Showing papers in "Discrete and Continuous Dynamical Systems in 1997"
TL;DR: In this article, the evolutions of the interfaces between gases and the vacuum were studied for both inviscid and viscous one-dimensional gas motions, and the local existence of solutions was proved.
Abstract: In this paper we study the evolutions of the interfaces between
gases and the vacuum for both inviscid and viscous one dimensional
isentropic gas motions. The local (in time) existence of solutions
for both inviscid and viscous models with initial data containing
vacuum states is proved and some singular properties on the free
surfaces separating the gas and the vacuum are obtained. It is
found that the Euler equations are better behaved near the vacuum than
the compressible Navier-Stokes equations. The Navier-Stokes
equations with viscosity depending on density are introduced,
which is shown to be well-posed (at least
locally) and yield the desired solutions near vacuum.
261 citations
TL;DR: In this paper, the authors investigate how the non-analytic solitary wave solutions of an integrable bi-Hamiltonian system arising in fluid mechanics, can be recovered as limits of classical solitary wave solution forming analytic homoclinic orbits for the reduced dynamical system.
Abstract: We investigate how the non-analytic solitary wave solutions -- peakons and
compactons -- of an integrable bi-Hamiltonian system arising in fluid mechanics, can be
recovered as limits of classical solitary wave solutions forming analytic homoclinic orbits for
the reduced dynamical system. This phenomenon is examined to understand the important
effect of linear dispersion terms on the analyticity of such homoclinic orbits.
101 citations
TL;DR: In this paper, the existence of random attractors of randomly perturbed dynamical systems has been proved using Lyapunov's direct method and the Hopf bifurcation behavior of the random======Duffing-van der Pol equation.
Abstract: This paper is concerned with attractors of randomly perturbed
dynamical systems, called random attractors. The framework used is
provided by the theory of random dynamical systems.
We first define, analyze, and prove existence of random attractors. The
main result is a technique, similar to Lyapunov's direct method, to
ensure existence of random attractors for random differential
equations. This method is formulated as a generally applicable
procedure. As an illustration we shall apply it to the random
Duffing-van der Pol equation.
We then show, by the same example, that random attractors provide an
important tool to analyze the bifurcation behavior of stochastically
perturbed dynamical systems. We introduce new methods and techniques,
and we investigate the Hopf bifurcation behavior of the random
Duffing-van der Pol equation in detail. In addition, the
relationship of random attractors to invariant measures and unstable
sets is studied.
76 citations
TL;DR: In this paper, it was shown that the solitary wave solutions investigated in part of this paper can be extended as analytic functions in the complex plane, except for countably many branch points and branch lines.
Abstract: In this part, we prove that the solitary wave solutions investigated in part
I are
extended as analytic functions in the complex plane, except for
at most countably many branch points and branch lines. We describe
in detail how the
limiting behavior of the complex singularities allows the
creation of non-analytic solutions with corners and/or compact
support.
62 citations
TL;DR: In this article, the quasistatic evolution of an elastic-viscoplastic body in bilateral contact with a rigid foundation is considered, where the contact involves viscous friction of the Tresca type.
Abstract: The quasistatic evolution of an elastic-viscoplastic body in
bilateral contact with a
rigid foundation is considered. The contact involves viscous friction of
Tresca type. Two variational
formulations of the problem are proposed, followed by existence and
uniqueness results. Some
properties involving the equivalence between the previous variational
formulations, the continuous
dependence of the solution with respect to the data as well as a
convergence result with respect to
the friction yield limit are also obtained.
47 citations
TL;DR: In this paper, the authors consider a nonlinear wave equation in three dimensions with zero mass and a negative potential, and show that if the potential is sufficiently short-range, then it does not alter the global existence of small-amplitude solutions.
Abstract: Consider a nonlinear wave equation in three space dimensions with zero
mass together with a negative potential. If the potential is sufficiently short-range,
then it does not alter the global existence of small-amplitude solutions. On the other
hand, if the potential is sufficiently large, it will force some solutions to blow up in
a finite time.
45 citations
TL;DR: In this article, the authors consider optimal control problems governed by semilinear par-consuming equations with nonlinear boundary conditions and pointwise constraints on the state variable, and derive optimality conditions by means of a Lagrange multiplier theorem in Banach spaces.
