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Discrete & Continuous Dynamical Systems - A

May 2010 , Volume 27 , Issue 2

A special issue
Trends and Developments in DE/Dynamics
Part I

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The Cauchy problem for Schrödinger flows into Kähler manifolds
Carlos Kenig, Tobias Lamm, Daniel Pollack, Gigliola Staffilani and Tatiana Toro
2010, 27(2): 389-439 doi: 10.3934/dcds.2010.27.389 +[Abstract](762) +[PDF](487.6KB)
Abstract:
We prove local well-posedness of the Schrödinger flow from $\RR^n$ into a compact Kähler manifold $N$ with initial data in $H^{s+1}(\RR^n,N)$ for $s\geq[\frac{n}{2}]+4$.
Mathematical strategies for filtering turbulent dynamical systems
Andrew J. Majda, John Harlim and Boris Gershgorin
2010, 27(2): 441-486 doi: 10.3934/dcds.2010.27.441 +[Abstract](821) +[PDF](1261.9KB)
Abstract:
The modus operandi of modern applied mathematics in developing very recent mathematical strategies for filtering turbulent dynamical systems is emphasized here. The approach involves the synergy of rigorous mathematical guidelines, exactly solvable nonlinear models with physical insight, and novel cheap algorithms with judicious model errors to filter turbulent signals with many degrees of freedom. A large number of new theoretical and computational phenomena such as "catastrophic filter divergence" in finite ensemble filters are reviewed here with the intention to introduce mathematicians, applied mathematicians, and scientists to this remarkable emerging scientific discipline with increasing practical importance.
Inelastic interaction of nearly equal solitons for the BBM equation
Yvan Martel and Frank Merle
2010, 27(2): 487-532 doi: 10.3934/dcds.2010.27.487 +[Abstract](650) +[PDF](509.8KB)
Abstract:
This paper is concerned with the interaction of two solitons of nearly equal speeds for the (BBM) equation. This work is an extension of [31] addressing the same question for the quartic (gKdV) equation. We consider the (BBM) equation, for $\lambda \in [0,1)$,

$ (1-\lambda \partial_x^2) \partial_t u + \partial_x (\partial_x^2 u - u + u^2) =0. $     (BBM)

Solitons are solutions of the form $ R_{\mu,x_0}(t,x)=Q_{\mu}(x-\mu t -x_0), $ for $\mu > -1$, $x_0\in \RR$.
   For $\mu_0>0$ small, let $U(t,x)$ be the unique solution of (BBM) such that

$ \lim_{t\to -\infty} \||U(t) - Q_{-\mu_0}(.+\mu_0 t) - Q_{\mu_0}(.-\mu_0 t )\||_{H^1} = 0. $

First, we prove that $U(t)$ remains close to the sum of two solitons, for all time $t\in \RR$,

$ U(t,x) = Q_{\mu_1(t)}(x-y_1(t)) + Q_{\mu_2(t)}(x-y_2(t)) + $ε$\ (t) \quad where \quad |\|\varepsilon(t)\| | \leq \mu_0^{2^-}, $

with $ y_1(t)-y_2(t)> 2 |\ln \mu_0| + O(1), $ which means that at the main order the situation is similar to the integrable KdV case. However, we show that the collision is perfectly elastic if and only if $\lambda=0$ (i.e. only in the integrable case).

Mathematical theory of viscous fluids: Retrospective and future perspectives
Eduard Feireisl
2010, 27(2): 533-555 doi: 10.3934/dcds.2010.27.533 +[Abstract](583) +[PDF](326.0KB)
Abstract:
We review the recent state of art of the mathematical theory of viscous, compressible, and heat conducting fluids. We emphasize the significant role of the Second law of thermodynamics in our approach. Qualitative properties of solutions and relations between different models are also discussed.
One-dimensional dynamics in the new millennium
Sebastian van Strien
2010, 27(2): 557-588 doi: 10.3934/dcds.2010.27.557 +[Abstract](879) +[PDF](488.7KB)
Abstract:
In the early 60's Sarkovskii discovered his famous theorem on the coexistence of periodic orbits for interval maps. Then, in the mid 70's, Li & Yorke rediscovered this result and somewhat later the papers by Feigenbaum and Coullet & Tresser on renormalisation and by Guckenheimer and Misiurewicz on sensitive dependence and existence of invariant measures, kicked off one of the most exciting areas within dynamical systems: iterations in dimension one. The purpose of this paper is to survey some of the recent developments, and pose some of the challenges and questions that keep this subject so intriguing.
Open problems in the theory of non-uniform hyperbolicity
Yakov Pesin and Vaughn Climenhaga
2010, 27(2): 589-607 doi: 10.3934/dcds.2010.27.589 +[Abstract](824) +[PDF](268.4KB)
Abstract:
This is a survey-type article whose goal is to review some recent developments in studying the genericity problem for non-uniformly hyperbolic dynamical systems with discrete time on compact smooth manifolds. We discuss both cases of systems which are conservative (preserve the Riemannian volume) and dissipative (possess hyperbolic attractors). We also consider the problem of coexistence of hyperbolic and regular behaviour.
Rigidity of real-analytic actions of $SL(n,\Z)$ on $\T^n$: A case of realization of Zimmer program
Anatole Katok and Federico Rodriguez Hertz
2010, 27(2): 609-615 doi: 10.3934/dcds.2010.27.609 +[Abstract](705) +[PDF](140.6KB)
Abstract:
We prove that any real-analytic action of $SL(n,\Z),n\ge 3$ with standard homotopy data that preserves an ergodic measure $\mu$ whose support is not contained in a ball, is analytically conjugate on an open invariant set to the standard linear action on the complement to a finite union of periodic orbits.
An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles
Kimie Nakashima, Wei-Ming Ni and Linlin Su
2010, 27(2): 617-641 doi: 10.3934/dcds.2010.27.617 +[Abstract](791) +[PDF](295.3KB)
Abstract:
We study the following Neumann problem

