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Discrete & Continuous Dynamical Systems - A

2012 , Volume 32 , Issue 5

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Dynamic growth estimates of maximum vorticity for 3D incompressible Euler equations and the SQG model
Thomas Y. Hou and  Zuoqiang Shi
2012, 32(5): 1449-1463 doi: 10.3934/dcds.2012.32.1449 +[Abstract](40) +[PDF](475.3KB)
By performing estimates on the integral of the absolute value of vorticity along a local vortex line segment, we establish a relatively sharp dynamic growth estimate of maximum vorticity under some assumptions on the local geometric regularity of the vorticity vector. Our analysis applies to both the 3D incompressible Euler equations and the surface quasi-geostrophic model (SQG). As an application of our vorticity growth estimate, we apply our result to the 3D Euler equation with the two anti-parallel vortex tubes initial data considered by Hou-Li [12]. Under some additional assumption on the vorticity field, which seems to be consistent with the computational results of [12], we show that the maximum vorticity can not grow faster than double exponential in time. Our analysis extends the earlier results by Cordoba-Fefferman [6, 7] and Deng-Hou-Yu [8, 9].
Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems
Freddy Dumortier
2012, 32(5): 1465-1479 doi: 10.3934/dcds.2012.32.1465 +[Abstract](29) +[PDF](388.0KB)
In [1] and [2] upperbounds have been given for the number of large amplitude limit cycles in polynomial Liénard systems of type $(m,n)$ with $m<2n+1$, $m$ and $n$ odd. In the current paper we improve the upperbounds from [1] and [2] by one unity, obtaining sharp results. We therefore introduce the "method of cloning variables" that might be useful in other cyclicity problems.
Front tracking approximations for slow erosion
Debora Amadori and  Wen Shen
2012, 32(5): 1481-1502 doi: 10.3934/dcds.2012.32.1481 +[Abstract](36) +[PDF](437.4KB)
In this paper we study an integro-differential equation describing slow erosion, in a model of granular flow. In this equation the flux is non local and depends on $x$, $t$. We define approximate solutions by using a front tracking technique, adapted to this special equation. Convergence of the approximate solutions is established by means of suitable a priori estimates. In turn, these yield the global existence of entropy solutions in BV. Such entropy solutions are shown to be unique.
    We also prove the continuous dependence on initial data and on the erosion function, for the approximate as well as for the exact solutions. This establishes the well-posedness of the Cauchy problem.
Dimensional reduction for supremal functionals
Jean-François Babadjian , Francesca Prinari and  Elvira Zappale
2012, 32(5): 1503-1535 doi: 10.3934/dcds.2012.32.1503 +[Abstract](57) +[PDF](548.4KB)
A 3D-2D dimensional reduction analysis for supremal functionals is performed in the realm of $\Gamma^*$-convergence. We show that the limit functional still admits a supremal representation, and we provide a precise identification of its density in some particular cases. Our results rely on an abstract representation theorem for the $\Gamma^*$-limit of a family of supremal functionals.
Stable manifolds with optimal regularity for difference equations
Luis Barreira and  Claudia Valls
2012, 32(5): 1537-1555 doi: 10.3934/dcds.2012.32.1537 +[Abstract](38) +[PDF](392.1KB)
We obtain stable invariant manifolds with optimal $C^k$ regularity for a nonautonomous dynamics with discrete time. The dynamics is obtained from a sufficiently small perturbation of a nonuniform exponential dichotomy, which includes the notion of (uniform) exponential dichotomy as a very special case. We emphasize that we do not require the dynamics to be of class $C^{k+\epsilon}$, in strong contrast to former results in the context of nonuniform hyperbolicity. We use the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, our method also allows linear perturbations, and thus the results readily apply to the robustness problem of nonuniform exponential dichotomies.
Periodic and subharmonic solutions for duffing equation with a singularity
Zhibo Cheng and  Jingli Ren
2012, 32(5): 1557-1574 doi: 10.3934/dcds.2012.32.1557 +[Abstract](41) +[PDF](412.3KB)
This paper is devoted to the existence and multiplicity of periodic and subharmonic solutions for a superlinear Duffing equation with a singularity. In this manner, various preceding theorems are improved and sharpened. Our proof is based on a generalized version of the Poincaré-Birkhoff twist theorem.
