pinecone/core-2020-05-10-deduplication huggingface dataset:
Waveguide with non-periodically alternating Dirichlet and Robin
  conditions: homogenization and asymptotics

We consider a magnetic Schroedinger operator in a planar infinite strip with
frequently and non-periodically alternating Dirichlet and Robin boundary
conditions. Assuming that the homogenized boundary condition is the Dirichlet
or the Robin one, we establish the uniform resolvent convergence in various
operator norms and we prove the estimates for the rates of convergence. It is
shown that these estimates can be improved by using special boundary
correctors. In the case of periodic alternation, pure Laplacian, and the
homogenized Robin boundary condition, we construct two-terms asymptotics for
the first band functions, as well as the complete asymptotics expansion (up to
an exponentially small term) for the bottom of the band spectrum
The rolling ball problem on the plane revisited

Agraïments: The third author thanks the FCT (Portugal) for the partial support through Program POCTI/FEDER and PDCT/MAT/56476/2004.By a sequence of rollings without slipping or twisting along segments of an straight line of the plane a spherical ball of unit radius has to be transferred from an initial state to an arbitrary final state taking into account the orientation of the ball. We provide a new proof that with at most 3 moves we can go from a given initial state to an arbitrary final state. The first proof of this result is due to Hammersley [3]. His proof is more algebraic than ours which is more geometric
Entropy formulation of degenerate parabolic equation with zero-flux boundary condition

19 pagesWe consider the general degenerate hyperbolic-parabolic equation: \begin{equation}\label{E}\tag{E} u_t+\div f(u)-\Delta\phi(u)=0 \mbox{ in } Q = (0,T)\times\Omega,\;\;\;\; T>0,\;\;\;\Omega\subset\mathbb R^N ; \end{equation} with initial condition and the zero flux boundary condition. Here $\phi$ is a continuous non decreasing function. Following [B\"{u}rger, Frid and Karlsen, J. Math. Anal. Appl, 2007], we assume that $f$ is compactly supported (this is the case in several applications) and we define an appropriate notion of entropy solution. Using vanishing viscosity approximation, we prove existence of entropy solution for any space dimension $N\geq 1$ under a partial genuine nonlinearity assumption on $f$. Uniqueness is shown for the case $N=1$, using the idea of [Andreianov and Bouhsiss, J. Evol. Equ., 2004], nonlinear semigroup theory and a specific regularity result for one dimension
Long-time behavior of an angiogenesis model with flux at the tumor
  boundary

This paper deals with a nonlinear system of partial differential equations
modeling a simplified tumor-induced angiogenesis taking into account only the
interplay between tumor angiogenic factors and endothelial cells. Considered
model assumes a nonlinear flux at the tumor boundary and a nonlinear
chemotactic response. It is proved that the choice of some key parameters
influences the long-time behaviour of the system. More precisely, we show the
convergence of solutions to different semi-trivial stationary states for
different range of parameters.Comment: 17 page
Well-Posedness of Transonic Characteristic Discontinuities in
  Two-Dimensional Steady Compressible Euler Flows

In our previous work, we have established the existence of transonic
characteristic discontinuities separating supersonic flows from a static gas in
two-dimensional steady compressible Euler flows under a perturbation with small
total variation of the incoming supersonic flow over a solid right-wedge. It is
a free boundary problem in Eulerian coordinates and, across the free boundary
(characteristic discontinuity), the Euler equations are of elliptic-hyperbolic
composite-mixed type. In this paper, we further prove that such a transonic
characteristic discontinuity solution is unique and $L^1$--stable with respect
to the small perturbation of the incoming supersonic flow in Lagrangian
coordinates.Comment: 18 pages, 1 figur
Algebro-geometric solutions for the two-component Hunter-Saxton
  hierarchy

This paper is dedicated to provide theta function representations of
algebro-geometric solutions and related crucial quantities for the
two-component Hunter-Saxton (HS2) hierarchy through studying an
algebro-geometric initial value problem. Our main tools include the polynomial
recursive formalism, the hyperelliptic curve with finite number of genus, the
Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations
for auxiliary divisors, and the associated trace formulas. With the help of
these tools, the explicit representations of the algebro-geometric solutions
are obtained for the entire HS2 hierarchy.Comment: 46 pages. accepted for publication J Nonl Math Phys, 2014. arXiv
  admin note: substantial text overlap with arXiv:1406.6153, arXiv:1207.0574,
  arXiv:1205.6062; and with arXiv:nlin/0105021 by other author
On the stability loss for an euler beam resting on a tensionless pasternak foundation