$ d\Delta u+g(x)u^{2}(1-u)=0 \ $ in Ω ,
$ 0\leq u\leq 1 $in Ω and $ \frac{\partial u}{\partial\nu}=0 $ on ∂Ω,

where $\Delta$ is the Laplace operator, $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$ with $\nu$ as its unit outward normal on the boundary $\partial\Omega$, and $g$ changes sign in $\Omega$. This equation models the "complete dominance" case in population genetics of two alleles. We show that the diffusion rate $d$ and the integral $\int_{\Omega}g\ \d x$ play important roles for the existence of stable nontrivial solutions, and the sign of $g(x)$ determines the limiting profile of solutions as $d$ tends to $0$. In particular, a conjecture of Nagylaki and Lou has been largely resolved.
   Our results and methods cover a much wider class of nonlinearities than $u^{2}(1-u)$, and similar results have been obtained for Dirichlet and Robin boundary value problems as well.

An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity
Yuan Lou, Wei-Ming Ni and Linlin Su
2010, 27(2): 643-655 doi: 10.3934/dcds.2010.27.643 +[Abstract](690) +[PDF](214.0KB)
Abstract:
We study a genetic model with two alleles $A_{1}$ and $A_{2}$ in a bounded smooth habitat $\Omega$. The frequency $u$ of the allele $A_{1}$, under the combined influence of migration and selection, obeys a parabolic equation of the type

$ u_{t}=d\Delta u+g(x)f(u),~0\leq u\leq 1 $ in Ω × (0, ∞),
$ \frac{\partial u}{\partial\nu}=0 $ on ∂ Ω × (0, ∞),

where $\Delta$ denotes the Laplace operator, $g$ may change sign in $\Omega$, and $f(0)=f(1)=0$, $f(s)>0$ for $s\in(0,1)$. Our main results include stability/instability of the trivial steady states $u\equiv 0$ and $u\equiv 1$, and the multiplicity of nontrivial steady states. This is a continuation of our work [12]. In particular, the conjecture of Nagylaki and Lou [11, p. 152] has been largely resolved. Similar results are obtained for Dirichlet and Robin boundary value problems as well.

The Pullback equation for degenerate forms
Saugata Bandyopadhyay, Bernard Dacorogna and Olivier Kneuss
2010, 27(2): 657-691 doi: 10.3934/dcds.2010.27.657 +[Abstract](662) +[PDF](340.1KB)
Abstract:
We discuss the existence of a diffeomorphism $\varphi:\mathbb{R} ^{n}\rightarrow\mathbb{R}^{n}$ such that

φ (g) =f,

where $f,g:\mathbb{R}^{n}\rightarrow\Lambda^{k}$ are closed differential forms and $2\leq k\leq n-1.$

Shock formation and breaking in granular avalanches
Michael Shearer and Nicholas Giffen
2010, 27(2): 693-714 doi: 10.3934/dcds.2010.27.693 +[Abstract](563) +[PDF](2691.1KB)
Abstract:
In this paper, we explore properties of shock wave solutions of the Gray-Thornton model for particle size segregation in granular avalanches. The model equation is a nonlinear scalar conservation law expressing conservation of mass under shear for the concentration of small particles in a bidisperse mixture. Shock waves are weak solutions of the partial differential equation across which the concentration jumps. We give precise criteria on smooth initial conditions under which a shock wave forms in the interior of the avalanche in finite time. Shocks typically lose stability as they are sheared by the flow, giving way to a complex structure in which a two-dimensional rarefaction wave interacts dynamically with a pair of shocks. The rarefaction represents a mixing zone, in which small and large particles are mixed as they are transported up and down (respectively) through the zone. The mixing zone expands and twice changes its detailed structure before reaching the boundary.
On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies
Thomas Chen and Nataša Pavlović
2010, 27(2): 715-739 doi: 10.3934/dcds.2010.27.715 +[Abstract](840) +[PDF](315.0KB)
Abstract:
We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$, $d\geq1$, for cubic, quintic, focusing and defocusing interactions. For both the focusing and defocusing case, and any $d\geq1$, we prove local existence and uniqueness of solutions in certain Sobolev type spaces $\H_\xi^\alpha$ of sequences of marginal density matrices which satisfy the space-time bound conjectured by Klainerman and Machedon for the cubic GP hierarchy in $d=3$. The regularity is accounted for by

