pinecone/core-2020-05-10-deduplication huggingface dataset:
On the initial- and boundary-value problem for 2D micropolar equations
  with only angular velocity dissipation

This paper focuses on the initial- and boundary-value problem for the
two-dimensional micropolar equations with only angular velocity dissipation in
a smooth bounded domain. The aim here is to establish the global existence and
uniqueness of solutions by imposing natural boundary conditions and minimal
regularity assumptions on the initial data. Besides, the global solution is
shown to possess higher regularity when the initial datum is more regular. To
obtain these results, we overcome two main difficulties, one due to the lack of
full dissipation and one due to the boundary conditions. In addition to the
global regularity problem, we also examine the large-time behavior of solutions
and obtain explicit decay rates.Comment: 29 page
A Convergent Finite Difference Scheme for the Variational Heat Equation

The variational heat equation is a nonlinear, parabolic equation not in
divergence form that arises as a model for the dynamics of the director field
in a nematic liquid crystal. We present a finite difference scheme for a
transformed, possibly degenerate version of this equation and prove that a
subsequence of the numerical solutions converges to a weak solution. This
result is supplemented by numerical examples that show that weak solutions are
not unique and give some intuition about how to obtain the physically relevant
solution
Periodic traveling interfacial hydroelastic waves with or without mass

We study the motion of an interface between two irrotational, incompressible
fluids, with elastic bending forces present; this is the hydroelastic wave
problem. We prove a global bifurcation theorem for the existence of families of
spatially periodic traveling waves on infinite depth. Our traveling wave
formulation uses a parameterized curve, in which the waves are able to have
multi-valued height. This formulation and the presence of the elastic bending
terms allows for the application of an abstract global bifurcation theorem of
"identity plus compact" type. We furthermore perform numerical computations of
these families of traveling waves, finding that, depending on the choice of
parameters, the curves of traveling waves can either be unbounded, reconnect to
trivial solutions, or end with a wave which has a self-intersection. Our
analytical and computational methods are able to treat in a unified way the
cases of positive or zero mass density along the sheet, the cases of
single-valued or multi-valued height, and the cases of single-fluid or
interfacial waves.Comment: 29 pages, 4 figure
Cylindrically Symmetric Ground State Solutions for Curl-Curl Equations
  with Critical Exponent

We study the following nonlinear critical curl-curl equation
\begin{equation}\label{eq0.1}\nabla\times \nabla\times U +V(x)U=|U|^{p-2}U+
|U|^4U,\quad x\in \mathbb{R}^3,\end{equation} where $V(x)=V(r, x_3)$ with
$r=\sqrt{x_1^2+x_2^2}$ is 1-periodic in $x_3$ direction and belongs to
$L^\infty(\R^3)$. When $0\not\in \sigma(-\Delta+\frac{1}{r^2}+V)$ and
$p\in(4,6)$, we prove the existence of nontrivial solution for (\ref{eq0.1}),
which is indeed a ground state solution in a suitable cylindrically symmetric
space. Especially, if $ \sigma(-\Delta+\frac{1}{r^2}+V)>0$, a ground state
solution is obtained for any $p\in(2,6)$
A free boundary approach to the Rosensweig instability of ferrofluids

We establish the existence of saddle points for a free boundary problem
describing the two-dimensional free surface of a ferrofluid which undergoes
normal field instability (also known as Rosensweig instability). The starting
point consists in the ferro-hydrostatic equations for the magnetic potentials
in the ferrofluid and air, and the function describing their interface. The
former constitute the strong form for the Euler-Lagrange equations of a
convex-concave functional. We extend this functional in order to include
interfaces that are not necessarily graphs of functions. Saddle points are then
found by iterating the direct method of the calculus of variations and by
applying classical results of convex analysis. For the existence part we assume
a general (arbitrary) non linear magnetization law. We also treat the case of a
linear law: we show, via convex duality arguments, that the saddle point is a
constrained minimizer of the relevant energy functional of the physical
problem
Asymptotic stability of shear-flow solutions to incompressible viscous
  free boundary problems with and without surface tension

