pinecone/core-2020-05-10-deduplication huggingface dataset:
Global Regularity and Long-time Behavior of the Solutions to the 2D
  Boussinesq Equations without Diffusivity in a Bounded Domain

New results are obtained for global regularity and long-time behavior of the
solutions to the 2D Boussinesq equations for the flow of an incompressible
fluid with positive viscosity and zero diffusivity in a smooth bounded domain.
Our first result for global boundedness of the solution $(u, \theta) \in
D(A)\times H^1$ improves considerably the main result of the recent article
[7]. Our second result on global regularity of the solution $(u, \theta) \in V
\times H^1$ for both bounded domain and the whole space ${\mathbb R}^2$ is a
new one. It has been open and also seems much more challenging than the first
result. Global regularity of the solution $(u, \theta) \in D(A) \times H^2$ is
also proved
The Dynamics of Flat Surface Internal Geophysical Waves with Currents

A two-dimensional water wave system is examined consisting of two discrete
incompressible fluid domains separated by a free common interface. In a
geophysical context this is a model of an internal wave, formed at a pycnocline
or thermocline in the ocean. The system is considered as being bounded at the
bottom and top by a flatbed and wave-free surface respectively. A current
profile with depth-dependent currents in each domain is considered. The
Hamiltonian of the system is determined and expressed in terms of canonical
wave-related variables. Limiting behaviour is examined and compared to that of
other known models. The linearised equations as well as long-wave
approximations are presented.Comment: LaTeX, 21 pages, 1 figure, available online in J. Math. Fluid Mech.
  (2016
Dimension reduction for the full Navier-Stokes-Fourier system

It is well known that the full Navier-Stokes-Fourier system does not possess
a strong solution in three dimensions which causes problems in applications.
However, when modeling the flow of a fluid in a thin long pipe, the influence
of the cross section can be neglected and the flow is basically
one-dimensional. This allows us to deal with strong solutions which are more
convenient for numerical computations. The goal of this paper is to provide a
rigorous justification of this approach. Namely, we prove that any suitable
weak solution to the three-dimensional NSF system tends to a strong solution to
the one-dimensional system as the thickness of the pipe tends to zero.Comment: arXiv admin note: text overlap with arXiv:1111.4256 by other author
Uniqueness Results for Weak Leray-Hopf Solutions of the Navier-Stokes
  System with Initial Values in Critical Spaces

The main subject of this paper concerns the establishment of certain classes
of initial data, which grant short time uniqueness of the associated weak
Leray-Hopf solutions of the three dimensional Navier-Stokes equations. In
particular, our main theorem that this holds for any solenodial initial data,
with finite $L_2(\mathbb{R}^3)$ norm, that also belongs to to certain subsets
of $VMO^{-1}(\mathbb{R}^3)$. As a corollary of this, we obtain the same
conclusion for any solenodial $u_{0}$ belonging to $L_{2}(\mathbb{R}^3)\cap
\mathbb{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$, for any
$3<p<\infty$. Here,
$\mathbb{\dot{B}}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$ denotes the
closure of test functions in the critical Besov space
${\dot{B}}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$. Our results rely on the
establishment of certain continuity properties near the initial time, for weak
Leray-Hopf solutions of the Navier-Stokes equations, with these classes of
initial data. Such properties seem to be of independent interest. Consequently,
we are also able to show if a weak Leray-Hopf solution $u$ satisfies certain
extensions of the Prodi-Serrin condition on $\mathbb{R}^3 \times ]0,T[$, then
it is unique on $\mathbb{R}^3 \times ]0,T[$ amongst all other weak Leray-Hopf
solutions with the same initial value. In particular, we show this is the case
if $u\in L^{q,s}(0,T; L^{p,s}(\mathbb{R}^3))$ or if it's $L^{q,\infty}(0,T;
L^{p,\infty}(\mathbb{R}^3))$ norm is sufficiently small, where $3<p< \infty$,
$1\leq s<\infty$ and $3/p+2/q=1$.Comment: 44 pages. Submitted. Corollary 1.4, Proposition 1.6 and Section 5
  have been added. Additional remarks included in the introduction and at the
  end of section 4. Another subsection has been added to the 'Preliminaries'.
  Minor typos have also been correcte
Linear Inviscid Damping for Couette Flow in Stratified Fluid