In the present work, the tensionless contact problem of an Euler–Bernoulli beam of finite length resting on a two-parameter Pasternak-type foundation is investigated. Owing to the tensionless character of the contact, the beam may lift-off the foundation and the point where contact ceases and detachment begins, named contact locus, needs be assessed. In this situation, a one-dimensional free boundary problem is dealt with. An extra condition, in the form of a homogeneous second-order equation in the displacement and its derivatives, is demanded to set the contact locus and it gives the problem its nonlinear feature. Conversely, the loading and the beam length may be such that the beam rests entirely supported on the foundation, which situation is governed by a classical linear boundary value problem. In this work, contact evolution is discussed for a continuously varying loading condition, starting from a symmetric layout and at a given beam length, until overturning is eventually reached. In particular, stability is numerically assessed through the energy criterion, which is shown to stand for the free boundary situation as well. At overturning, a descending pathway in the system energy appears and stability loss is confirmed
Inverse Jacobi multipliers and first integrals for nonautonomous differential systems

In this paper we consider nonautonomous differential systems of arbitrary dimension and first find expressions for their inverse Jacobi multipliers and first integrals in some nonautonomous invariant set in terms of the solutions of the differential system. Given an inverse Jacobi multiplier $V$, we find a relation between the Poincar\'{e} translation map $\Pi$ at time $T$ that extends to arbitrary dimensions the fundamental relation for scalar equations, $V(T,\Pi(x))=V(0,x)\Pi'(x)$, found in Trans. Amer. Math. Soc. 362 (2010), 3591-3612. The main result guarantees the existence of continua of $T$-periodic solutions for $T$-periodic systems in the presence of $T$-periodic first integrals and inverse Jacobi multipliers.The authors are partially supported by a MCYT/FEDER grant number MTM2008-00694
and by a CIRIT grant number 2014 SGR 1204
Existence and non-existence of Blow-up solutions for a non-autonomous
  problem with indefinite and gradient terms

We deal with existence and non-existence of non-negative entire solutions
that blow-up at infinity for a quasilinear problem depending on a non-negative
real parameter. Our main objectives in this paper are to provide far more
general conditions for existence and non-existence of solutions. To this end,
we explore an associated $\mu$-parameter convective ground state problem, sub
and super solutions method combined and an approximation arguments to show
existence of solutions. To show the result of non-existence of solutions, we
follow an idea due to Mitidieri-Pohozaev
Improving the averaging theory for computing periodic solutions of the differential equations

Agraïments: FEDER-UNAB-10-4E-378. The  second  author  is  partially  supported  by  a  FAPESP–BRAZIL grant 2013/16492–0.  The two authors are also supported by a CAPES CSF–PVE grant 88881.030454/ 2013-01.For m = 1, 2, 3, we consider differential systems of the form x0 = F0(t, x) +Xmi=1εiFi(t, x) + εm+1R(t, x, ε), where Fi: R × D → Rn, and R : R × D × (−ε0, ε0) → Rn are Cm+1 functions, and T–periodic in the first variable, being D an open subset of Rn, and ε a small parameter. For such system we assume that the unperturbed system x0 = F0(t, x) has a k–dimensional manifold of periodic solutions with k ≤ n. We weaken the sufficient assumptions for studying the periodic solutions of the perturbed system when (ε) > 0 is sufficiently small
The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity,
  planar polyconvexity and existence of minimizers

We consider a family of isotropic volumetric-isochoric decoupled strain
energies $$ F\mapsto W_{\rm eH}(F):=\widehat{W}_{\rm
eH}(U):=\left\{\begin{array}{lll} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log
{U}\|^2}+\frac{\kappa}{2\hat{k}}\,e^{\hat{k}\,[{\rm tr}(\log U)]^2}&\text{if}&
{\rm det}\, F>0,\\ +\infty &\text{if} &{\rm det} F\leq 0,
\end{array}\right.\quad $$ based on the Hencky-logarithmic (true, natural)
strain tensor $\log U$, where $\mu>0$ is the infinitesimal shear modulus,
$\kappa=\frac{2\mu+3\lambda}{3}>0$ is the infinitesimal bulk modulus with
$\lambda$ the first Lam\'{e} constant, $k,\hat{k}$ are dimensionless
parameters, $F=\nabla \varphi$ is the gradient of deformation, $U=\sqrt{F^T F}$
is the right stretch tensor and ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n} {\rm
tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part (the projection onto the
traceless tensors) of the strain tensor $\log U$. For small elastic strains the
energies reduce to first order to the classical quadratic Hencky energy $$
  F\mapsto W{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U):={\mu}\,\|{\rm dev}_n\log
U\|^2+\frac{\kappa}{2}\,[{\rm tr}(\log U)]^2, $$
  which is known to be not rank-one convex.
  The main result in this paper is that in plane elastostatics the energies of
the family $W_{_{\rm eH}}$ are polyconvex for $k\geq \frac{1}{3}$,
$\widehat{k}\geq \frac{1}{8}$, extending a previous finding on its rank-one
convexity. Our method uses a judicious application of Steigmann's polyconvexity
criteria based on the representation of the energy in terms of the principal
invariants of the stretch tensor $U$. These energies also satisfy suitable
growth and coercivity conditions. We formulate the equilibrium equations and we
prove the existence of minimizers by the direct methods of the calculus of
variations
Bifurcations for a Coupled Schr\"odinger System with Multiple Components

