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Discrete & Continuous Dynamical Systems - A

2008 , Volume 21 , Issue 3

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Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\R^2$
J. Colliander, M. Keel, Gigliola Staffilani, H. Takaoka and T. Tao
2008, 21(3): 665-686 doi: 10.3934/dcds.2008.21.665 +[Abstract](382) +[PDF](300.6KB)
Abstract:
The initial value problem for the cubic defocusing nonlinear Schrödinger equation $i \partial_t u + \Delta u = |u|^2 u$ on the plane is shown to be globally well-posed for initial data in $H^s ( \R^2)$ provided $s>1/2$. The same result holds true for the analogous focusing problem provided the mass of the initial data is smaller than the mass of the ground state. The proof relies upon an almost conserved quantity constructed using multilinear correction terms. The main new difficulty is to control the contribution of resonant interactions to these correction terms. The resonant interactions are significant due to the multidimensional setting of the problem and some orthogonality issues which arise.
Nearly optimal patchy feedbacks
Alberto Bressan and Fabio S. Priuli
2008, 21(3): 687-701 doi: 10.3934/dcds.2008.21.687 +[Abstract](292) +[PDF](221.8KB)
Abstract:
The paper is concerned with a general optimization problem for a nonlinear control system, in the presence of a running cost and a terminal cost, with free terminal time. We prove the existence of a patchy feedback whose trajectories are all nearly optimal solutions, with pre-assigned accuracy.
Convergence to self-similar solutions for a semilinear parabolic equation
Marek Fila, Michael Winkler and Eiji Yanagida
2008, 21(3): 703-716 doi: 10.3934/dcds.2008.21.703 +[Abstract](348) +[PDF](197.8KB)
Abstract:
We study the behavior of solutions of the Cauchy problem for a parabolic equation with power nonlinearity. Our concern is the rate of convergence of solutions to forward self-similar solutions. We determine the exact rate of convergence which turns out to depend on the spatial decay rate of initial data.
On partial regularity for the Navier-Stokes equations
Igor Kukavica
2008, 21(3): 717-728 doi: 10.3934/dcds.2008.21.717 +[Abstract](506) +[PDF](179.0KB)
Abstract:
We consider the partial regularity of suitable weak solutions of the Navier-Stokes equations in a domain $D$. We prove that the parabolic Hausdorff dimension of space-time singularities in $D$ is less than or equal to 1 provided the force $f$ satisfies $f\in L^{2}(D)$. Our argument simplifies the proof of a classical result of Caffarelli, Kohn, and Nirenberg, who proved the partial regularity under the assumption $f\in L^{5/2+\delta}$ where $\delta>0$.
Large deviations for short recurrence
Miguel Abadi and Sandro Vaienti
2008, 21(3): 729-747 doi: 10.3934/dcds.2008.21.729 +[Abstract](361) +[PDF](240.3KB)
Abstract:
Over a $\psi$-mixing dynamical system we consider the function $\tau(C_n)$ $/n$ in the limit of large $n$, where $\tau(C_n)$ is the first return of a cylinder of length $n$ to itself. Saussol et al. ([30]) proved that this function is constant almost everywhere if the $C_n$ are chosen in a descending sequence of cylinders around a given point. We prove upper and lower general bounds for its large deviation function. Under mild assumptions we compute the large deviation function directly and show that the limit corresponds to the Rényi's entropy of the system. We finally compute the free energy function of $\tau(C_n)$ $/n$. We illustrate our results with a few examples.
Finite rank approximations of expanding maps with neutral singularities
Michael Blank
2008, 21(3): 749-762 doi: 10.3934/dcds.2008.21.749 +[Abstract](357) +[PDF](189.1KB)
Abstract:
For a class of expanding maps with neutral singularities we prove the validity of a finite rank approximation scheme for the analysis of Sinai-Ruelle-Bowen measures. Earlier results of this sort were known only in the case of hyperbolic systems.
Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions
Mariane Bourgoing
2008, 21(3): 763-800 doi: 10.3934/dcds.2008.21.763 +[Abstract](444) +[PDF](454.4KB)
Abstract:
In this article, we are interested in viscosity solutions for second-order fully nonlinear parabolic equations having a $L^1$ dependence in time and associated with nonlinear Neumann boundary conditions. The main contributions of our study are, not only to treat the case of nonlinear Neumann boundary conditions, but also to revisit the theory of viscosity solutions for such equations and to extend it in order to take in account singular geometrical equations. In particular, we provide comparison results, both for the cases of standard and geometrical equations, which extend the known results for Neumann boundary conditions even in the framework of continuous equations.
Perturbations of embedded eigenvalues for the bilaplacian on a cylinder
Gianne Derks, Sara Maad and Björn Sandstede
2008, 21(3): 801-821 doi: 10.3934/dcds.2008.21.801 +[Abstract](322) +[PDF](255.6KB)
Abstract:
Perturbation problems for operators with embedded eigenvalues are generally challenging since the embedded eigenvalues cannot be separated from the rest of the spectrum. In this paper we study a perturbation problem for embedded eigenvalues for the bilaplacian with an added potential, when the underlying domain is a cylinder. We show that the set of nearby potentials, for which a simple embedded eigenvalue persists, forms a smooth manifold of finite codimension.
Discrete and continuous spectra on laminations over Aubry-Mather sets
Bassam Fayad
2008, 21(3): 823-834 doi: 10.3934/dcds.2008.21.823 +[Abstract](328) +[PDF](200.3KB)
Abstract:
We study a class of completely integrable Hamiltonian system with two degrees of freedom for which the perturbed flow displays, on some energy level, invariant sets that are laminations over Aubry-Mather sets of a Poincaré section of the flow. Each one of these laminations carries a unique invariant probability measure for the flow and it is interesting therefore to understand the statistical properties of this measure. From a result of Kocergin in [13], we know that mixing is a priori impossible. In this paper, we investigate on the possible occurrence of weak mixing.