We study the inviscid damping of Couette flow with an exponentially
stratified density. The optimal decay rates of the velocity field and the
density are obtained for general perturbations with minimal regularity. For
Boussinesq approximation model, the decay rates we get are consistent with the
previous results in the literature. We also study the decay rates for the full
Euler equations of stratified fluids, which were not studied before. For both
models, the decay rates depend on the Richardson number in a very similar way.
Besides, we also study the dispersive decay due to the exponential
stratification when there is no shear
Stability Results for Idealised Shear Flows on a Rectangular Periodic
  Domain

We present a new linearly stable solution of the Euler fluid flow on a torus.
On a two-dimensional rectangular periodic domain $[0,2\pi)\times[0,2\pi /
\kappa)$ for $\kappa\in\mathbb{R}^+$, the Euler equations admit a family of
stationary solutions given by the vorticity profiles $\Omega^*(\mathbf{x})=
\Gamma \cos(p_1x_1+ \kappa p_2x_2)$. We show linear stability for such flows
when $p_2=0$ and $\kappa \geq |p_1|$ (equivalently $p_1=0$ and
$\kappa{|p_2|}\leq{1}$). The classical result due to Arnold is that for $p_1 =
1, p_2 = 0$ and $\kappa \ge 1$ the stationary flow is {nonlinearly} stable via
the energy-Casimir method. We show that for $\kappa \ge |p_1| \ge 2, p_2 = 0$
the flow is linearly stable, but one cannot expect a similar nonlinear
stability result. Finally we prove nonlinear instability for all equilibria
satisfying $p_1^2+\kappa^2{p_2^2}>\frac{{3(\kappa^2+1)}}{4(7-4\sqrt{3})}$. The
modification and application of a structure-preserving Hamiltonian truncation
is discussed for the $\kappa\neq 1$ case. This leads to an explicit Lie-Poisson
integrator for the truncated system
Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

The main aim of the paper is to investigate the transitions of the
thermohaline circulation in a spherical shell in a parameter regime which only
allows transitions to multiple equilibria. We find that the first transition is
either continuous (Type-I) or drastic (Type-II) depending on the sign of the
transition number. The transition number depends on the system parameters and
$l_c$, which is the common degree of spherical harmonics of the first critical
eigenmodes, and it can be written as a sum of terms describing the nonlinear
interactions of various modes with the critical modes. We obtain the exact
formulas of this transition number for $l_c=1$ and $l_c=2$ cases. Numerically,
we find that the main contribution to the transition number is due to nonlinear
interactions with modes having zero wave number and the contribution from the
nonlinear interactions with higher frequency modes is negligible. In our
numerical experiments we encountered both types of transition for $Le<1$ but
only continuous transition for $Le>1$. In the continuous transition scenario,
we rigorously prove that an attractor in the phase space bifurcates which is
homeomorphic to the 2$l_c$ dimensional sphere and consists entirely of
degenerate steady state solutions
A Hele-Shaw-Cahn-Hilliard model for incompressible two-phase flows with
  different densities

Topology changes in multi-phase fluid flows are difficult to model within a
traditional sharp interface theory. Diffuse interface models turn out to be an
attractive alternative to model two-phase flows. Based on a
Cahn-Hilliard-Navier-Stokes model introduced by Abels, Garcke and Gr\"{u}n
(Math. Models Methods Appl. Sci. 2012), which uses a volume averaged velocity,
we derive a diffuse interface model in a Hele-Shaw geometry, which in the case
of non-matched densities, simplifies an earlier model of Lee, Lowengrub and
Goodman (Phys. Fluids 2002). We recover the classical Hele-Shaw model as a
sharp interface limit of the diffuse interface model. Furthermore, we show the
existence of weak solutions and present several numerical computations
including situations with rising bubbles and fingering instabilities.Comment: 41 pages, 75 figure
Stability analysis for the incompressible Navier-Stokes equations with
  Navier boundary conditions

This paper concerns the instability and stability of the trivial steady
states of the incompressible Navier-Stokes equations with Navier-slip boundary
conditions in a slab domain in dimension two. The main results show that the
stability (or instability) of this constant equilibrium depends crucially on
whether the boundaries dissipate energy and the strengthen of the viscosity and
slip length. It is shown that in the case that when all the boundaries are
dissipative, then nonlinear asymptotic stability holds true, otherwise, there
is a sharp critical viscosity, which distinguishes the nonlinear stability from
instability.Comment: 34 page
Solvability of the initial value problem to the Isobe-Kakinuma model for
  water waves

