# Journal of Differential Equations

Published by Elsevier

Online ISSN: 1090-2732

Print ISSN: 0022-0396

Published by Elsevier

Online ISSN: 1090-2732

Print ISSN: 0022-0396

Publications

The study of dynamics of gene regulatory networks is of increasing interest in systems biology. A useful approach to the study of these complex systems is to view them as decomposed into feedback loops around open loop monotone systems. Key features of the dynamics of the original system are then deduced from the input-output characteristics of the open loop system and the sign of the feedback. This paper extends these results, showing how to use the same framework of input-output systems in order to prove existence of oscillations, if the slowly varying strength of the feedback depends on the state of the system.

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In this paper, we improve on classical averaging theorems for functional differential equations by proposing new averaged models

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We consider an inverse spectral problem for a class of singular Sturm–Liouville operators on the unit interval with explicit singularity a(a+1)/x2, a∈N, related to the Schrödinger operator with a radially symmetric potential. The purpose of this paper is the global parametrization of potentials by the spectral data λa and some norming constants κa. For a=0 or 1, λa×κa is already known to be a global coordinate system on . Using some transformation operators, we show that this result holds for any non-negative integer a; moreover, we give a description of the isospectral sets.

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For a polynomial planar vector field of degree n⩾2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1+(n−1)(n−2)/2 when n is even, and (n−1)(n−2)/2 when n is odd. Furthermore, these upper bounds are reached.

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The weak Hilbert 16th problem for n=2 was solved by Horozov and Iliev (Proc. London Math. Soc. 69 (1994) 198–244), Zhang and Li (Adv. in Math. 26 (1997) 445–460), (Gavrilov Invent. Math. 143 (2001) 449–497), and Li and Zhang (Nonlinearity 15 (2002) 1775–1992), by using different methods for different cases. The aim of this paper is to give a unified and easier proof for all cases. The proof is restricted to the real domain, combines geometric and analytical methods, and uses deformation arguments.

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We discuss here a systematic approach towards a positive answer to Hilbert′s 16th problem for quadratic systems, namely the existence of a uniform bound for the number of limit cycles of a quadratic system. The method is the following: describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity. In this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features. We also indicate how to find or where to find concrete examples of these limit periodic sets.

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We consider a linear wave equation, on the interval (0,1), with bilinear control and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory. We prove that the following results hold generically.•For every T>2, this system is locally controllable in H3×H2, in time T, with controls in L2((0,T),R).•For T=2, this system is locally controllable up to codimension one in H3×H2, in time T, with controls in L2((0,T),R): the reachable set is (locally) a non-flat submanifold of H3×H2 with codimension one.•For every T<2, this system is not locally controllable, more precisely, the reachable set, with controls in L2((0,T),R), is contained in a non-flat submanifold of H3×H2, with infinite codimension. The proof of these results relies on the inverse mapping theorem and second order expansions.

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This paper concerns periodic solutions for a 1D-model with nonlocal velocity
given by the periodic Hilbert transform. There is a rich literature showing
that this model presents singular behavior of solutions via numerics and
mathematical approaches. For instance, they can blow up by forming
mass-concentration. We develop a global well-posedness theory for periodic
measure initial data that allows, in particular, to analyze how the model
evolves from those singularities. Our results are based on periodic mass
transport theory and the abstract gradient flow theory in metric spaces
developed by Ambrosio et al. [2]. A viscous version of the model is also
analyzed and inviscid limit properties are obtained.

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We study the well-posedness of the Dirac-Klein-Gordon system in one space
dimension with initial data that are analytic in a strip around the real axis.
It is proved that for short times $t$ the radius of analyticity $\sigma(t)$ of
the solutions remains constant while for $t \to \infty$ we obtain a lower bound
$\sigma(t) \ge c/t^{5+}$ in the case of positive Klein-Gordon mass and
$\sigma(t) \ge c/t^{8+}$ in the massless case.

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We analyze a one-dimensional fluid–particle interaction model, composed by the Burgers equation for the fluid velocity and an ordinary differential equation which governs the particle movement. The coupling is achieved through a friction term. One of the novelties is to consider entropy weak solutions involving shock waves. The difficulty is the interaction between these shock waves and the particle. We prove that the Riemann problem with arbitrary data always admits a solution, which is explicitly constructed. Besides, two asymptotic behaviors are described: the long-time behavior and the behavior for large friction coefficients.