In this paper, we study local bifurcations of an indefinite elliptic system
with multiple components: \begin{equation*}
  \left\{\begin{array}{ll}
  -\Delta u_j + au_j = \mu_ju_j^3+\beta\sum_{k\ne j}u_k^2u_j,
  u_j>0\ \ \hbox{in}\ \Omega, u_j=0 \ \ \hbox{on}\ \partial\Omega,\
j=1,\dots,n.
  \end{array}
  \right. \end{equation*} Here $\Omega\subset{\mathbb{R}}^N$ is a smooth and
bounded domain, $n\ge3$, $a<-\Lambda_1$ where $\Lambda_1$ is the principal
eigenvalue of $(-\Delta, H_0^1(\Omega))$; $\mu_j$ and $\beta$ are real
constants. Using the positive and non-degenerate solution of the scalar
equation $-\Delta\omega-\omega=-\omega^3$, $\omega\in H_0^1(\Omega)$, we
construct a synchronized solution branch $\mathcal{T}_\omega$. Then we find a
sequence of local bifurcations with respect to $\mathcal{T}_\omega$, and we
find global bifurcation branches of partially synchronized solutions.Comment: 16 pages, 2 figure
Boundedness in a quasilinear fully parabolic Keller-Segel system with
  logistic source

This paper deals with the Neumann boundary value problem for the system
  $$u_t=\nabla\cdot\left(D(u)\nabla u\right)-\nabla\cdot\left(S(u)\nabla
v\right)+f(u) ,\quad x\in\Omega,\ t>0$$
  $$v_t=\Delta v-v+u,\quad x\in\Omega,\ t>0$$
  in a smooth bounded domain $\Omega\subset\mathbb{R}^n$ $(n\geq1)$, where the
functions $D(u)$ and $S(u)$ are supposed to be smooth satisfying $D(u)\geq
Mu^{-\alpha}$ and $S(u)\leq Mu^{\beta}$ with $M>0$, $\alpha\in\mathbb{R}$ and
$\beta\in\mathbb{R}$ for all $u\geq1$, and the logistic source $f(u)$ is smooth
fulfilling $f(0)\geq0$ as well as $f(u)\leq a-\mu u^{\gamma}$ with $a\geq0$,
$\mu>0$ and $\gamma\geq1$ for all $u\geq0$. It is shown that if
  $\alpha+2\beta<\gamma-1+\frac{2}{n}$, for $1\leq\gamma<2$ and
$\alpha+2\beta<\gamma-1+\frac{4}{n+2}$, for $\gamma\geq2$,
  then for sufficiently smooth initial data the problem possesses a unique
global classical solution which is uniformly bounded.Comment: 10 page
Existence and large time behavior for a stochastic model of modified magnetohydrodynamic equations

In this paper, we study a system of nonlinear stochastic partial differential equations describing the motion of
turbulent non-Newtonian media in the presence of fluctuating magnetic field. The system is basically obtained by a coupling
of the dynamical equations of a non-Newtonian fluids having p-structure and the Maxwell equations. We mainly show the
existence of weak martingale solutions and their exponential decay when time goes to infinity.Austrian Science Foundation through the Lise Meitner Program M1487 and the.National
Research Foundation of South Africa.http://link.springer.com/journal/332016-10-26hb201
A semilinear parabolic system with a free boundary