We consider the initial value problem to the Isobe-Kakinuma model for water
waves and the structure of the model. The Isobe-Kakinuma model is the
Euler-Lagrange equations for an approximate Lagrangian which is derived from
Luke's Lagrangian for water waves by approximating the velocity potential in
the Lagrangian. The Isobe-Kakinuma model is a system of second order partial
differential equations and is classified into a system of nonlinear dispersive
equations. Since the hypersurface $t=0$ is characteristic for the
Isobe-Kakinuma model, the initial data have to be restricted in an infinite
dimensional manifold for the existence of the solution. Under this necessary
condition and a sign condition, which corresponds to a generalized
Rayleigh-Taylor sign condition for water waves, on the initial data, we show
that the initial value problem is solvable locally in time in Sobolev spaces.
We also discuss the linear dispersion relation to the model
On critical spaces for the Navier-Stokes equations

The abstract theory of critical spaces developed in [22] and [20] is applied
to the Navier-Stokes equations in bounded domains with Navier boundary
conditions as well as no-slip conditions. Our approach unifies, simplifies and
extends existing work in the $L_p$-$L_q$ setting, considerably. As an essential
step, it is shown that the strong and weak Stokes operators with Navier
conditions admit an $\mathcal{H}^\infty$-calculus with
$\mathcal{H}^\infty$-angle 0, and the real and complex interpolation spaces of
these operators are identified.Comment: 21 page
Navier-Stokes flow past a rigid body: attainability of steady solutions
  as limits of unsteady weak solutions, starting and landing cases

Consider the Navier-Stokes flow in 3-dimensional exterior domains, where a
rigid body is translating with prescribed translational velocity
$-h(t)u_\infty$ with constant vector $u_\infty\in \mathbb R^3\setminus\{0\}$.
Finn raised the question whether his steady slutions are attainable as limits
for $t\to\infty$ of unsteady solutions starting from motionless state when
$h(t)=1$ after some finite time and $h(0)=0$ (starting problem). This was
affirmatively solved by Galdi, Heywood and Shibata for small $u_\infty$. We
study some generalized situation in which unsteady solutions start from large
motions being in $L^3$. We then conclude that the steady solutions for small
$u_\infty$ are still attainable as limits of evolution of those fluid motions
which are found as a sort of weak solutions. The opposite situation, in which
$h(t)=0$ after some finite time and $h(0)=1$ (landing problem), is also
discussed. In this latter case, the rest state is attainable no matter how
large $u_\infty$ is
Compressible-incompressible two-phase flows with phase transition: model
  problem

We study the compressible and incompressible two-phase flows separated by a
sharp interface with a phase transition and a surface tension. In particular,
we consider the problem in $\mathbb{R}^N$, and the Navier-Stokes-Korteweg
equations is used in the upper domain and the Navier-Stokes equations is used
in the lower domain. We prove the existence of $\mathcal{R}$-bounded solution
operator families for a resolvent problem arising from its model problem.
According to Shibata \cite{GS2014}, the regularity of $\rho_+$ is $W^1_q$ in
space, but to solve the kinetic equation: $\mathbf{u}_\Gamma\cdot\mathbf{n}_t =
[[\rho\mathbf{u}]]\cdot\mathbf{n}_t /[[\rho]]$ on $\Gamma_t$ we need
$W^{2-1/q}_q$ regularity of $\rho_+$ on $\Gamma_t$, which means the regularity
loss. Since the regularity of $\rho_+$ dominated by the Navier-Stokes-Korteweg
equations is $W^3_q$ in space, we eliminate the problem by using the
Navier-Stokes-Korteweg equations instead of the compressible Navier-Stokes
equations.Comment: Typos correcte
Solitary wave solutions to a class of modified Green-Naghdi systems

We provide the existence and asymptotic description of solitary wave
solutions to a class of modified Green-Naghdi systems, modeling the propagation
of long surface or internal waves. This class was recently proposed by
Duch{\^e}ne, Israwi and Talhouk (Stud. Appl. Math.,137 (2016)) in order to
improve the frequency dispersion of the original Green-Naghdi system while
maintaining the same precision. The solitary waves are constructed from the
solutions of a constrained minimization problem. The main difficulties stem
from the fact that the functional at stake involves low order non-local
operators, intertwining multiplications and convolutions through Fourier
multipliers
Derivation of the Navier - Stokes - Poisson system with radiation for an
  accretion disk