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We analyse the asymptotic behaviour of solutions to the one dimensional
fractional version of the porous medium equation introduced by Caffarelli and
V\'azquez, where the pressure is obtained as a Riesz potential associated to
the density. We take advantage of the displacement convexity of the Riesz
potential in one dimension to show a functional inequality involving the
entropy, entropy dissipation, and the Euclidean transport distance. An argument
by approximation shows that this functional inequality is enough to deduce the
exponential convergence of solutions in self-similar variables to the unique
steady states.

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In this paper, we prove that the 1D Cauchy problem of the compressible
Navier-Stokes equations admits a unique global classical solution $(\rho,\rm
u)$ if the viscosity $\mu(\rho)=1+\rho^{\beta}$ with $\beta\geq0$. The initial
data can be arbitrarily large and may contain vacuum.
Some new weighted estimates of the density and velocity are obtained when
deriving higher order estimates of the solution.

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We study a 1D transport equation with nonlocal velocity. First, we prove
eventual regularization of the viscous regularization when dissipation is in
the supercritical range with non-negative initial data. Next, we will prove
global regularity for solutions when dissipation is slightly supercritical.
Both results utilize a nonlocal maximum principle.

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We first consider an elastic thin heterogeneous cylinder of radius of order
epsilon: the interior of the cylinder is occupied by a stiff material (fiber)
that is surrounded by a soft material (matrix). By assuming that the elasticity
tensor of the fiber does not scale with epsilon and that of the matrix scales
with epsilon square, we prove that the one dimensional model is a nonlocal
system.
We then consider a reference configuration domain filled out by periodically
distributed rods similar to those described above. We prove that the
homogenized model is a second order nonlocal problem.
In particular, we show that the homogenization problem is directly connected
to the 3D-1D dimensional reduction problem.

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We consider the 1D massless Dirac operator on the real line with compactly
supported potentials. We study resonances as the poles of scattering matrix or
equivalently as the zeros of modified Fredholm determinant. We obtain the
following properties of the resonances: 1) asymptotics of counting function, 2)
estimates on the resonances and the forbidden domain, 3) the trace formula in
terms of resonances.

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We present a general framework for deriving continuous dependence estimates for, possibly polynomially growing, viscosity solutions of fully nonlinear degenerate parabolic integro-PDEs. We use this framework to provide explicit estimates for the continuous dependence on the coefficients and the “Lévy measure” in the Bellman/Isaacs integro-PDEs arising in stochastic control/differential games. Moreover, these explicit estimates are used to prove regularity results and rates of convergence for some singular perturbation problems. Finally, we illustrate our results on some integro-PDEs arising when attempting to price European/American options in an incomplete stock market driven by a geometric Lévy process. Many of the results obtained herein are new even in the convex case where stochastic control theory provides an alternative to our pure PDE methods.

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In this article we study the global regularity of 2D generalized
magnetohydrodynamic equations (2D GMHD), in which the dissipation terms are $-
\nu (- \triangle)^{\alpha} u$ and $- \kappa (-\triangle)^{\beta} b$. We show
that smooth solutions are global in the following three cases: $\alpha
\geqslant 1 / 2, \beta \geqslant 1$; $0 \leqslant \alpha < 1 / 2, 2 \alpha +
\beta > 2$; $\alpha \geqslant 2, \beta = 0$. We also show that in the inviscid
case $\nu = 0$, if $\beta > 1$, then smooth solutions are global as long as the
direction of the magnetic field remains smooth enough.

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We consider self-similar solutions of the 2d incompressible Euler equations.
We construct a class of solutions with vorticity forming algebraic spirals near
the origin, in analogy to vortex sheets rolling up into algebraic spirals.

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In this paper, we first prove the unique global strong solution with vacuum
to the two dimensional nonhomogeneous incompressible MHD system, as long as the
initial data satisfies some compatibility condition. As a corollary, the global
existence of strong solution with vacuum to the 2D nonhomogeneous
incompressible Navier-Stokes equations is also established. Our main result
improves all the previous results where the initial density need to be strictly
positive. The key idea is to use some critical Sobolev inequality of
logarithmic type, which is originally due to Brezis-Wainger.