Abstract: We consider optimal control problems governed by semilinear par-
abolic equations with nonlinear boundary conditions and pointwise constraints
on the state variable. In Robin boundary conditions considered here, the nonlinear term is neither necessarily monotone nor Lipschitz with respect to the
state variable. We derive optimality conditions by means of a Lagrange multiplier theorem in Banach spaces. The adjoint state must satisfy a parabolic
equation with Radon measures in Robin boundary conditions, in the terminal
condition and in the distributed term. We give a precise meaning to the adjoint equation with measures as data and we prove the existence of a unique
weak solution for this equation in an appropriate space.
41 citations
TL;DR: In this article, the Cauchy problem was shown to have a unique "viscosity solution for a nonlinear hyperbolic system, where each characteristic field is either linearly degenerate or genuinely nonlinear.
Abstract: This paper is concerned with
the Cauchy problem
$(*) \quad \quad u_t+[F(u)]_x=g(t,x,u),\quad
u(0,x)=\overline{u}(x),$
for a nonlinear $2\times 2$ hyperbolic system of
inhomogeneous balance laws
in one space dimension. As usual,
we assume that the system is strictly hyperbolic and that each
characteristic field is either
linearly degenerate or genuinely nonlinear.
Under suitable assumptions on $g$, we prove that there exists
$T>0$ such that, for every $\overline{u}$ with sufficiently small
total variation, the Cauchy problem ($*$)
has a unique "viscosity solution",
defined for $t\in [0,T]$,
depending continuously on the initial data.
38 citations
TL;DR: In this article, the spectral and large-time properties of positive operator semigroups have been studied and the spectral radii of a completely monotonic operator family have been shown to form a superconvex function.
Abstract: We consider positive perturbations $A = B+ C $
of resolvent positive operators $B$ by positive
operators $C: D(A) \to X$ and in particular study their spectral
properties. We characterize the spectral bound of $A$, $s(A)$, in
terms of the resolvent outputs $F(\lambda) = C (\lambda - B)^{-1}$
and derive conditions for $s(A)$ to be an eigenvalue of $A$ and a
(first order) pole of the resolvent of $A$.
On our way we show that the spectral radii of a completely monotonic
operator family form a superconvex function. Our results will be
used in forthcoming publications to study the spectral and large-time
properties of positive operator semigroups.
33 citations
TL;DR: In this paper, the authors study the local Cauchy problem for semilinear parabolic equations in Sobolev spaces of fractional sizes and prove some smoothing effects.
Abstract: We study the local
Cauchy problem for the semilinear parabolic equations
$\partial _t U-\Delta U=P(D)F(U), \quad (t,x)
\in [0,T[ \times \mathbb{R}^n $
with initial data in Sobolev spaces of fractional
order $H^s_p(\mathbb{R}^n)$. The techniques that
we use allow us to consider measures but also distributions
as initial data ($s<0$). We also prove some smoothing effects
and $L^q([0,T[,L^p)$ estimates for
the solutions of such equations.
31 citations
TL;DR: In this paper, the authors consider discretized random perturbations of hyperbolic dynamical systems and prove that when perturbation tends to zero invariant measures, corresponding Markov chains converge to the Sinai-Bowen-Ruelle measure of the dynamical system.
Abstract: I consider discretized random perturbations
of hyperbolic dynamical systems
and prove that when perturbation
parameter tends to zero invariant measures
of corresponding Markov chains converge to
the Sinai-Bowen-Ruelle measure of the dynamical system. This
provides a robust method for computations of
such measures and for visualizations of some hyperbolic
attractors by modeling randomly perturbed
dynamical systems on a computer.
Similar results are true for
discretized random perturbations
of maps of the interval satisfying the Misiurewicz
condition considered in [KK].
TL;DR: In this paper, the qualitative dynamics of the local flow in a neighborhood of zero regular and singular points of the discontinuity surface were analyzed for vector fields around the origin in dimension three, which are approximations of discontinuous ones.