We study the 3-D compressible barotropic radiation fluid dynamics system
describing the motion of the compressible rotating viscous fluid with
gravitation and radiation confined to a straight layer. We show that weak
solutions in the 3-D domain converge to the strong solution of the rotating 2-D
Navier-Stokes-Poisson system with radiation for all times less than the maximal
life time of the strong solution of the 2-D system when the Froude number is
small or to the strong solution of the rotating pure 2-D Navier- Stokes system
with radiation
Dynamic Transitions and Baroclinic Instability for 3D Continuously
  Stratified Boussinesq Flows

The main objective of this article is to study the nonlinear stability and
dynamic transitions of the basic (zonal) shear flows for the three-dimensional
continuously stratified rotating Boussinesq model. The model equations are
fundamental equations in geophysical fluid dynamics, and dynamics associated
with their basic zonal shear flows play a crucial role in understanding many
important geophysical fluid dynamical processes, such as the meridional
overturning oceanic circulation and the geophysical baroclinic instability. In
this paper, first we derive a threshold for the energy stability of the basic
shear flow, and obtain a criteria for nonlinear stability in terms of the
critical horizontal wavenumbers and the system parameters such as the Froude
number, the Rossby number, the Prandtl number and the strength of the shear
flow. Next we demonstrate that the system always undergoes a dynamic transition
from the basic shear flow to either a spatiotemporal oscillatory pattern or
circle of steady states, as the shear strength $\Lambda$ of the basic flow
crosses a critical threshold $\Lambda_c$. Also we show that the dynamic
transition can be either continuous or catastrophic, and is dictated by the
sign of a transition parameter $A$, fully characterizing the nonlinear
interactions of different modes. A systematic numerical method is carried out
to explore transition in different flow parameter regimes. We find that the
system admits only critical eigenmodes with horizontal wave indices $(0,m_y)$.
Such modes, horizontally have the pattern consisting of $m_y$-rolls aligned
with the x-axis. Furthermore, numerically we encountered continuous transitions
to multiple steady states, continuous and catastrophic transitions to
spatiotemporal oscillations.Comment: 20 pages, 7 figure
Evolutionary Oseen model for generalized Newtonian fluid with Multivalued Nonmonotone Friction Law

The paper deals with the non-stationary Oseen system of equations for the generalized Newtonian incompressible fluid with multivalued and nonmonotone frictional slip boundary conditions. First, we provide a result on existence of a unique solution to an abstract evolutionary inclusion involving the Clarke subdifferential term for a nonconvex function. We employ a method based on a surjectivity theorem for multivalued L-pseudomonotone operators. Then, we exploit the abstract result to prove the weak unique solvability of the Oseen system
Spatial dynamics methods for solitary waves on a ferrofluid jet

This paper presents existence theories for several families of axisymmetric
solitary waves on the surface of an otherwise cylindrical ferrofluid jet
surrounding a stationary metal rod. The ferrofluid, which is governed by a
general (nonlinear) magnetisation law, is subject to an azimuthal magnetic
field generated by an electric current flowing along the rod.
  The ferrohydrodynamic problem for axisymmetric travelling waves is formulated
as an infinite-dimensional Hamiltonian system in which the axial direction is
the time-like variable. A centre-manifold reduction technique is employed to
reduce the system to a locally equivalent Hamiltonian system with a finite
number of degrees of freedom, and homoclinic solutions to the reduced system,
which correspond to solitary waves, are detected by dynamical-systems methods
A Floating Cylinder on An Unbounded Bath

In this paper, we reconsider a circular cylinder horizontally floating on an
unbounded reservoir in a gravitational field directed downwards, which was
studied by Bhatnargar and Finn in 2006. We follow their approach but with some
modifications. We establish the relation between the total energy relative to
the undisturbed state and the total force. There is a monotone relation between
the height of the centre and the wetting angle. We study the number of
equilibria, the floating configurations and their stability for all parameter
values. We find that the system admits at most two equilibrium points for
arbitrary contact angle, the one with smaller wetting angle is stable and the
one with larger wetting angle is unstable. The initial model has a limitation
that the fluid interfaces may intersect. We show that the stable equilibrium
point never lies in the intersection region, while the unstable equilibrium
point may lie in the intersection region.Comment: 40 pages, 27 figure
Quaternionic Kaehler manifolds with Hermitian and Norden metrics