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We find a smooth solution of the 2D Euler equation on a bounded domain which
exists and is unique in a natural class locally in time, but blows up in finite
time in the sense of its vorticity losing continuity. The domain's boundary is
smooth except at two points, which are interior cusps.

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We prove existence and uniqueness of a weak solution to the incompressible 2D
Euler equations in the exterior of a bounded smooth obstacle when the initial
data is a bounded divergence-free velocity field having bounded scalar curl.
This work completes and extends the ideas outlined by P. Serfati for the same
problem in the whole-plane case. With non-decaying vorticity, the Biot-Savart
integral does not converge, and thus velocity cannot be reconstructed from
vorticity in a straightforward way. The key to circumventing this difficulty is
the use of the Serfati identity, which is based on the Biot-Savart integral,
but holds in more general settings.

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We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H1-critical semilinear wave equation on a smooth bounded domain Ω⊂R2. First, we prove an appropriate Strichartz type estimate using the Lq spectral projector estimates of the Laplace operator. Our proof follows Burq, Lebeau and Planchon (2008) [4]. Then, we show the global well-posedness when the energy is below or at the threshold given by the sharp Moser–Trudinger inequality. Finally, in the supercritical case, we prove an instability result using the finite speed of propagation and a quantitative study of the associated ODE with oscillatory data.

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The existence of an attractor for a 2D-Navier–Stokes system with delay is proved. The theory of pullback attractors is successfully applied to obtain the results since the abstract functional framework considered turns out to be nonautonomous. However, on some occasions, the attractors may attract not only in the pullback sense but in the forward one as well. Also, this formulation allows to treat, in a unified way, terms containing various classes of delay features (constant, variable, distributed delays, etc.). As a consequence, some results for the autonomous model are deduced as particular cases of our general formulation.

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This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the Lr-norm of the vertical velocity v for any 1<r<∞ is globally bounded and that the L∞-norm of v controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace δ(−Δ) for δ>0 would guarantee the global regularity of classical solutions.

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We consider the elliptic equation −Δu+u=0 in a bounded, smooth domain Ω in R2, subject to the nonlinear Neumann boundary condition . Here p>1 is a large parameter. We prove that given any integer m⩾1 there exist at least two families of solutions up developing exactly m peaks ξi∈∂Ω, in the sense that , as p→∞.

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Some results on the pathwise exponential stability of the weak solutions to a stochastic 2D-Navier–Stokes equation are established. The first ones are proved as a consequence of the exponential mean square stability of the solutions. However, some of them are improved by avoiding the previous mean square stability in some more particular and restrictive situations. Also, some results and comments concerning the stabilizability and stabilization of these equations are stated.

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Stochastic Navier-Stokes equations in 2D and 3D possibly unbounded domains
driven by a multiplicative Gaussian noise are considered. The noise term
depends on the unknown velocity and its spatial derivatives. The existence of a
martingale solution is proved. The construction of the solution is based on the
classical Faedo-Galerkin approximation, the compactness method and the
Jakubowski version of the Skorokhod Theorem for non-metric spaces. Moreover,
some compactness and tightness criteria in non-metric spaces are proved.
Compactness results are based on a certain generalization of the classical
Dubinsky Theorem.

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There are two results within this paper. The one is the regularity of trajectory attractor and the trajectory asymptotic smoothing effect of the incompressible non-Newtonian fluid on 2D bounded domains, for which the solution to each initial value could be non-unique. The other is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from Ωm to Ω=R×(−L,L), where is an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that Ωm→Ω as m→+∞. That is, let A and Am be the global attractors of the fluid corresponding to Ω and Ωm, respectively, we establish that for any neighborhood O(A) of A, the global attractor Am enters O(A) if m is large enough.

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It has long been known, for the autonomous 2D Navier-Stokes equations with
singular forcing, that there exist unique solutions which gain one derivative,
globally. On the other hand, if the forcing term smooth enough, it is known
that the solution gains two derivatives globally. In this paper, we explore
classical techniques to show that if the force is sufficiently smooth, then the
solution gains two derivatives globally. These methods break down when the
force becomes singular. In this scenario, we use a Littlewood-Paley
decomposition in Fourier space to show that a solutions gain two derivatives
locally in time and the interval of time depends only on the size of the
initial data.