Abstract: In this paper vector fields around the origin in dimension three
which are approximations of discontinuous ones are studied. In a
former work of Sotomayor and Teixeira [6] it is shown, via
regularization, that Filippov's conditions are the natural ones to
extend the orbit solutions through the discontinuity set for vector
fields in dimension two. In this paper we show that this is also the
case for discontinuous vector fields in dimension three. Moreover, we
analyse the qualitative dynamics of the local flow in a neighborhood
of the codimension zero regular and singular points of the
discontinuity surface.
TL;DR: In this article, the authors studied the asymptotic behavior in time of solutions to the derivative nonlinear Schrodinger equation (DNLSE) and showed that the solution of DNLS satisfies the time decay estimate.
Abstract: We continue to study the asymptotic behavior in time of
solutions
to the derivative nonlinear Schrodinger equation
$ i u_t + u_{x x} + ia(|u|^2u)_x = 0,
\quad (t,x) \in \mathbf{R}\times \mathbf{R},$
$ u(0,x) = u_0 (x), \quad x\in \mathbf{R},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$(DNLS)
where $a \in \mathbf{R}$. We prove that if
$ ||u_0||_{ H^{1,\gamma}} + ||u_0||_{ H^{1+\gamma,0}}$
is sufficiently small with $\gamma > 1/2$,
then the solution of (DNLS)
satisfies the time decay estimate
$
||u(t)||_{L^\infty} + ||u_x(t)||_{L^\infty}\le C(1+|t|)^{-1/2},
$
where
$H^{m,s}= \{f\in \mathcal{S}'; ||f||_{m,s}=
||(1+|x|^2)^{s/2}(1-\partial_x^2)^{m/2}f||_{L^2}
< \infty\}$, $m,s\in \mathbf{R}$.
In the previous paper [4,Theorem 1.1] we showed the same result
under the condition that $\gamma \ge 2$. Furthermore we show
the asymptotic behavior in time of solutions involving
the previous result [4,Theorem 1.2].
TL;DR: In this article, the authors used the method of moving planes to fill the gap between the critical Sobolev exponent (n+2}/{n-1} ) and the critical ODE (n + 2/n-2) exponent.
Abstract: In this paper, we study the elliptic boundary value problem in a bounded
domain $\Omega$ in $R^n$, with smooth boundary:
$-\Delta u = R(x) u^p \quad \quad u > 0 x \in \Omega$
$u(x) = 0 \quad \quad x \in \partial \Omega.$
where $R(x)$ is a smooth function that may change signs.
In [2], using a blowing up argument,
Berestycki, Dolcetta, and Nirenberg, obtained a priori
estimates and hence the existence of solutions for the problem when
the exponent $1 < p < {n+2}/{n-1}$. Inspired by their result, in this
article, we use the method of moving planes to fill the gap between
${n+2}/{n-1}$ and the critical Sobolev exponent ${n+2}/{n-2}$. We
obtain a priori estimates for the solutions for all $1 < p < {n+2}/{n-2}$.
TL;DR: In this paper, it was shown that in the weak topology of measure preserving transformations, a dense shift-invariant measure is generated by transformations with purely singular continuous spectrum.
Abstract: We prove that for a dense $G_{\delta}$ of shift-invariant measures
on $A^{\ZZ^d}$, all $d$ shifts have purely singular continuous spectrum
and give a new proof that in the weak topology of measure preserving $\ZZ^d$
transformations, a dense $G_{\delta}$ is generated by transformations
with purely singular continuous spectrum.
We also give new examples of smooth unitary cocycles over an
irrational rotation which have purely singular continuous spectrum.
Quantitative weak mixing properties are related by results of Strichartz and
Last to spectral properties of the unitary Koopman operators.
TL;DR: In this article, the authors prove the local simultaneous linearizability of a pair of commuting functions at a shared fixed point under a very general -consuming -diophantine condition.
Abstract: We prove the local simultaneous linearizability of a pair of commuting
holomorphic functions at a shared fixed point under a very general -
we conjecture optimal - diophantine condition.
Let $f,g :\mathbb{C} \to \mathbb{C}$ with
a common fixed point at the origin and suppose that
$f(z) = \lambda z + \cdots$ and
$\lambda
e 0$. The map, $f,$ is called linearizable if there
is an analytic diffeomorphism, $h$, which conjugates $f$ with its linear part
in a neighborhood of the origin,
i.e., $h^{-1} \circ f \circ h (z) = \lambda z$ where $\lambda = f'(0).$
Two such diffeomorphisms are simultaneously linearizable
if they are linearized by the same map, $h$.