Almost hypercomplex manifolds with Hermitian and Norden metrics and more
specially the corresponding quaternionic Kaehler manifolds are considered. Some
necessary and sufficient conditions the investigated manifolds be isotropic
hyper-Kaehlerian and flat are found. It is proved that the quaternionic Kaehler
manifolds with the considered metric structure are Einstein for dimension at
least 8. The class of the non-hyper-Kaehler quaternionic Kaehler manifold of
the considered type is determined.Comment: 10 page
Codes from Hall planes of even order

Codes from Hall planes of even order, J. D. Key, T. P. McDonough and V. C. Mavron, Journal of Geometry, volume 105, issue 1, April 2014, pages 33?41.We show that the binary code C of the projective Hall plane Hq2 of even order q 2 where q = 2 t , for t?2 has words of weight 2q 2 in its hull that are not the difference of the incidence vectors of two lines of Hq2 ; together with an earlier result for the dual Hall planes of even order, this shows that for all t?2 the Hall plane and its dual are not tame. We also deduce that dim(C)>32t+1, the dimension of the binary code of the desarguesian projective plane of order 22t , thus supporting the Hamada?Sachar conjecture for this infinite class of planes.authorsversionPeer reviewe
The Euler and Grace-Danielsson inequalities for nested triangles and
  tetrahedra: a derivation and generalisation using quantum information theory

We derive several results in classical Euclidean elementary geometry using
the steering ellipsoid formalism from quantum mechanics. This gives a
physically motivated derivation of very non-trivial geometric results, some of
which are entirely new. We consider a sphere of radius $r$ contained inside
another sphere of radius $R$, with the sphere centres separated by distance
$d$. When does there exist a nested tetrahedron circumscribed about the smaller
sphere and inscribed in the larger? We derive the Grace-Danielsson inequality
$d^2 \leq (R+r)(R-3r)$ as the sole necessary and sufficient condition for the
existence of a nested tetrahedron. Our method also gives the condition $d^2
\leq R(R-2r)$ for the existence of a nested triangle in the analogous
2-dimensional scenario. These results imply the Euler inequality in 2 and 3
dimensions. Furthermore, we formulate a new inequality that applies to the more
general case of ellipses and ellipsoids.Comment: 8 pages, 1 figure. Published versio
Geodesics in generalized Wallach spaces

We study geodesics in generalized Wallach spaces which are expressed as
orbits of products of three exponential terms. These are homogeneous spaces
$M=G/K$ whose isotropy representation decomposes into a direct sum of three
submodules $\frak{m}=\frak{m}_1\oplus\frak{m}_2\oplus\frak{m}_3$, satisfying
the relations $[\frak{m}_i,\frak{m}_i]\subset \frak{k}$. Assuming that the
submodules $\frak{m}_i$ are pairwise non isomorphic, we study geodesics on such
spaces of the form $\gamma (t)=\exp (tX)\exp (tY)\exp (tZ)\cdot o$, where
$X\in\fr{m}_1, Y\in\fr{m}_2, Z\in\fr{m}_3$ ($o=eK$), with respect to a
$G$-invariant metric. Our investigation imposes certain restrictions on the
$G$-invariant metric, so the geodesics turn out to be orbits of two exponential
terms.
  We give a point of view using Riemannian submersions.
  As an application, we describe geodesics in generalized flag manifolds with
three isotropy summands and with second Betti number $b_2(M)=2$, and in the
Stiefel manifolds $SO(n+2)/S(n)$. We relate our results to geodesic orbit
spaces (g.o. spaces).Comment: Journal of Geometry (2015
Measure Partitions via Fourier Analysis II: Center Transversality in the
  $L^2$-norm for Complex Hyperplanes

Applications of harmonic analysis on finite groups were recently introduced
to measure partition problems, with a variety of equipartition types by convex
fundamental domains obtained as the vanishing of prescribed Fourier transforms.
Considering the circle group, we extend this approach to the compact Lie group
setting, in which case the annihilation of transforms in the classical Fourier
series produces measure transversality similar in spirit to the classical
centerpoint theorem of Rado: for any $q\geq 2$, the existence of a complex
hyperplane whose surrounding regular $q$-fans are close -- in an $L^2$-sense --
to equipartitioning a given set of measures. The proofs of these results
represent the first application of continuous as opposed to finite group
actions in the usual equivariant topological reductions prevalent in
combinatorial geometry.Comment: 7 page