This paper deals with a semilinear parabolic system with free boundary in one
space dimension. We suppose that unknown functions $u$ and $v$ undergo
nonlinear reactions $u^q$ and $v^p$, and exist initially in a interval $\{0\leq
x\leq s(0)\}$, but expand to the right with spreading front $\{x=s(t)\}$, with
$s(t)$ evolving according to the free boundary condition $s'(t)=-\mu (u_x+\rho
v_x)$, where $p,\, q,\, \mu, \,\rho$ are given positive constants. The main
purpose of this paper is to understand the existence, uniqueness, regularity
and long time behavior of positive solution or maximal positive solution.
Firstly, we prove that this problem has a unique positive solution $(u,v,s)$
defined in the maximal existence interval $[0,T_{\max})$ when $p,\,q\geq 1$,
while it has a unique maximal positive solution $(u,v,s)$ defined in the
maximal existence interval $[0,T_{\max})$ when $p<1$ or $q<1$. Moreover,
$(u,v,s)$ and $T_{\max}$ have property that either (i) $T_{\max}=+\infty$, or
(ii) $T_{\max}<+\infty$ and
  $$ \limsup_{T\nearrow
T_{\max}}\|u,\,v\|_{L^{\infty}([0,T]\times[0,s(t)])}=+\infty.$$ Then we study
the regularity of $(u,v)$ and $s$. At last, we discuss the global existence
($T_{\max}=+\infty$), finite time blow-up ($T_{\max}<+\infty$), and long time
behavior of bounded global solution.Comment: 26 page
Boundedness in a three-dimensional chemotaxis-haptotaxis model

This paper studies the chemotaxis-haptotaxis system \begin{equation}\nonumber
\left\{ \begin{array}{llc} u_t=\Delta u-\chi\nabla\cdot(u\nabla
v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), &(x,t)\in \Omega\times (0,T),\\
v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T),\\ w_t=-vw,&(x,t)\in \Omega\times
(0,T) \end{array} \right.\quad\quad(\star) \end{equation} under Neumann
boundary conditions. Here $\Omega\subset\mathbb{R}^3$ is a bounded domain with
smooth boundary and the parameters $\xi,\chi,\mu>0$. We prove that for
nonnegative and suitably smooth initial data $(u_0,v_0,w_0)$, if $\chi/\mu$ is
sufficiently small, ($\star$) possesses a global classical solution which is
bounded in $\Omega\times(0,\infty)$. We underline that the result fully
parallels the corresponding parabolic-elliptic-ODE system.Comment: correct Lemma 2.5 in version 1 due to an error in the proof, and
  reform Sec.3 to be more clear, main results and arguments remain unchange
Optimal Decay Rates of Classical Solutions for the Full Compressible MHD
  Equations

In this paper, we are concerned with optimal decay rates for higher order
spatial derivatives of classical solutions to the full compressible MHD
equations in three dimensional whole space. If the initial perturbation are
small in $H^3$-norm and bounded in $L^q(q\in \left[1,
\frac{6}{5}\right))$-norm, we apply the Fourier splitting method by
Schonbek[Arch. Rational Mech. Anal. 88 (1985)] to establish optimal decay rates
for the second order spatial derivatives of solutions and the third order
spatial derivatives of magnetic field in $L^2$-norm. These results improve the
work of Pu and Guo [Z. Angew. Math. Phys. 64 (2013) 519-538].Comment: 26 pages. arXiv admin note: substantial text overlap with
  arXiv:1505.0042
Data dependence for the amplitude equation of surface waves

We consider the amplitude equation for nonlinear surface wave solutions of hyperbolic conservation laws. This is an asymptotic nonlocal, Hamiltonian evolution equation with quadratic nonlinearity. For example, this equation describes the propagation of nonlinear Rayleigh waves, surface waves on current-vortex sheets in incompressible MHD and on the  incompressible plasma-vacuum interface. \ud
The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables was shown in a previous article. In the present paper we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard
Incompressible Limit of Solutions of Multidimensional Steady
  Compressible Euler Equations

A compactness framework is formulated for the incompressible limit of
approximate solutions with weak uniform bounds with respect to the adiabatic
exponent for the steady Euler equations for compressible fluids in any
dimension. One of our main observations is that the compactness can be achieved
by using only natural weak estimates for the mass conservation and the
vorticity. Another observation is that the incompressibility of the limit for
the homentropic Euler flow is directly from the continuity equation, while the
incompresibility of the limit for the full Euler flow is from a combination of
all the Euler equations. As direct applications of the compactness framework,
we establish two incompressible limit theorems for multidimensional steady
Euler flows through infinitely long nozzles, which lead to two new existence
theorems for the corresponding problems for multidimensional steady
incompressible Euler equations.Comment: 17 pages; 2 figures. arXiv admin note: text overlap with
  arXiv:1311.398
Multiple positive solutions for nonlinear critical fractional elliptic
  equations involving sign-changing weight functions