If $|\lambda| = 1$ then the situation is delicate. Nonlinearizable maps are
topologically abundant, i.e., for $\lambda$ in a dense co-meager
set in $\mathbb{S}^1$ there exist nonlinearizable analytic maps
with linear coefficient $\lambda$. In contrast there is
a diophantine condition on $\lambda$ that is satisfied by a set of full
measure in $\mathbb{S}^1$ which assures linearizability
of the map $f$.
TL;DR: In this article, the authors established coincidence of major types of dimensions for a broad class of separable metric spaces with finite borel measures, called tight spaces, for which these dimensions coincide naturally.
Abstract: We establish coincidence of major types of dimensions for a broad
class of separable metric spaces with finite borel measures.
To do this we introduce a new type of separable metric spaces,
so called tight spaces, for which these dimensions coincide naturally.
This class includes, for example, all manifolds of the curvature bounded
from below and any their subsets with induced metric.
In particular, we prove that Hentshel-Procaccia and Renyi spectra
for dimensions are equal in tight spaces for any measure.
We also give the examples that demonstrate that all known dimensions
can differ for bad enough metric spaces.
TL;DR: In this article, the existence of bounded solutions to second order differential equations of Lienard type under asymmetric conditions was proved under generalizing recent results of Ahmad and Ortega.
Abstract: We prove the existence of bounded solutions to
second order differential equations of Lienard type under
asymptotic conditions generalizing recent results of Ahmad and
Ortega.
TL;DR: In this paper, an asymptotically linear Hamiltonian system with strong resonance at the ϵ-infinity was considered and the existence of multiple periodic solutions was proved via variational methods in an equivariant setting.
Abstract: An asymptotically linear Hamiltonian system with strong resonance at
infinity is considered. The existence of multiple periodic solutions is proved via
variational methods in an equivariant setting.
TL;DR: In this article, the authors considered a class of integro-differential equations of parabolic type arising in the study of a quasi-static thermoelastic contact problem involving a critical parameter.
Abstract: In this paper we consider a class of integro-differential equations
of parabolic type arising in the study of a quasi-static thermoelastic contact
problem involving a critical parameter $\alpha$.
For $\alpha <1$, the problem is first transformed
into an equivalent standard parabolic equation with non-local
and non-linear boundary conditions. Then the existence, uniqueness and
continuous dependence of the solution upon the data are demonstrated via
solution representation techniques and the maximum principle. Finally
the asymptotic behavior of the solution as $ t \rightarrow \infty$ is examined, and we
show that the non-local term has no impact on the asymptotic behavior
for $ \alpha 1$.
TL;DR: In this article, a simplified model of fracture propagation based on the evolution by minimizing movements of the Mumford-Shah energy was proposed. But the model was not considered in the two-dimensional case, and it was shown that under a few additional assumptions on the "fracture" the movement solves the heat equation, with weak Neumann boundary conditions.
Abstract: We study a simplified model of fracture propagation introduced by L. Ambrosio and A. Braides, based on the evolution by minimizing movements of the Mumford-Shah energy. In the two-dimensional case, we show that under a few additional assumptions on the "fracture" the movement solves the heat equation, with (weak) Neumann
boundary conditions, and we are able to give some estimate on the decrease of the Mumford-Shah energy.
TL;DR: In this paper, the existence of periodic solutions for perturbations of second order nonlinear differential equations was proved by using the Poincare-Birkhoff fixed point theorem.
Abstract: We prove the existence of periodic solutions for perturbations of some
autonomous second order nonlinear differential equations by the use of the Poincare-
Birkhoff fixed point theorem.
TL;DR: In this paper, the Implicit-Function-Theorem was used to prove the existence of time-periodic solutions at high frequencies for infinite-dimensional second-order systems.
Abstract: We study the existence of periodic
solutions for
the infinite-dimensional second order system $\ddot x=V_{x},\
x\in\mathbb{T}^{\mathbb{Z}_+}.$
Using the Implicit-Function-Theorem, we prove the
existence of time-periodic solutions at "high frequencies"; no "smallness condition" on $V(x)$ is required.