This paper concerns the dynamics of a layer of incompressible viscous fluid
lying above a rigid plane and with an upper boundary given by a free surface.
The fluid is subject to a constant external force with a horizontal component,
which arises in modeling the motion of such a fluid down an inclined plane,
after a coordinate change. We consider the problem both with and without
surface tension for horizontally periodic flows. This problem gives rise to
shear-flow equilibrium solutions, and the main thrust of this paper is to study
the asymptotic stability of the equilibria in certain parameter regimes. We
prove that there exists a parameter regime in which sufficiently small
perturbations of the equilibrium at time $t=0$ give rise to global-in-time
solutions that return to equilibrium exponentially in the case with surface
tension and almost exponentially in the case without surface tension. We also
establish a vanishing surface tension limit, which connects the solutions with
and without surface tension.Comment: 33 page
A thermodynamically consistent model of magneto-elastic materials under
  diffusion at large strains and its analysis

The theory of elastic magnets is formulated under possible diffusion and heat
flow governed by Fick's and Fourier's laws in the deformed (Eulerian)
configuration, respectively. The concepts of nonlocal nonsimple materials and
viscous Cahn-Hilliard equations are used. The formulation of the problem uses
Lagrangian (reference) configuration while the transport processes are pulled
back. Except the static problem, the demagnetizing energy is ignored and only
local non-selfpenetration is considered. The analysis as far as existence of
weak solutions of the (thermo)dynamical problem is performed by a careful
regularization and approximation by a Galerkin method, suggesting also a
numerical strategy. Either ignoring or combining particular aspects, the model
has numerous applications as ferro-to-paramagnetic transformation in elastic
ferromagnets, diffusion of solvents in polymers possibly accompanied by
magnetic effects (magnetic gels), or metal-hydride phase transformation in some
intermetalics under diffusion of hydrogen accompanied possibly by magnetic
effects (and in particular ferro-to-antiferromagnetic phase transformation),
all in the full thermodynamical context under large strains
Moffatt vortices: Concerns and Finiteness

Till date, the sequence of vortices present in the solid corners of steady
internal viscous incompressible flows, widely known as Moffatt vortices was
thought to be infinite. However, the already existing and most recent geometric
theories on incompressible viscous flows that express vortical structures in
terms of critical points in bounded domains, indicate a strong opposition to
this notion of infiniteness. In this study, we endeavor to bridge the gap
between the two opposing stream of thoughts by addressing what might have gone
wrong and pinpoint the shortcomings on the assumptions of the existing theorems
on Moffatt vortices. We provide our own set of proofs for establishing the
finiteness of the sequence of Moffatt vortices by making use of the continuum
hypothesis and Kolmogorov scale, which guarantee a non-zero scale for the
smallest vortex structure possible in incompressible viscous flows. We point
out that the notion of infiniteness resulting from discrete self-similarity of
the vortex structures is not physically feasible. The centers of these vortices
have been quantified by us as fixed points through Brouwer fixed-point theorem
and boundary of a vortex as circle cell. With the aid of these new developments
and making use of some existing theorems in topology along with some elementary
concept of mathematical analysis, we provide several approaches to delve into
this issue. All these approaches converge to the same conclusion that the
sequence of Moffatt vortices cannot be infinite; in fact, it is at most finite.Comment: 21 pages, 4 figures. New representation with different title of the
  previous version of the article "The Finiteness of Moffatt vortices
Global, finite energy, weak solutions for the NLS with rough,
  time-dependent magnetic potentials

We prove the existence of weak solutions in the space of energy for a class
of non-linear Schroedinger equations in the presence of a external rough
magnetic potential. Under our assumptions it is not possible to study the
problem by means of usual arguments like resolvent techniques or Fourier
integral operators, for example. We use a parabolic regularization and we solve
the approximating Cauchy problem. This is achieved by obtaining suitable
smoothing estimates for the dissipative evolution. The total mass and energy
bounds allow to extend the solution globally in time. We then infer sufficient
compactness properties in order to produce a global-in-time finite energy weak
solution to our original problem.Comment: Submitted to Zeitschrift f\"ur angewandte Mathematik und Physi
Convergence to equilibrium of renormalised solutions to nonlinear
  chemical reaction-diffusion systems

