# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

2003 , Volume 9 , Issue 1

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2003, 9(1): 1-30 doi: 10.3934/dcds.2003.9.1 +[Abstract](396) +[PDF](308.8KB)
Abstract:
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2003, 9(1): 31-54 doi: 10.3934/dcds.2003.9.31 +[Abstract](342) +[PDF](283.2KB)
Abstract:
We study the long-time behaviour of the focusing cubic NLS on $\mathbf R$ in the Sobolev norms $H^s$ for $0 < s < 1$. We obtain polynomial growth-type upper bounds on the $H^s$ norms, and also limit any orbital $H^s$ instability of the ground state to polynomial growth at worst; this is a partial analogue of the $H^1$ orbital stability result of Weinstein [27], [26]. In the sequel to this paper we generalize this result to other nonlinear Schrödinger equations. Our arguments are based on the "$I$-method" from earlier papers [9]-[15] which pushes down from the energy norm, as well as an "upside-down $I$-method" which pushes up from the $L^2$ norm.
2003, 9(1): 55-68 doi: 10.3934/dcds.2003.9.55 +[Abstract](418) +[PDF](216.3KB)
Abstract:
We discuss the existence of positive solutions of perturbation to a class of quasilinear elliptic equations on $\mathbb R$.
2003, 9(1): 69-95 doi: 10.3934/dcds.2003.9.69 +[Abstract](468) +[PDF](293.0KB)
Abstract:
We consider the monotone twist map $\bar f$ on $(\mathbb R/\mathbb Z)\times R$, itslift $f$ on $R^2$ and its associated variational principle $h:\mathbb R^2\to\mathbb R$ through its generating function. By working with the variationalprinciple $h$, we first show that for an adjacent minimal chain$\{(u^k, v^k)\}_{k=s}^t$ of fixed points of $f$, if there exists abarrier $B_k$ for each adjacent minimal pair $u^k < u^{k+1}$, $s \le k \le {t-1}$, then there exists a heteroclinic orbit between $(u^s, v^s)$ and$(u^t, v^t)$, then by assuming that there is a barrier for any twoneighboring globally minimal critical points and $m$ is sufficientlylarge, we construct an invariant set $\Lambda^m\subset (\mathbb R/\mathbb Z)\times\mathbb R$ such that the shift map of the $n$-symbol space is a factor of$\bar f^m|_{\Lambda^m}$, where $n$ is the total number of the globallyminimal fixed points of $\bar f$.
2003, 9(1): 97-131 doi: 10.3934/dcds.2003.9.97 +[Abstract](356) +[PDF](325.6KB)
Abstract:
In this article we study the global existence of strong solutions of the Primitive Equations (PEs) for the large scale ocean under the small depth hypothesis. The small depth hypothesis implies that the domain $M_\varepsilon$ occupied by the ocean is a thin domain, its thickness parameter $\varepsilon$ is the aspect ratio between its vertical and horizontal scales. Using and generalizing the methods developed in [23], [24], we establish the global existence of strong solutions for initial data and volume and boundary 'forces', which belong to large sets in their respective phase spaces, provided $\varepsilon$ is sufficiently small. Our proof of the existence results for the PEs is based on precise estimates of the dependence of a number of classical constants on the thickness $\varepsilon$ of the domain. The extension of the results to the atmosphere or the coupled ocean and atmosphere or to other relevant boundary conditions will appear elsewhere.
2003, 9(1): 133-166 doi: 10.3934/dcds.2003.9.133 +[Abstract](391) +[PDF](377.2KB)
Abstract:
We consider the problem of writing Glimm type interaction estimates for the hyperbolic system

$u_t + A(u) u_x = 0.\qquad\qquad (0.1)$

The aim of these estimates is to prove that there is Glimm-type functional $Q(u)$ such that

Tot.Var.$(u) + C_1 Q(u)$ is lower semicontinuous w.r.t. $L^1-$ norm, $\qquad\qquad (0.2)$

with $C_1$ sufficiently large, and $u$ with small BV norm.
In the first part we analyze the more general case of quasilinear hyperbolic systems. We show that in general this result is not true if the system is not in conservation form: there are Riemann solvers, identified by selecting an entropic conditions on the jumps, which do not satisfy the Glimm interaction estimate (0.2). Next we consider hyperbolic systems in conservation form, i.e. $A(u) = Df(u)$. In this case, there is only one entropic Riemann solver, and we prove that this particular Riemann solver satisfies (0.2) for a particular functional $Q$, which we construct explicitly. The main novelty here is that we suppose only the Jacobian matrix $Df(u)$ strictly hyperbolic, without any assumption on the number of inflection points of $f$.
These results are achieved by an analysis of the growth of Tot.Var.$(u)$ when nonlinear waves of (0.1) interact, and the introduction of a Glimm type functional $Q$, similar but not equivalent to Liu's interaction functional [11].

2003, 9(1): 167-192 doi: 10.3934/dcds.2003.9.167 +[Abstract](444) +[PDF](299.2KB)
Abstract:
The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations with convex Hamiltonians $H=H(Du)$ is established, provided the discontinuous initial value function $\varphi(x)$ is continuous outside a set $\Gamma$ of measure zero and satisfies

(*)$\qquad\qquad \varphi(x)\ge\varphi_{\star \star}(x) \equiv \lim$inf$_{y\rightarrow x, y\in\mathbb R^d\backslash\Gamma}\varphi(y). The regularity of discontinuous solutions to Hamilton-Jacobi equations with locally strictly convex Hamiltonians is proved: The discontinuous solutions with almost everywhere continuous initial data satisfying (*) become Lipschitz continuous after finite time. The$L^1$-accessibility of initial data and a comparison principle for discontinuous solutions are shown. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions,$L$-solutions, minimax solutions, and$L^\infty\$-solutions is also clarified.

2003, 9(1): 193-208 doi: 10.3934/dcds.2003.9.193 +[Abstract](355) +[PDF](265.2KB)
Abstract:
We show that volume-preserving perturbations of some product actions of property (T) groups exhibit a "foliation rigidity" property, which reduces the partially hyperbolic action to a family of hyperbolic actions. This is used to show that certain partially hyperbolic actions are locally rigid.
2003, 9(1): 209-224 doi: 10.3934/dcds.2003.9.209 +[Abstract](426) +[PDF](252.2KB)
Abstract:
We study the phenomenon of stability breakdown for non-autonomous differential equations whose time dependence is determined by a minimal, strictly ergodic flow. We find that, under appropriate assumptions, a new attractor may appear. More generally, almost automorphic minimal sets are found.

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