TL;DR: In this paper, it was shown that the expansion rate of a positively invariant set for a C 1 endomorphism is the same as the infimum of the Lyapunov exponents for ergodic measures with support in the set.
Abstract: The logarithmic expansion rate of a positively invariant set for a C 1 endomorphism is shown to equal the infimum of the Lyapunov exponents for ergodic measures with support in the invariant set. Using this result, aperiodic flows of the two torus are shown to have an expansion rate of zero and the eects of conjugacies on expansion rates are investigated.
TL;DR: In this article, the authors proved instant extinction of the Dirichlet and Neumann boundary value solutions for quasilinear parabolic equations whose diffusion coefficient is singular when the spatial gradient of unknown function is zero.
Abstract: In this paper we prove instant extinction of the solutions to Dirichlet and
Neumann boundary value problem for some quasilinear parabolic equations whose
diffusion coefficient is singular when the spatial gradient of unknown function is zero.
TL;DR: A global existence theorem for two semilinear diffusion equations is proved in this paper, where the coefficients of the diffusion operator are not uniformly elliptic and the coefficients are coupled.
Abstract: A global existence theorem for two semilinear
diffusion equations is proved. The equations are
coupled and the diffusion coefficients are not
uniformly elliptic. They arise in the study of
a simple zonally averaged climate model (See
also [8, 9, 13, 14]).
The sectoriality of the diffusion operator is
proved with the help of a technique of
F. Ali Mehmeti and S. Nicaise [2].
Some imbedding results for weighted Sobolev spaces
and sign conditions for the nonlinearities
allow the application of a result due to Amann
[3], which proves the global result.
TL;DR: In this paper, an existence theorem about invariant hyperbolic tori for Hamiltonian systems with respect to perturbation was proved. But the existence theorem was not proved for the unperturbed systems.
Abstract: This paper deals with a problem, when the invariant hyperbolic tori for Hamiltonian systems persist under perturbation. An existence theorem about such invariant tori is
proved. Because the unperturbed systems possess the stronger degeneracy, this generalizes
the classical KAM theorem and a well known result of Graff.
TL;DR: In this article, the authors constructed several approximate inertial manifold for weakly damped nonlinear Schrodinger equation, and introduced suitable smooth approximations for the solution of the equation and its time derivatives.
Abstract: We construct several approximate inertial
manifolds for a weakly damped nonlinear
Schrodinger equation. For that purpose,
we introduce suitable smooth approximations
for the solution of the equation and
for its time derivatives.
TL;DR: In this article, the existence and multiplicity of homoclinic solutions for periodic solutions of second order Hamiltonian systems with nonconvex superpotentials was proved. But these solutions are constructed as critical points of the corresponding non-autonomous and nonsmooth Hamiltonian energy functional.
Abstract: This paper contains some existence and multiplicity
results for periodic solutions of second order
nonautonomous and nonsmooth Hamiltonian systems
involving nonconvex superpotentials. This study is
achieved by proving the existence of homoclinic
solutions. These solutions are constructed as critical
points of the corresponding nonconvex and nonsmooth
energy functional.
TL;DR: In this article, the existence of homoclinic orbits was investigated in systems with an expanding fixed point (snap-back repeller) and a perturbed system. And they showed that, if a certain analytical condition is satisfied, homoclineic orbits of the perturbed equation occur in pair on one side of the equation while they are not present on the other side.
Abstract: We consider the problem of existence of homoclinic orbits
in systems like $x_{n+1}=f(x_n)+\mu g(x_n,\mu )$, $x\in \mathbb{R} ^N$,
$\mu \in \mathbb{R}$, when the unperturbed system $x_{n+1}=f(x_n)$ has an
orbit ${ \gamma _n } _{n\in \mathbb{Z}}$ homoclinic to an expanding fixed
point (snap-back repeller) and such that $f'(\gamma _n)$ is
invertible for any $n
e 0$ but $f'(\gamma _0)$ is not. We show that,
if a certain analytical condition is satisfied, homoclinic orbits of
the perturbed equation occur in pair on one side of $\mu =0$ while
are not present on the other side.