The convergence to equilibrium for renormalised solutions to nonlinear
reaction-diffusion systems is studied. The considered reaction-diffusion
systems arise from chemical reaction networks with mass action kinetics and
satisfy the complex balanced condition. By applying the so-called entropy
method, we show that if the system does not have boundary equilibria, then any
renormalised solution converges exponentially to the complex balanced
equilibrium with a rate, which can be computed explicitly up to a finite
dimensional inequality. This inequality is proven via a contradiction argument
and thus not explicitly. An explicit method of proof, however, is provided for
a specific application modelling a reversible enzyme reaction by exploiting the
specific structure of the conservation laws.
  Our approach is also useful to study the trend to equilibrium for systems
possessing boundary equilibria. More precisely, to show the convergence to
equilibrium for systems with boundary equilibria, we establish a sufficient
condition in terms of a modified finite dimensional inequality along
trajectories of the system. By assuming this condition, which roughly means
that the system produces too much entropy to stay close to a boundary
equilibrium for infinite time, the entropy method shows exponential convergence
to equilibrium for renormalised solutions to complex balanced systems with
boundary equilibria.Comment: 25 page
Measure-valued solutions to the complete Euler system revisited

We consider the complete Euler system describing the time evolution of a
general inviscid compressible fluid. We introduce a new concept of
measure-valued solution based on the total energy balance and entropy
inequality for the physical entropy without any renormalization. This class of
so-called dissipative measure-valued solutions is large enough to include the
vanishing dissipation limits of the Navier-Stokes-Fourier system. Our main
result states that any sequence of weak solutions to the Navier-Stokes-Fourier
system with vanishing viscosity and heat conductivity coefficients generates a
dissipative measure-valued solution of the Euler system under some physically
grounded constitutive relations. Finally, we discuss the same asymptotic limit
for the bi-velocity fluid model introduced by H.Brenner
On the convergence of a fully discrete scheme of LES type to physically
  relevant solutions of the incompressible Navier-Stokes

Obtaining reliable numerical simulations of turbulent fluids is a challenging
problem in computational fluid mechanics. The Large Eddy Simulations (LES)
models are efficient tools to approximate turbulent fluids and an important
step in the validation of these models is the ability to reproduce relevant
properties of the flow. In this paper we consider a fully discrete
approximation of the Navier-Stokes-Voigt model by an implicit Euler algorithm
(with respect to the time variable) and a Fourier-Galerkin method (in the space
variables). We prove the convergence to weak solutions of the incompressible
Navier-Stokes equations satisfying the natural local entropy condition, hence
selecting the so-called physically relevant solution
Quasi-neutral limit of Euler-Poisson system of compressible fluids
  coupled to a magnetic field

In this paper, we consider the quasi-neutral limit of a three dimensional
Euler-Poisson system of compressible fluids coupled to a magnetic field. We
prove that, as Debye length tends to zero, periodic initial-value problems of
the model have unique smooth solutions existing in the time interval where the
ideal incompressible magnetohydrodynamic equations has smooth solution.
Meanwhile, it is proved that smooth solutions converge to solutions of
incompressible magnetohydrodynamic equations with a sharp convergence rate in
the process of quasi-neutral limit
Energy decay of a viscoelastic wave equation with supercritical
  nonlinearities

This paper presents a study of the asymptotic behavior of the solutions for
the history value problem of a viscoelastic wave equation which features a
fading memory term as well as a supercritical source term and a frictional
damping term: \begin{align*} \begin{cases} u_{tt}- k(0) \Delta u -
\int_0^{\infty} k'(s) \Delta u(t-s) ds +|u_t|^{m-1}u_t =|u|^{p-1}u, \quad
\text{ in } \Omega \times (0,T), \\ u(x,t)=u_0(x,t), \quad \text{ in } \Omega
\times (-\infty,0], \end{cases} \end{align*} where $\Omega$ is a bounded domain
in $\mathbb R^3$ with a Dirichl\'et boundary condition and $u_0$ represents the
history value. A suitable notion of a potential well is introduced for the
system, and global existence of solutions is justified provided that the
history value $u_0$ is taken from a subset of the potential well. Also, uniform
energy decay rate is obtained which depends on the relaxation kernel $-k'(s)$
as well as the growth rate of the damping term. This manuscript complements our
previous work [Guo et al. in J Differ Equ 257, 3778-3812(2014), J Differ Equ
262, 1956-1979(2017)] where Hadamard well-posedness and the singularity
formulation have been studied for the system. It is worth stressing the special
features of the model, namely the source term here has a supercritical growth
rate and the memory term accounts to the full past history that goes back to
$-\infty$
The fractional diffusion limit of a kinetic model with biochemical
  pathway

