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.
2,364 questions
5
votes
0answers
55 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
43 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
159 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
58 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
0answers
27 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
40 views
Exact solution of two coupled transport equations
I want to solve the following system
$$\eqalign{
& y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr
& z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr
& y(0,x) = y_0,\,\,z(...
0
votes
1answer
61 views
Reference request: Moving source to initial condition and vice versa in PDE problem
I am trying to find references in the literature that connect solutions of two problems given bellow. They deal with deterministic conservation laws.
Inhomogeneous Cauchy problem:
$$(1) \hspace{1cm} ...
5
votes
0answers
120 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
292 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
298 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
89 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
62 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
53 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 (...
0
votes
0answers
71 views
Linearization around a traveling wave
In [1], Remark 2.1., the authors say the following: ".. in moving coordinates, the linearization of (37) around a traveling wave profile is given by
\begin{equation}
v_t = x \partial^2_x v + \frac{2}{...
6
votes
1answer
119 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}{...
4
votes
1answer
225 views
Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$
Consider the following PDE:
\begin{equation}
p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$}
\end{equation}
where $g$ is a flat function at the point (...
1
vote
1answer
55 views
Is $H_0^1$ a redundant assumption in the 2D Agmon inequality?
The Wikipedia article on Agomon's inequality states the following:
Let $u\in H^2(\Omega)\cap H^1_0(\Omega)$ where $\Omega\subset\mathbb{R}^2$. Then Agmon's inequality in 2D states that there ...
0
votes
0answers
53 views
Vector fields whose divergence is Gaussian
Let f be the pdf of a $n$ dimensional $N(0,C)$ distribution i.e up to a multiplicative constant, $f(x) = \exp(-\frac{1}{2} x'C^{-1}x)$.
Which vector fields $F$ are so that ${\rm div} (F)= f$ ?
0
votes
0answers
59 views
A Generalized Bernstein's Problem
Suppose $\Sigma\subset \mathbb{R}^n$ be an area-minimizing hypersurfaces, the standard results in GMT tell that when $3\leq n\leq 7$, $\Sigma$ is a hyperplane; while for $n\geq 8$, there are examples ...
3
votes
2answers
201 views
Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability
We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth ...
2
votes
1answer
92 views
Optimal control theory of PDEs
This is a somewhat openly phrased question because I am not quite sure what has been done in that direction.
Imagine one has two evolution equations
$$\partial_t u = p(x,\partial_x,f)u$$
$$\...
2
votes
1answer
130 views
Quantitative finite speed of propagation property for ODE (cone of dependence)
Consider the following ODE initial value problem
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \...
2
votes
0answers
54 views
Smooth dependence of convex functions on Monge-Ampère measure
Let $\Omega\subset\mathbb{R}^2$ be a bounded convex domain, $f$ be a positive smooth function on $\Omega$ and $\phi:\partial\Omega\rightarrow\mathbb{R}$ be a continuous function. It is known that as ...
0
votes
2answers
76 views
Reference request for fractional Poincare inequality
Suppose we consider in $\mathbb R^n$, then how to show $\Vert f \Vert_{L^{p}} \leq C\Vert \nabla^{s}f \Vert_{L^{q}}^{\alpha}$, where $s>0$ is noninteger and $\alpha \in (0,1)$?
3
votes
1answer
68 views
Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$
Is there any Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$ in the literature?
More precisely I would like to know if there is an answer to the following
QUESTION: Let $f : \...
0
votes
1answer
85 views
Separation of variables for PDE
Consider the PDE
$$ \partial_t f(t,x) = \Phi(x) f(t,x)+ \Psi(x)f(t,x-a)$$
$$f(0,x)=1$$
$$f(t,0)=1$$
where $a$ is a constant and $\Phi$ and $\Psi$ are some differentiable functions. To solve this I ...
2
votes
1answer
216 views
+50
Meaning of Alberti rank-one theorem
Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$?
Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...
2
votes
1answer
104 views
Liouville theorem for fractional Laplacian
Is there any Liouville type theorem for the half space problem
\begin{equation}
\ \ \left\{\begin{aligned}
(-\Delta)^s v &= 0 &&\text{in } \mathbb R^N_+\\
v & =0 &&\text{...
0
votes
0answers
43 views
linear fractional laplacian problem
If $U(x)$ is a classical solution of
\begin{equation}
\ \ \left\{\begin{aligned}
(-\Delta)^s U+ U &=m(r)U &&\text{in } B \\
U &= 0 &&\text{in } \mathbb R^N \...
1
vote
1answer
74 views
Extended Global approximation theorem
In Evans,
$\textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $\partial U$ is $C^1$. Suppose as well that $u \in W^{k,p}(U)$ for some $1\leq p < \infty$. Then, there ...