Global linearization of periodic difference equations
Anna Cima , Armengol Gasull and  Francesc Mañosas
2012, 32(5): 1575-1595 doi: 10.3934/dcds.2012.32.1575 +[Abstract](35) +[PDF](472.4KB)
We deal with $m$-periodic, $n$-th order difference equations and study whether they can be globally linearized. We give an affirmative answer when $m=n+1$ and for most of the known examples appearing in the literature. Our main tool is a refinement of the Montgomery-Bochner Theorem.
Central limit theorem for stationary products of toral automorphisms
Jean-Pierre Conze , Stéphane Le Borgne and  Mikaël Roger
2012, 32(5): 1597-1626 doi: 10.3934/dcds.2012.32.1597 +[Abstract](30) +[PDF](556.6KB)
Let $(A_n(\omega))$ be a stationary process in ${\mathcal M}_d^*(\mathbb{Z})$. For a Hölder function $f$ on $\mathbb{T}^d$ we consider the sums $\sum_{k=1}^n f(^t\hskip -3pt A_k(\omega) \, ^t\hskip -3pt A_{k-1}(\omega) \cdots ^t\hskip -3pt A_1(\omega) \, x {\rm \ mod \ } 1)$ and prove a Central Limit Theorem for a.e. $\omega$ in different situations in particular for "kicked" stationary processes. We use the method of multiplicative systems of Komlòs and the Multiplicative Ergodic Theorem.
The center--focus problem and small amplitude limit cycles in rigid systems
Fabio Scalco Dias and  Luis Fernando Mello
2012, 32(5): 1627-1637 doi: 10.3934/dcds.2012.32.1627 +[Abstract](44) +[PDF](334.0KB)
In this paper we study the center--focus problem in families of rigid systems. We give explicit necessary and sufficient conditions to the unique equilibrium to be a center. We also study small amplitude limit cycles in these families of systems.
Extending $T^p$ automorphisms over $\mathbb{R}^{p+2}$ and realizing DE attractors
Fan Ding , Yi Liu , Shicheng Wang and  Jiangang Yao
2012, 32(5): 1639-1655 doi: 10.3934/dcds.2012.32.1639 +[Abstract](39) +[PDF](288.4KB)
We show that for every expanding self-map of a connected, closed $p$-dimensional manifold $M$, and for every codimension $q\geq p+1$, there exists a corresponding $(p,q)$-type DE attractor realized by a compactly-supported self-diffeomorphsm of $\mathbb{R}^{p+q}$. Moreover, when $M$ is the standard smooth $p$-dimensional torus $T^p$, the codimension $q$ can be taken as two. As a key ingredient of the construction, for the standard unknotted embedding $\imath_p:T^p\hookrightarrow\mathbb{R}^{p+2}$, we show the automorphisms that diffeomorphically extend over $\mathbb{R}^{p+2}$ form a subgroup of $Aut(T^p)$ of index at most $2^p-1$.
Minimal skew products with hypertransitive or mixing properties
Matúš Dirbák
2012, 32(5): 1657-1674 doi: 10.3934/dcds.2012.32.1657 +[Abstract](42) +[PDF](423.5KB)
Let $X$ be an infinite compact metric space and let $Z$ be a compact metric space admitting an arc-wise connected group $\mathcal H_0(Z)$ of homeomorphisms whose natural action on $Z$ is topologically transitive. We show that every map $f$ on $X$ with a hypertransitive property $\Lambda$ admits a skew product extension $F=(f,g_x)$ on $X\times Z$ which also has the property $\Lambda$ and whose all fibre maps $g_x$ lie in the closure $\overline{\mathcal H_0(Z)}$ of $\mathcal H_0(Z)$ in the space $\mathcal H(Z)$ of all homeomorphisms on $Z$.
    If we additionally assume that both the map $f$ and the action of $\mathcal H_0(Z)$ on $Z$ are minimal then we can guarantee the existence of such an extension $F$ in the class of minimal maps. In particular case when $\Lambda$= topological transitivity, such a theorem was known before (for invertible $f$ it was proved by Glasner and Weiss already in 1979).
    Finally, we show that if one imposes further restrictions on the group $\mathcal H_0(Z)$ then the analogues of the mentioned results for hypertransitive properties $\Lambda$ hold also for $\Lambda$= strong mixing.
Measure valued solutions of sub-linear diffusion equations with a drift term
Simona Fornaro , Stefano Lisini , Giuseppe Savaré and  Giuseppe Toscani
2012, 32(5): 1675-1707 doi: 10.3934/dcds.2012.32.1675 +[Abstract](46) +[PDF](442.3KB)
In this paper we study nonnegative, measure-valued solutions of the initial value problem for one-dimensional drift-diffusion equations when the nonlinear diffusion is governed by a strictly increasing $C^1$ function $\beta$ with $\lim_{r\to +\infty} \beta(r)<+\infty$. By using tools of optimal transport, we will show that this kind of problems is well posed in the class of nonnegative Borel measures with finite mass $m$ and finite quadratic momentum and it is the gradient flow of a suitable entropy functional with respect to the so called $L^2$-Wasserstein distance.