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The 2d Boussinesq equations model large scale atmospheric and oceanic flows. Whether its solutions develop a singularity in finite-time remains a classical open problem in mathematical fluid dynamics. In this work, blowup from smooth nontrivial initial velocities in stagnation-point form solutions of this system is established. On an infinite strip $\Omega=\{(x,y)\in[0,1]\times\mathbb{R}^+\}$, we consider velocities of the form $u=(f(t,x),-yf_x(t,x))$, with scalar temperature\, $\theta=y\rho(t,x)$. Assuming $f_x(0,x)$ attains its global maximum only at points $x_i^*$ located on the boundary of $[0,1]$, general criteria for finite-time blowup of the vorticity $-yf_{xx}(t,x_i^*)$ and the time integral of $f_x(t,x_i^*)$ are presented. Briefly, for blowup to occur it is sufficient that $\rho(0,x)\geq0$ and $f(t,x_i^*)=\rho(0,x_i^*)=0$, while $-yf_{xx}(0,x_i^*)\neq0$. To illustrate how vorticity may suppress blowup, we also construct a family of global exact solutions. A local-existence result and additional regularity criteria in terms of the time integral of $\left\|f_x(t,\cdot)\right\|_{L^\infty([0,1])}$ are also provided.

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We consider the generalized two-dimensional Zakharov-Kuznetsov equation
$u_t+\partial_x \Delta u+\partial_x(u^{k+1})=0$, where $k\geq3$ is an integer
number. For $k\geq8$ we prove local well-posedness in the $L^2$-based Sobolev
spaces $H^s(\mathbb{R}^2)$, where $s$ is greater than the critical scaling
index $s_k=1-2/k$. For $k\geq 3$ we also establish a sharp criteria to obtain
global $H^1(\R^2)$ solutions. A nonlinear scattering result in $H^1(\R^2)$ is
also established assuming the initial data is small and belongs to a suitable
Lebesgue space.

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In this paper, we study the global well-posedness of classical solutions to
the 2D compressible MHD equations with large initial data and vacuum. With the
assumption $\mu=const.$ and $\lambda=\rho^\beta,~\beta>1$
(Va\v{i}gant-Kazhikhov Model) for the viscosity coefficients, the global
existence and uniqueness of classical solutions to the initial value problem is
established on the torus $\mathbb{T}^2$ and the whole space $\mathbb{R}^2$
(with vacuum or non-vacuum far fields). These results generalize the previous
ones for the Va\v{i}gant-Kazhikhov model of compressible Navier-Stokes.

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We show that the stochastic flow generated by the 2-dimensional Stochastic Navier-Stokes equations with rough noise on a Poincare-like domain has a unique random attractor. One of the technical problems associated with the rough noise is overcomed by the use of the corresponding Cameron-Martin (or reproducing kernel Hilbert) space. Our results complement the result by Brzezniak and Li (2006) [10] who showed that the corresponding flow is asymptotically compact and also generalize Caraballo et al. (2006) [12] who proved existence of a unique attractor for the time-dependent deterministic Navier-Stokes equations. (C) 2013 Published by Elsevier Inc.

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We investigate the global in time stability of regular solutions with large velocity vectors to the evolutionary Navier-Stokes equation in ${\bf R}^3$. The class of stable flows contains all two dimensional weak solutions. The only assumption which is required is smallness of the $L_2$-norm of initial perturbation or its derivative with respect to the `$z$'-coordinate in the same norm. The magnitude of the rest of the norm of initial datum is not restricted.

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This paper is devoted to the study of the low Mach number limit for the 2D
isentropic Euler system associated to ill-prepared initial data with slow blow
up rate on $\log\varepsilon^{-1}$. We prove in particular the strong
convergence to the solution of the incompressible Euler system when the
vorticity belongs to some weighted $BMO$ spaces allowing unbounded functions.
The proof is based on the extension of the result of \cite{B-K} to a
compressible transport model.

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We consider the boundary value problem (∗)Lu(x)=p(x)u(x)+g(x,u(0)(x),…,u(2m−1)(x))u(x),x∈(0,π),where (i) L is a 2mth order, self-adjoint, disconjugate ordinary differential operator on [0,π], together with separated boundary conditions at 0 and π; (ii) p is continuous and p⩾0 on [0,π], while p≢0 on any interval in [0,π]; (iii) is continuous and there exist increasing functions such that with (the non-linear term in (∗) is superlinear as |u(x)|→∞). We obtain a global bifurcation result for a related bifurcation problem. We then use this to obtain infinitely many solutions of (∗) having specified nodal properties.