In this article, we prove the existence and multiplicity of positive
solutions for the following fractional elliptic equation with sign-changing
weight functions: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l}
(-\Delta)^\alpha u= a_\lambda(x)|u|^{q-2}u+b(x)|u|^{2^*_\alpha-1}u &{\rm
in}\,\,\Omega, u=0\,\,&{\rm in}\,\,\R^N\setminus\Omega, \end{array} \right.
\end{eqnarray*} where $0<\alpha<1$, $ \Omega $ is a bounded domain with smooth
boundary in $ \R^N $ with $N>2\alpha$ and $ 2^*_\alpha=2N/(N-2\alpha)$ is the
fractional critical Sobolev exponent. Our multiplicity results are based on
studying the decomposition of the Nehari manifold and the
Ljusternik-Schnirelmann category
Qualitative analysis of stationary Keller-Segel chemotaxis models with
  logistic growth

We study the stationary Keller--Segel chemotaxis models with logistic
cellular growth over a one-dimensional region subject to the Neumann boundary
condition. We show that nonconstant solutions emerge in the sense of Turing's
instability as the chemotaxis rate $\chi$ surpasses a threshold number. By
taking the chemotaxis rate as the bifurcation parameter, we carry out
bifurcation analysis on the system to obtain the explicit formulas of
bifurcation values and small amplitude nonconstant positive solutions. Moreover
we show that solutions stay strictly positive in the continuum of each branch.
The stabilities of these steady state solutions are well studied when the
creation and degradation rate of the chemical is assumed to be a linear
function. Finally we investigate the asymptotic behaviors of the monotone
steady states. We construct solutions with interesting patterns such as a
boundary spike when the chemotaxis rate is large enough and/or the cell
motility is small.Comment: Zeitschrift f\"ur Angewandte Mathematik und Physik, 201
Multiscale analysis of the acoustic scattering by many scatterers of
  impedance type

We are concerned with the acoustic scattering problem, at a frequency
$\kappa$, by many small obstacles of arbitrary shapes with impedance boundary
condition. These scatterers are assumed to be included in a bounded domain
$\Omega$ in $\mathbb{R}^3$ which is embedded in an acoustic background
characterized by an eventually locally varying index of refraction. The
collection of the scatterers $D_m, \; m=1,...,M$ is modeled by four parameters:
their number $M$, their maximum radius $a$, their minimum distance $d$ and the
surface impedances $\lambda_m, \; m=1,...,M$. We consider the parameters $M, d$
and $\lambda_m$'s having the following scaling properties: $M:=M(a)=O(a^{-s})$,
$d:=d(a)\approx a^t$ and $\lambda_m:=\lambda_m(a)=\lambda_{m,0}a^{-\beta}$, as
$a \rightarrow 0$, with non negative constants $s, t$ and $\beta$ and complex
numbers $\lambda_{m, 0}$'s with eventually negative imaginary parts.
  We derive the asymptotic expansion of the farfields with explicit error
estimate in terms of $a$, as $a\rightarrow 0$. The dominant term is the
Foldy-Lax field corresponding to the scattering by the point-like scatterers
located at the centers $z_m$'s of the scatterers $D_m$'s with $\lambda_m \vert
\partial D_m\vert$ as the related scattering coefficients.Comment: 27 pages, 1figur
Polynomial approximations of a class of stochastic multiscale elasticity
  problems

We consider a class of elasticity equations in ${\mathbb R}^d$ whose elastic
moduli depend on $n$ separated microscopic scales, are random and expressed as
a linear expansion of a countable sequence of random variables which are
independently and identically uniformly distributed in a compact interval. The
multiscale Hellinger-Reissner problem that allows for computing the stress
directly, and the multiscale mixed problem for nearly incompressible isotropic
materials are considered. The stochastic problems are studied via deterministic
problems that depend on a countable number of real parameters. We study the
multiscale homogenized problems that contain all the macroscopic and
microscopic information, whose solutions are written as generalized polynomial
chaos (gpc) expansions. We approximate these solutions by semidiscrete Galerkin
approximating problems that project into the spaces of functions with only a
finite number of $N$ gpc modes. We deduce bounds and summability properties for
the solutions' gpc expansion coefficients, which imply explicit rates of
convergence in terms of $N$ when the gpc modes used for the Galerkin
approximation are chosen to correspond to the best $N$ terms in the gpc
expansion. For the mixed problem for nearly incompressible materials, the rate
of convergence for the best $N$ term approximation is independent of the Lam\'e
constants' ratio. We establish parametric correctors in terms of the
semidiscrete Galerkin approximations. For two scale problems, an explicit
homogenization rate is deduced. Together with the best $N$ term rate, it
provides an explicit convergence rate for the correctors of the parametric
multiscale problems. For nearly incompressible materials, we obtain a
homogenization rate that is independent of the ratio of the Lam\'e constants,
so that the error for the corrector is also independent of this ratio