    Due to the degeneracy of diffusion for large densities, concentration of masses can occur, whose support is transported by the drift. We shall show that the large-time behavior of solutions depends on a critical mass $m_{\rm c}$, which can be explicitly characterized in terms of $\beta$ and of the drift term. If the initial mass is less then $m_{\rm c}$, the entropy has a unique minimizer which is absolutely continuous with respect to the Lebesgue measure. Conversely, when the total mass $m$ of the solutions is greater than the critical one, the stationary solution has a singular part in which the exceeding mass $m- m_{\rm c}$ is accumulated.
Expansive and fixed point free homeomorphisms of the plane
Jorge Groisman
2012, 32(5): 1709-1721 doi: 10.3934/dcds.2012.32.1709 +[Abstract](34) +[PDF](172.8KB)
The aim of this work is to describe the set of fixed point free homeomorphisms of the plane (preserving orientation or not) under certain expansive conditions. We find necessary and sufficient conditions for a fixed point free homeomorphism of the plane to be topologically conjugate to a translation.
On a nonlocal parabolic problem arising in electrostatic MEMS control
Jong-Shenq Guo and  Nikos I. Kavallaris
2012, 32(5): 1723-1746 doi: 10.3934/dcds.2012.32.1723 +[Abstract](134) +[PDF](525.0KB)
We consider a nonlocal parabolic equation associated with Dirichlet boundary and initial conditions arising in MEMS control. First, we investigate the structure of the associated steady-state problem for a general star-shaped domain. Then we classify radially symmetric stationary solutions and their radial Morse indices. Finally, we study under which circumstances the solution of the time-dependent problem is global-in-time or quenches in finite time.
Solitary waves in critical Abelian gauge theories
Emmanuel Hebey
2012, 32(5): 1747-1761 doi: 10.3934/dcds.2012.32.1747 +[Abstract](27) +[PDF](428.9KB)
We prove existence of standing waves solutions for electrostatic Klein-Gordon-Maxwell systems in arbitrary dimensional compact Riemannian manifolds with boundary for zero Dirichlet boundary conditions. We prove that phase compensation holds true when the dimension $n = 3$ or $4$. In these dimensions, existence of a solution is obtained when the mass of the particle field, balanced by the phase, is small in a geometrically quantified sense. In particular, existence holds true for sufficiently large phases. When $n \ge 5$, existence of a solution is obtained when the mass of the particle field is sufficiently small.
Schubart-like orbits in the Newtonian collinear four-body problem: A variational proof
Hsin-Yuan Huang
2012, 32(5): 1763-1774 doi: 10.3934/dcds.2012.32.1763 +[Abstract](41) +[PDF](383.1KB)
The Schubart-like orbits in the collinear four-body problem are similar to those discovered numerically by Schubart[12] in the collinear three-body problem. Schubart-like orbits are periodic solutions with exactly two binary collisions and one simultaneous binary collision per period. The proof of the existence of these orbits given in this paper is based on the direct method of Calculus of Variations. We exploit the variational structure of the problem and show that the minimizers of the Lagrangian action functional in a suitably chosen space have the desired properties.
Breather continuation from infinity in nonlinear oscillator chains
Guillaume James and  Dmitry Pelinovsky
2012, 32(5): 1775-1799 doi: 10.3934/dcds.2012.32.1775 +[Abstract](35) +[PDF](547.4KB)
Existence of large-amplitude time-periodic breathers localized near a single site is proved for the discrete Klein--Gordon equation, in the case when the derivative of the on-site potential has a compact support. Breathers are obtained at small coupling between oscillators and under nonresonance conditions. Our method is different from the classical anti-continuum limit developed by MacKay and Aubry, and yields in general branches of breather solutions that cannot be captured with this approach. When the coupling constant goes to zero, the amplitude and period of oscillations at the excited site go to infinity. Our method is based on near-identity transformations, analysis of singular limits in nonlinear oscillator equations, and fixed-point arguments.