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We consider the Navier–Stokes equations on thin 3D domains Qε=Ω×(0, ε), supplemented mainly with purely periodic boundary conditions or with periodic boundary conditions in the thin direction and homogeneous Dirichlet conditions on the lateral boundary. We prove global existence and uniqueness of solutions for initial data and forcing terms, which are larger and less regular than in previous works on thin domains. An important tool in the proofs are some Sobolev embeddings into anisotropic Lp-type spaces. Better results are proved in the purely periodic case, where the conservation of enstrophy property is used. For example, when the forcing term vanishes, we prove global existence and uniqueness of solutions if ‖(I−M) u0‖H1/2(Qε) exp(C−1ε−1/s ‖Mu0 ‖2/sL2(Qε))⩽C for both boundary conditions or ‖Mu0‖H1(Qε)⩽Cε−β, ‖(Mu0)3‖L2(Qε)⩽Cεβ, ‖(I−M) u0‖H1/2(Qε)⩽Cε1/4−β/2 for purely periodic boundary conditions, where 1/2<s<1 and 0⩽β⩽1/2 are arbitrary, C is a prescribed positive constant independent of ε, and M denotes the average operator in the thin direction. We also give a new uniqueness criterium for weak Leray solutions.

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We obtain a dispersive long-time decay in weighted energy norms for solutions of the 3D Klein–Gordon equation with generic potential. The decay extends the results obtained by Jensen and Kato for the 3D Schrödinger equation. For the proof we modify the spectral approach of Jensen and Kato to make it applicable to relativistic equations.

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This work establishes two regularity criteria for the 3D incompressible MHD equations. The first one is in terms of the derivative of the velocity field in one direction while the second one requires suitable boundedness of the derivative of the pressure in one direction.

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In view of the possibility that the 3D Navier–Stokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multi-valued dissipative evolutionary systems with respect to two topologies—weak and strong. Each such system possesses a global attractor in the weak topology, but not necessarily in the strong. In case the latter exists and is weakly closed, it coincides with the weak global attractor. We give a sufficient condition for the existence of the strong global attractor, which is verified for the 3D NSE when all solutions on the weak global attractor are strongly continuous. We also introduce and study a two-parameter family of models for the Navier–Stokes equations, with similar properties and open problems. These models always possess weak global attractors, but on some of them every solution blows up (in a norm stronger than the standard energy one) in finite time.

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We prove that solutions of the 3D relativistic Vlasov-Maxwell system can be
extended, as long as the quantity $\sigma_{-1}(t, x) = \max_{|\omega|=1}
\,\int_{R^3} \frac{dp}{\sqrt{1+p^2}}\, \frac{1}{(1+v\cdot\omega)}\, f(t, x, p)$
is bounded in $L^2_x$.

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We construct pullback attractors to the weak solutions of the
three-dimensional Dirichlet problem for the incompressible Navier-Stokes
equations in the case when the external force may become unbounded as time goes
to plus or minus infinity.

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A global existence, uniqueness and regularity theorem is proved for the simplest Markovian Wigner–Poisson–Fokker–Planck model incorporating friction and dissipation mechanisms. The proof relies on Green function and energy estimates under mild formulation, making essential use of the Husimi function and the elliptic regularization of the Fokker–Planck operator.

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We obtain results on the convergence of Galerkin solutions and continuous dependence on data for the spectrally-hyperviscous Navier–Stokes equations. Let uN denote the Galerkin approximates to the solution u, and let wN=u−uN. Then our main result uses the decomposition wN=PnwN+QnwN where (for fixed n) Pn is the projection onto the first n eigenspaces of A=−Δ and Qn=I−Pn. For assumptions on n that compare well with those in related previous results, the convergence of ‖QnwN(t)Hβ‖ as N→∞ depends linearly on key parameters (and on negative powers of λn), thus reflective of Kolmogorov-theory predictions that in high wavenumber modes viscous (i.e. linear) effects dominate. Meanwhile ‖PnwN(t)Hβ‖ satisfies a more standard exponential estimate, with positive, but fractional, dependence on λn. Modifications of these estimates demonstrate continuous dependence on data.