Kinetic-transport equations that take into account the intra-cellular
pathways are now considered as the correct description of bacterial chemotaxis
by run and tumble. Recent mathematical studies have shown their interest and
their relations to more standard models. Macroscopic equations of Keller-Segel
type have been derived using parabolic scaling. Due to the randomness of
receptor methylation or intra-cellular chemical reactions, noise occurs in the
signaling pathways and affects the tumbling rate. Then, comes the question to
understand the role of an internal noise on the behavior of the full
population. In this paper we consider a kinetic model for chemotaxis which
includes biochemical pathway with noises. We show that under proper scaling and
conditions on the tumbling frequency as well as the form of noise, fractional
diffusion can arise in the macroscopic limits of the kinetic equation. This
gives a new mathematical theory about how long jumps can be due to the internal
noise of the bacteria
Large time behavior of entropy solutions to one-dimensional unipolar
  hydrodynamic model for semiconductor devices

We are concerned with the global existence and large time behavior of entropy
solutions to the one dimensional unipolar hydrodynamic model for semiconductors
in the form of Euler-Poisson equations in a bounded interval. In this paper, we
first prove the global existence of entropy solution by vanishing viscosity and
compensated compactness framework. In particular, the solutions are uniformly
bounded with respect to space and time variables by introducing modified
Riemann invariants and the theory of invariant region. Based on the uniform
estimates of density, we further show that the entropy solution converges to
the corresponding unique stationary solution exponentially in time. No any
smallness condition is assumed on the initial data and doping profile.
Moreover, the novelty in this paper is about the unform bound with respect to
time for the weak solutions of the isentropic Euler-Possion system
Asymptotic decay and non-rupture of viscous sheets

For a nonlinear system of coupled PDEs, that describes evolution of a viscous
thin liquid sheet and takes account of surface tension at the free surface, we
show exponential $(H^1,\,L^2)$ asymptotic decay to the flat profile of its
solutions considered with general initial data. Additionally, by transforming
the system to Lagrangian coordinates we show that the minimal thickness of the
sheet stays positive for all times. This result proves the conjecture formally
accepted in the physical literature [1], that a viscous sheet can not rupture
in finite time in absence of external forcing. Moreover, in absence of surface
tension we find a special class of initial data for which the Lagrangian
solution exhibits $L^2$-exponential decay to the flat profile
Line and Point Defects in Nonlinear Anisotropic Solids

In this paper, we present some analytical solutions for the stress fields of
nonlinear anisotropic solids with distributed line and point defects. In
particular, we determine the stress fields of i) a parallel
cylindrically-symmetric distribution of screw dislocations in infinite
orthotropic and monoclinic media, ii) a cylindrically-symmetric distribution of
parallel wedge disclinations in an infinite orthotropic medium, iii) a
distribution of edge dislocations in an orthotropic medium, and iv) a
spherically-symmetric distribution of point defects in a transversely isotropic
spherical ball
Groundstates of the Choquard equations with a sign-changing
  self-interaction potential

We consider a nonlinear Choquard equation $$
  -\Delta u+u= (V * |u|^p )|u|^{p-2}u \qquad \text{in }\mathbb{R}^N, $$ when
the self-interaction potential $V$ is unbounded from below. Under some
assumptions on $V$ and on $p$, covering $p =2$ and $V$ being the one- or
two-dimensional Newton kernel, we prove the existence of a nontrivial
groundstate solution $u\in H^1 (\mathbb{R}^N)\setminus\{0\}$ by solving a
relaxed problem by a constrained minimization and then proving the convergence
of the relaxed solutions to a groundstate of the original equation.Comment: 16 page
On physical-constraints-preserving schemes for special relativistic
  magnetohydrodynamics with a general equation of state