Multiple periodic solutions of state-dependent threshold delay equations
Benjamin B. Kennedy
2012, 32(5): 1801-1833 doi: 10.3934/dcds.2012.32.1801 +[Abstract](60) +[PDF](633.9KB)
We prove the existence of multiple periodic solutions for scalar-valued state-dependent delay equations of the form $x'(t) = f(x(t - d(x_t)))$, where $d(x_t)$ is given by a threshold condition and $f$ is close, in a suitable sense, to the step function $h(x) = -\mbox{sign}(x)$. We construct maps whose fixed points correspond to periodic solutions and show that these maps have nontrivial fixed points via homotopy to constant maps.
    We also describe part of the global dynamics of the model equation $x'(t) = h(x(t - d(x_t)))$.
Blow-up phenomena for the 3D compressible MHD equations
Ming Lu , Yi Du and  Zheng-An Yao
2012, 32(5): 1835-1855 doi: 10.3934/dcds.2012.32.1835 +[Abstract](36) +[PDF](398.0KB)
In this paper, we study the three-dimensional(3D) compressible magnetohydrodynamic equations. Firstly, we obtain a blow-up criterion for the local strong solutions in terms of the gradient of the velocity, which is similar to the Beal-Kato-Majda criterion(see [1]) for the ideal incompressible flow. Secondly, we extend the well-known Serrin's blow-up criterion for the 3D incompressible Navier-Stokes equations to the 3D compressible magnetohydrodynamic equations. In addition, initial vacuum is allowed in our cases.
Prescribing the scalar curvature problem on higher-dimensional manifolds
Randa Ben Mahmoud and  Hichem Chtioui
2012, 32(5): 1857-1879 doi: 10.3934/dcds.2012.32.1857 +[Abstract](48) +[PDF](466.1KB)
In this paper we consider the problem of existence of conformal metrics with prescribed scalar curvature on n-dimensional Riemannian manifolds, $n \geq 5 $. Using precise estimates on the losses of compactness, we characterize the critical points at infinity of the associated variational problem and we prove existence results for curvatures satisfying an assumption of Bahri-Coron type.
Periodic perturbation of quadratic systems with two infinite heteroclinic cycles
Marcelo Messias
2012, 32(5): 1881-1899 doi: 10.3934/dcds.2012.32.1881 +[Abstract](31) +[PDF](8724.1KB)
We study periodic perturbations of planar quadratic vector fields having infinite heteroclinic cycles, consisting of an invariant straight line joining two saddle points at infinity and an arc of orbit also at infinity. The global study concerning the infinity of the perturbed system is performed by means of the Poincaré compactification in polar coordinates, from which we obtain a system defined on a set equivalent to a solid torus in $\mathbb{R}^3$, whose boundary plays the role of the infinity. It is shown that for certain type of periodic perturbation, there exist two differentiable curves in the parameter space for which the perturbed system presents heteroclinic tangencies and transversal intersections between the stable and unstable manifolds of two normally hyperbolic lines of singularities at infinity. The transversality of the manifolds is proved using the Melnikov method and implies, via the Birkhoff-Smale Theorem, in a complex dynamical behavior of the perturbed system solutions in a finite part of the phase space. Numerical simulations are performed for a particular example in order to illustrate this behavior, which could be called "the chaos arising from infinity", because it depends on the global structure of the quadratic system, including the points at infinity.
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion
Youshan Tao and  Michael Winkler
2012, 32(5): 1901-1914 doi: 10.3934/dcds.2012.32.1901 +[Abstract](76) +[PDF](419.0KB)
This paper deals with a boundary-value problem in two-dimensional smoothly bounded domains for the coupled chemotaxis-fluid model $$ \left\{ \begin{array}{l} n_t+ u\cdot \nabla n=\Delta n^m - \nabla \cdot (n\chi(c)\nabla c)\\ c_t+ u\cdot \nabla c=\Delta c-nf(c)\\ u_t +\nabla P-\eta \Delta u+n \nabla \phi=0 \\ \nabla \cdot u=0, \end{array} \right. $$ which describes the motion of oxygen-driven swimming bacteria in an incompressible fluid. The given functions $\chi$ and $f$ are supposed to be sufficiently smooth and such that $f(0)=0$.
    It is proved that global bounded weak solutions exist whenever $m>1$ and the initial data $(n_0,c_0,u_0)$ are sufficiently regular satisfying $n_0 \ge 0$ and $c_0\ge 0$. This extends a recent result by Di Francesco, Lorz and Markowich (Discrete Cont. Dyn. Syst. A 28 (2010)) which asserts global existence of weak solutions under the constraint $m \in (\frac{3}{2},2]$.

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