    The answer will essentially depend on the number of orbits of gaps in the Aubry-Mather set. More precisely, if the Aubry-Mather set has exactly one orbit of gaps and is hyperbolic then the special flow over it with any smooth ceiling function will be conjugate to a suspension with a constant ceiling function, failing hence to be weak mixing or even topologically weak mixing. To the contrary, if the Aubry-Mather set has more than one orbit of gaps with at least two in a general position then the special flow over it will in general be weak mixing.
Asymptotics of the solitary waves for the generalized Kadomtsev-Petviashvili equations
Philippe Gravejat
2008, 21(3): 835-882 doi: 10.3934/dcds.2008.21.835 +[Abstract](353) +[PDF](560.5KB)
Abstract:
We investigate the asymptotic behaviour of the localized solitary waves for the generalized Kadomtsev-Petviashvili equations. In particular, we compute their first order asymptotics in any dimension $N \geq 2$.
Global weak solutions to the Camassa-Holm equation
Zhenhua Guo, Mina Jiang, Zhian Wang and Gao-Feng Zheng
2008, 21(3): 883-906 doi: 10.3934/dcds.2008.21.883 +[Abstract](446) +[PDF](283.3KB)
Abstract:
The existence of a global weak solution to the Cauchy problem for a one-dimensional Camassa-Holm equation is established. In this paper, we assume that the initial condition $u_0(x)$ has end states $u_{\pm}$, which has much weaker constraints than that $u_0(x) \in H^1(\mathbb R)$ discussed in [30]. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution as a limit of viscous approximation under the assumption $u_- < u_+$.
Transversal families of hyperbolic skew-products
Eugen Mihailescu and Mariusz Urbański
2008, 21(3): 907-928 doi: 10.3934/dcds.2008.21.907 +[Abstract](300) +[PDF](253.3KB)
Abstract:
We study families of hyperbolic skew products with the transversality condition and in particular, the Hausdorff dimension of their fibers, by using thermodynamical formalism. The maps we consider can be non-invertible, and the study of their dynamics is influenced greatly by this fact.
    We introduce and employ probability measures (constructed from equilibrium measures on the natural extension), which are supported on the fibers of the skew product. A stronger condition, that of Uniform Transversality is then considered in order to obtain a general formula for Hausdorff dimension of fibers for all base points and almost all parameters.
    In the end we study a large class of examples of transversal hyperbolic families which locally depend linearly on the parameters, and also another class of examples related to complex dynamics.
Well-posedness and stability of a multidimensional moving boundary problem modeling the growth of tumor cord
Fujun Zhou and Shangbin Cui
2008, 21(3): 929-943 doi: 10.3934/dcds.2008.21.929 +[Abstract](479) +[PDF](204.3KB)
Abstract:
In this paper we study a multidimensional moving boundary problem modeling the growth of tumor cord. This problem contains two coupled elliptic equations defined in a bounded domain in $R^2$ whose boundary consists of two disjoint closed curves, one fixed and the other moving and a priori unknown. The evolution of the moving boundary is governed by a Stefan type equation. By using the functional analysis method based on applications of the theory of analytic semigroups, we prove that (1) this problem is locally well-posed in Hölder spaces, (2) it has a unique radially symmetric stationary solution, and (3) this radially symmetric stationary solution is asymptotically stable for arbitrary sufficiently small perturbations in these Hölder spaces.
Indices of singularities of robustly transitive sets
Shengzhi Zhu, Shaobo Gan and Lan Wen
2008, 21(3): 945-957 doi: 10.3934/dcds.2008.21.945 +[Abstract](442) +[PDF](197.9KB)
Abstract:
It was proved recently in [4] that any robustly transitive singular set that is strongly homogenous must be partially hyperbolic, as long as the indices of singularities and periodic orbits satisfy certain condition. We prove in this paper that this index-condition is automatically satisfied under the strongly homogenous condition, hence can be removed from the assumptions. Moreover, we prove that a robustly transitive singular set that is strongly homogenous is in fact singular hyperbolic.
A new problem of adiabatic invariance related to the rigid body dynamics
Giancarlo Benettin, Massimiliano Guzzo and Anatoly Neishtadt
2008, 21(3): 959-975 doi: 10.3934/dcds.2008.21.959 +[Abstract](328) +[PDF](665.1KB)
Abstract:
We study a new problem of adiabatic invariance, namely a nonlinear oscillator with slowly moving center of oscillation; the frequency of small oscillations vanishes when the center of oscillation passes through the origin (the fast motion is no longer fast), and this can produce nontrivial motions. Similar systems naturally appear in the study of the perturbed Euler rigid body, in the vicinity of proper rotations and in connection with the 1:1 resonance, as models for the normal form. In this paper we provide, on the one hand, a rigorous upper bound on the possible size of chaotic motions; on the other hand we work out, heuristically, a lower bound for the same quantity, and the two bounds do coincide up to a logarithmic correction. We also illustrate the theory by quite accurate numerical results, including, besides the size of the chaotic motions, the behavior of Lyapunov Exponents. As far as the system at hand is a model problem for the rigid body dynamics, our results fill the gap existing in the literature between the theoretically proved stability properties of proper rotations and the numerically observed ones, which in the case of the 1:1 resonance did not completely agree, so indicating a not yet optimal theory.

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