The paper studies the physical-constraints-preserving (PCP) schemes for
multi-dimensional special relativistic magnetohydrodynamics with a general
equation of state (EOS) on more general meshes. It is an extension of the work
[Math. Models Methods Appl. Sci., 27:1871-1928, 2017] which focuses on the
ideal EOS and uniform Cartesian meshes. The general EOS without a special
expression poses some additional difficulties in discussing the mathematical
properties of admissible state set with the physical constraints on the fluid
velocity, density and pressure. Rigorous analyses are provided for the PCP
property of finite volume or discontinuous Galerkin schemes with the
Lax-Friedrichs (LxF) type flux on a general mesh with non-self-intersecting
polytopes. Those are built on a more general form of generalized LxF splitting
property and a different convex decomposition technique. It is shown in theory
that the PCP property is closely connected with a discrete divergence-free
condition, which is proposed on the general mesh and milder than that in [Math.
Models Methods Appl. Sci., 27:1871-1928, 2017]
The Korteweg-de Vries equation on a metric star graph

We prove local well-posedness for the Cauchy problem associated to
Korteweg-de Vries equation on a metric star graph with three semi-infinite
edges given by one negative half-line and two positives half-lines attached to
a common vertex, for two classes of boundary conditions. The results are
obtained in the low regularity setting by using the Duhamel Boundary Forcing
Operator, in context of half-lines, introduced by Colliander, Kenig (2002),and
extended by Holmer (2006) and Cavalcante (2017).Comment: 24 pages, 1 figure. A final version. Published in Zeitschrift f\"ur
  angewandte Mathematik und Physi
Stability estimate for the Helmholtz equation with rapidly jumping
  coefficients

The goal of this paper is to investigate the stability of the Helmholtz
equation in the high- frequency regime with non-smooth and rapidly oscillating
coefficients on bounded domains. Existence and uniqueness of the problem can be
proved using the unique continuation principle in Fredholm's alternative.
However, this approach does not give directly a coefficient-explicit energy
estimate. We present a new theoretical approach for the one-dimensional problem
and find that for a new class of coefficients, including coefficients with an
arbitrary number of discontinuities, the stability constant (i.e., the norm of
the solution operator) is bounded by a term independent of the number of jumps.
We emphasize that no periodicity of the coefficients is required. By selecting
the wave speed function in a certain \resonant" way, we construct a class of
oscillatory configurations, such that the stability constant grows
exponentially in the frequency. This shows that our estimates are sharp.Comment: a) Added references, b) rewritten the introduction with a summary of
  the results/techniques of the paper, c) Corrected typo
A More Secure Chaotic Cryptographic Scheme based on The Dynamic Look-up Table

We have proposed a chaotic cryptographic scheme based on iterating the logistic map and updating the look-up table dynamically. However, it has been broken recently. In this paper, the weaknesses of the original dynamic look-up table scheme are analyzed and a more secure chaotic encryption scheme based on this dynamic look-up table concept is proposed. Theoretical analysis indicates that the modified scheme can resist the reported attacks. Moreover, simulation results show that this encryption scheme leads to a flat ciphertext distribution
Robust constrained model predictive control based on parameter-dependent Lyapunov functions

The problem of robust constrained model predictive control (MPC) of systems with polytopic uncertainties is considered in this paper. New sufficient conditions for the existence of parameter-dependent Lyapunov functions are proposed in terms of linear matrix inequalities (LMIs), which will reduce the conservativeness resulting from using a single Lyapunov function. At each sampling instant, the corresponding parameter-dependent Lyapunov function is an upper bound for a worst-case objective function, which can be minimized using the LMI convex optimization approach. Based on the solution of optimization at each sampling instant, the corresponding state feedback controller is designed, which can guarantee that the resulting closed-loop system is robustly asymptotically stable. In addition, the feedback controller will meet the specifications for systems with input or output constraints, for all admissible time-varying parameter uncertainties. Numerical examples are presented to demonstrate the effectiveness of the proposed techniques