$ \alpha $ > 1/2 if d=1
$ \alpha > \frac d2-\frac{1}{2(p-1)} if d\geq2 and (d,p)\neq(3,2) $
$ \alpha \geq 1 if (d,p)=(3,2) $

where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy; the parameter $\xi>0$ is arbitrary and determines the energy scale of the problem. For focusing GP hierarchies, we prove lower bounds on the blowup rate. Moreover, pseudoconformal invariance is established in the cases corresponding to $L^2$ criticality, both in the focusing and defocusing context. All of these results hold without the assumption of factorized initial conditions.

A Jang equation approach to the Penrose inequality
Hubert L. Bray and Marcus A. Khuri
2010, 27(2): 741-766 doi: 10.3934/dcds.2010.27.741 +[Abstract](718) +[PDF](252.0KB)
Abstract:
We introduce a generalized version of the Jang equation, designed for the general case of the Penrose Inequality in the setting of an asymptotically flat space-like hypersurface of a spacetime satisfying the dominant energy condition. The appropriate existence and regularity results are established in the special case of spherically symmetric Cauchy data, and are applied to give a new proof of the general Penrose Inequality for these data sets. When appropriately coupled with an inverse mean curvature flow, analogous existence and regularity results for the associated system of equations in the nonspherical setting would yield a proof of the full Penrose Conjecture. Thus it remains as an important and challenging open problem to determine whether this system does indeed admit the desired solutions.
Degenerate diffusion with a drift potential: A viscosity solutions approach
Inwon C. Kim and Helen K. Lei
2010, 27(2): 767-786 doi: 10.3934/dcds.2010.27.767 +[Abstract](771) +[PDF](267.9KB)
Abstract:
We introduce a notion of viscosity solutions for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solutions coincide with the weak solutions defined via integration by parts. As an application of the viscosity solutions theory, we show that the free boundary uniformly converges to the equilibrium as $t$ grows. In the case of a convex potential, an exponential rate of free boundary convergence is obtained.
Omega-limit sets for spiral maps
Bruce Kitchens and Michał Misiurewicz
2010, 27(2): 787-798 doi: 10.3934/dcds.2010.27.787 +[Abstract](693) +[PDF](186.5KB)
Abstract:
We investigate a class of homeomorphisms of a cylinder, with all trajectories convergent to the cylinder base and one fixed point in the base. Let A be a nonempty finite or countable family of sets, each of which can be a priori an $\omega$-limit set. Then there is a homeomorphism from our class, for which A is the family of all $\omega$-limit sets.
Weak Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress
Victor Isakov and Nanhee Kim
2010, 27(2): 799-825 doi: 10.3934/dcds.2010.27.799 +[Abstract](760) +[PDF](296.8KB)
Abstract:
We derive weak Carleman estimates with two large parameters for a general partial differential operator of second order under pseudo-convexity conditions on the weight function. We use these estimates to derive most natural Carleman type estimates for the (anisotropic) system of elasticity with residual stress and give applications to uniqueness and stability of the continuation and identification of the residual stress from boundary measurements. We give explicit sufficient pseudo-convexity conditions. Proofs use differential quadratic forms and Fourier analysis, combined with special (micro)localization arguments.
Front propagation problems with sub-diffusion
Yana Nec, Vladimir A Volpert and Alexander A Nepomnyashchy
2010, 27(2): 827-846 doi: 10.3934/dcds.2010.27.827 +[Abstract](636) +[PDF](251.7KB)
Abstract:
Front propagation is considered for two kinds of sub-diffusion - reaction systems: (i) systems with sub-diffusion limited reaction rate governed by models with fractional time derivatives; (ii) systems with activation limited reaction rate governed by integro-differential equations with two time variables. It is shown that in the former case the front is described by a travelling wave solution, while in the latter the velocity of the front decreases with time.
Eventual regularization of the slightly supercritical fractional Burgers equation
Chi Hin Chan, Magdalena Czubak and Luis Silvestre
2010, 27(2): 847-861 doi: 10.3934/dcds.2010.27.847 +[Abstract](745) +[PDF](217.3KB)
Abstract:
We prove that a weak solution of a slightly supercritical fractional Burgers equation becomes Hölder continuous for large time.

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