3
votes
1answer
211 views
Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator
For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...
4
votes
1answer
132 views
Interpolation of some Lebesgue spaces
When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that
$$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...
1
vote
1answer
97 views
Exponential decay of resolvent kernel
For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function ...
3
votes
1answer
46 views
Difference between linear and parabolic velocity profiles in Stokes flows of two fluids
For droplet interactions in low-Reynolds number flow, solutions are available when the underlying flow can be written as linear compositions of strain and rotation, see Batchelor & Green (1972a).
...
4
votes
1answer
99 views
Power series in functions other than monomials
I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example.
Consider the interval $[-\pi,\pi]$ let's say.
...
1
vote
1answer
173 views
Boundary regularity for the Monge-Ampère equation $\det D^2u=1$
Let $\Delta$ be an open triangle in $\mathbb{R}^2$ and $u\in C^0(\overline{\Delta})\cap C^\infty(\Delta)$ be the convex function satisfying
$$
\det D^2u=1,\quad u|_{\partial\Delta}=0.
$$
Classical ...
1
vote
0answers
58 views
Unique continuation from the boundary for inhomogeneous elliptic pde
Let $Lu = f$ be satisfied on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, where $L$ is a strongly elliptic second order differential operator with real ...
3
votes
0answers
142 views
Stationary Navier-Stokes solutions
Are there known nontrivial ($u\neq0$) stationary solutions to Navier-Stokes equations in $\mathbb R^3$ ? Not square integrable of course (that's impossible), but with self-similar amplitudes of ...
1
vote
0answers
70 views
asymptotic behaviour of principal eigenfunctions and Large Deviations
Dear Math Overflowers,
I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...
1
vote
0answers
47 views
On the non-existence of weak solutions to nonlinear evolutionary PDEs without time derivative
Imagine we have a nonlinear PDE, e.g. some velocity model
$$\begin{aligned} -\Delta v + (v \cdot \nabla) v + \nabla p = f \\ \text{div} \, v = 0 \end{aligned}$$
which has a weak solution for some ...
2
votes
0answers
53 views
The space $H_\Delta(\Omega)$
I'm looking for good references for the study of the following space
$$H_\Delta(\Omega)=\{u\in L^2(\Omega) : \Delta u \in L^2(\Omega)\},$$
especially the trace theorems with proofs, where $\Omega \...
1
vote
1answer
83 views
Compact Embedding Between Parabolic Holder Spaces
My question is about the following compact embedding:
\begin{equation}
C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T).
\end{equation}
what condition should be put ...
1
vote
1answer
109 views
Existence and regularity for fractional Poisson-type equation
According to Theorem 2.7 in the paper https://arxiv.org/pdf/1704.07560.pdf, we have the following classical results.
Let $s \in (0,1)$ and $1<p<\infty$. Then for any $F \in L^p(\mathbb{R}^n)$ ...
9
votes
2answers
434 views
Harmonic oscillator in spherical coordinates
It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry.
More precisely, the operator
$$-\frac{d^2}{dx^2}+x^2$$
can be ...
14
votes
3answers
311 views
What is known about sufficient conditions for the rigidity of a convex surface?
A convex surface is a connected open subset of the boundary of a convex body in $\mathbb{R}^3$.
An "infinitesimal bending" of a convex surface $S$ is a deformation of $S$ given by a velocity field $v:...
4
votes
2answers
382 views
Klein Gordon equation - references
The Klein Gordon equation of the form:
$\Delta u+ \lambda u^p=0$
is been studied for $p = 2$?
(i.e.$\Delta u+ \lambda u^2=0$)
If yes are there references?
2
votes
0answers
74 views
geodesic balls in the conformal change
Consider a compact smooth Riemannian manifold $(\mathcal{M}, g)$. We consider a conformal change of the metric. Let $\tilde{g}=\phi g$, where $\phi$ is smooth and positive. Moreover, it satisfies $...
0
votes
0answers
38 views
What are tools we need to prove scattering result for NLS when propagator is unitary?
Consider nonlinear Schrödinger equation (NLS)
$$i\partial_t u + \Delta u =F(u) , u(x,0)=u_0$$
where $F$ is some nonlinearity.
Assume that there exists Banach space $X$ so that $\|e^{it\Delta} f\|_{X}...
1
vote
0answers
53 views
Global interior estimate complex Monge-Ampere equation
Suppose u is a smooth strictly plurisubharmonic function satisfying $(i\partial \bar{\partial} u)^n =f dV_{Euc}$ on the unit ball centered at the origin, where $f>0$ is a smooth function. Let $C$ ...
3
votes
1answer
337 views
Does current follow the path(s) of least (total) resistance?
Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (...