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The asymptotic behavior of solutions of the three-dimensional Navier-Stokes
equations is considered on bounded smooth domains with no-slip boundary
conditions or on periodic domains. Asymptotic regularity conditions are
presented to ensure that the convergence of a Leray-Hopf weak solution to its
weak omega-limit set (weak in the sense of the weak topology of the space H of
square-integrable divergence-free velocity fields) are achieved also in the
strong topology of H. In particular, if a weak omega-limit set is bounded in
the space V of velocity fields with square-integrable vorticity then the
attraction to the set holds also in the strong topology of H. Corresponding
results for the strong convergence towards the weak global attractor of Foias
and Temam are also presented.

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In this paper, we study the 3D regularized Boussinesq equations.
The velocity equation is regularized \`a la Leray
through a smoothing kernel of order $\alpha$ in the nonlinear term and a
$\beta$-fractional Laplacian; we consider the critical case $\alpha+\beta=\frac{5}{4}$
and we assume $\frac 12 <\beta<\frac 54$.
The temperature equation is a pure transport equation, where the
transport velocity is regularized through the
same smoothing kernel of order $\alpha$.
We prove global well posedness when the initial velocity is in $H^r$
and the initial temperature is in $H^{r-\beta}$ for
$r>\max(2\beta,\beta+1)$.
This regularity is enough to prove uniqueness of solutions.
We also prove a continuous dependence of the solutions on the initial conditions.

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A solution of the Abel equation such that x(0)=x(1) is called a periodic orbit of the equation. Our main result proves that if there exist two real numbers a and b such that the function aA(t)+bB(t) is not identically zero, and does not change sign in [0,1] then the Abel differential equation has at most one non-zero periodic orbit. Furthermore, when this periodic orbit exists, it is hyperbolic. This result extends the known criteria about the Abel equation that only refer to the cases where either A(t)≢0 or B(t)≢0 does not change sign. We apply this new criterion to study the number of periodic solutions of two simple cases of Abel equations: the one where the functions A(t) and B(t) are 1-periodic trigonometric polynomials of degree one and the case where these two functions are polynomials with three monomials. Finally, we give an upper bound for the number of isolated periodic orbits of the general Abel equation , when A(t), B(t) and C(t) satisfy adequate conditions.

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Given two polynomials $P,q$ we consider the following question: "how large
can the index of the first non-zero moment $\tilde{m}_k=\int_a^b P^k q$ be,
assuming the sequence is not identically zero?". The answer $K$ to this
question is known as the moment Bautin index, and we provide the first general
upper bound: $K\leqslant 2+\mathrm{deg} q+3(\mathrm{deg} P-1)^2$. The proof is
based on qualitative analysis of linear ODEs, applied to Cauchy-type integrals
of certain algebraic functions.
The moment Bautin index plays an important role in the study of bifurcations
of periodic solution in the polynomial Abel equation $y'=py^2+\varepsilon qy^3$
for $p,q$ polynomials and $\varepsilon \ll 1$. In particular, our result
implies that for $p$ satisfying a well-known generic condition, the number of
periodic solutions near the zero solution does not exceed $5+\mathrm{deg}
q+3\mathrm{deg}^2 p$. This is the first such bound depending solely on the
degrees of the Abel equation.

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We investigate cubic systems which can be transformed to an equation of Abel form. The conditions for the origin to be a centre and, in particular, an isochronous centre are obtained. The maximum number of limit cycles which can bifurcate from a fine focus is determined and some information is obtained about the global phase portrait.

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This paper is devoted to prove two unexpected properties of the Abel equation dz/dt=z3+B(t)z2+C(t)z, where B and C are smooth, 2π-periodic complex valuated functions, t∈R and z∈C. The first one is that there is no upper bound for its number of isolated 2π-periodic solutions. In contrast, recall that if the functions B and C are real valuated then the number of complex 2π-periodic solutions is at most three. The second property is that there are examples of the above equation with B and C being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of B and C. This result is also in contrast with the characterization of the center variety for the examples of Abel equations dz/dt=A(t)z3+B(t)z2 studied in the literature, where the center variety is located in a finite number of connected components.

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