Vsevolod A. Solonnikov.
Lemma 3.3 Let $\Omega\subset\mathbb{R}^n$ satisfy a cone ...
2
votes
0answers
156 views
The use of dissipation in parabolic equations
I'm considering an equation in Sobolev spaces and stuck at a dissipation term. After constructing my desired Sobolev norm $W^{s,q}$, on the left hand side of the equation, I have $$\Vert \nabla (|\...
1
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0answers
41 views
Singular integral of the composition of the Hilbert transform and fractional Laplacian
Given $0<s<1$, we can define the Fractional Laplacian by
$$\Lambda^{-s}f(x):=(-\Delta)^{-s/2}(x)=\int_{-\infty}^{+\infty}|x-y|^{-1+s}f(y)dy$$
or by means of Fourier transform as $$\widehat{\...
5
votes
0answers
104 views
Gradient estimate for Poisson equation on manifold
In Gilbarg-Trudinger's book 'Elliptic Partial Differential Equations of Second order', the maximum principle is used to derive the following gradient estimates for Poisson equations on Euclidean ...
5
votes
0answers
65 views
Laplace Beltrami eigenvalues on surface of polytopes
The recently posted arxiv paper Spectrum of the Laplacian on Regular Polyhedra
by Evan Greif, Daniel Kaplan, Robert S. Strichartz, and Samuel C. Wiese, collects numerical evidence for conjectured ...
0
votes
0answers
55 views
Energy estimates involving test functions for weak solutions of PDE problems
I was reading an article on Arxiv.org about Navier-Stokes system ([Breit]) and I stumble on this sentence on the second page:
"A weak (in the PDE sense) solution satisfying the energy inequality ...
2
votes
1answer
284 views
Underdetermined system of linear PDEs
Let $a,b$ two smooth functions from the open square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$.
I ...
1
vote
0answers
61 views
Conditions for the embeddig of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$
Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.
If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...
1
vote
1answer
56 views
Local well-posedness of the quadratic NLS on the 1D torus
What is the proof of the local well-posedness of the quadratic nonlinear Schrödinger equation
$\mathrm{i} \,\partial_t u + \Delta u \pm \left|u\right| u = 0$
on the 1D torus in $H^s$ for $s > 1$ (a ...
1
vote
1answer
45 views
Exact solution of two coupled transport equations
I want to solve the following system
$$\eqalign{
& y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr
& z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr
& y(0,x) = y_0,\,\,z(...
0
votes
1answer
64 views
Reference request: Moving source to initial condition and vice versa in PDE problem
I am trying to find references in the literature that connect solutions of two problems given bellow. They deal with deterministic conservation laws.
Inhomogeneous Cauchy problem:
$$(1) \hspace{1cm} ...
5
votes
0answers
126 views
Level Sets of Harmonic Maps
Can anybody point me in the direction of some references in which the level sets of harmonic maps between Riemannian manifolds are studied. (Sorry I am unfamiliar with the area and would like some ...
10
votes
2answers
297 views
Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix?
Good morning,
I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v_1(x),\dots, v_n(...
5
votes
3answers
324 views
fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix
I am looking for the fundamental solution of the following PDE
$$\partial_i (a^{ij}\partial_j u)=f$$
where $a^{ij}(x)$ is a non-symmetric matrix with possibly non-constant coefficients.
I could ...
3
votes
1answer
94 views
Interpretation of the Schouten bracket as an integrability condition
The Schouten bracket is an extension of the Lie bracket to multivector fields. Given a multivector field $\Lambda$ the vanishing of the Schouten bracket $[\Lambda,\Lambda]=0$ is referred to as a sort ...
1
vote
0answers
68 views
$|\nabla \omega|^2= \alpha \omega^n+ \beta$ - References
I'm interested to know if anyone can give me some information (name, references, etc.) about this PDE
$$|\nabla \omega|^2= \alpha \omega^n+ \beta,$$
where $\omega$ is a scalar function of two or ...
0
votes
0answers
55 views
Strichartz estimates for fractional Schrodinger equations
A pair $(q,r)$ is $\alpha-$fractional admissible if $q\geq 2, r\geq 2$ and
$$\frac{\alpha}{q} = d \left( \frac{1}{2} - \frac{1}{r} \right).$$
We take fractional Schrodinger propagator $U(t)=e^{it (...
7
votes
2answers
156 views
“Overdetermined” Poisson equation
Consider the PDE $-\Delta u = f$ on a bounded domain $\Omega \subset \mathbb{R}^n$, where $f \in C^\infty(\bar{\Omega})$. I wish to consider both the boundary conditions $u = 0$ and $\frac{\partial u}{...
