American Institute of Mathematical Sciences

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1078-0947

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Discrete & Continuous Dynamical Systems - A

May 2020 , Volume 40 , Issue 5

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2020, 40(5): 2515-2559 doi: 10.3934/dcds.2020140 +[Abstract](938) +[HTML](68) +[PDF](486.23KB)
Abstract:

The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In the functional setting as close as possible to the physical energy spaces, we prove the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a complicated parabolic-hyperbolic system, and on a refined time-weighted inequality.

2020, 40(5): 2561-2591 doi: 10.3934/dcds.2020141 +[Abstract](604) +[HTML](68) +[PDF](454.97KB)
Abstract:

In this paper we consider the nonlinear beam equations accounting for rotational inertial forces. Under suitable hypotheses we prove the existence, regularity and finite dimensionality of a compact global attractor and an exponential attractor. The main purpose is to trace the behavior of solutions of the nonlinear beam equations when the effect of the rotational inertia fades away gradually. A natural question is whether there are qualitative differences would appear or not. To answer the question, we deal with the rotational inertia with a parameter \begin{document}$\alpha$\end{document} and consider the difference of behavior between the case \begin{document}$0<\alpha\le1$\end{document} and the case \begin{document}$\alpha = 0$\end{document}. The main novel contribution of this paper is to show the continuity of global attractors and exponential attractors with respect to \begin{document}$\alpha$\end{document} in some sense.

2020, 40(5): 2593-2613 doi: 10.3934/dcds.2020142 +[Abstract](491) +[HTML](89) +[PDF](405.83KB)
Abstract:

In this paper we construct two families of initial data being arbitrarily large under any scaling-invariant norm for which their corresponding weak solution to the three-dimensional Navier–Stokes equations become smooth on either \begin{document}$[0,T_1]$\end{document} or \begin{document}$[T_2,\infty)$\end{document}, respectively, where \begin{document}$T_1$\end{document} and \begin{document}$T_2$\end{document} are two times prescribed previously. In particular, \begin{document}$T_1$\end{document} can be arbitrarily large and \begin{document}$T_2$\end{document} can be arbitrarily small. Therefore, a possible formation of singularities would occur after a very long or short evolution time, respectively.

We further prove that if a large part of the kinetic energy is consumed prior to the first (possible) blow-up time, then the global-in-time smoothness of the solutions follows for the two families of initial data.

2020, 40(5): 2615-2639 doi: 10.3934/dcds.2020143 +[Abstract](519) +[HTML](54) +[PDF](429.95KB)
Abstract:

We show that a necessary and sufficient condition for a smooth function on the tangent bundle of a manifold to be a Lagrangian density whose action can be minimized is, roughly speaking, that it be the sum of a constant, a nonnegative function vanishing on the support of the minimizers, and an exact form.

We show that this exact form corresponds to the differential of a Lipschitz function on the manifold that is differentiable on the projection of the support of the minimizers, and its derivative there is Lipschitz. This function generalizes the notion of subsolution of the Hamilton-Jacobi equation that appears in weak KAM theory, and the Lipschitzity result allows for the recovery of Mather's celebrated 1991 result as a special case. We also show that our result is sharp with several examples.

Finally, we apply the same type of reasoning to an example of a finite horizon Legendre problem in optimal control, and together with the Lipschitzity result we obtain the Hamilton–Jacobi–Bellman equation and the Maximum Principle.

2020, 40(5): 2641-2669 doi: 10.3934/dcds.2020144 +[Abstract](589) +[HTML](86) +[PDF](430.14KB)
Abstract:

We establish equivalent conditions for the existence of an integrable or locally integrable fixed point for a Markov operator with the maximal support. Maximal support means that almost all initial points will concentrate on the support of the invariant density under the iteration of the process. One of the equivalent conditions for the existence of a locally integrable fixed point is weak almost periodicity of the jump operator with respect to some sweep-out set. This result includes the case of the existence of an absolutely continuous \begin{document}$\sigma$\end{document}-finite invariant measure when we consider a nonsingular transformation on a probability space. Weak almost periodicity implies the Jacobs-Deleeuw-Glicksberg splitting and we show that constrictive Markov operators which guarantee the spectral decomposition are typical weakly almost periodic operators.

2020, 40(5): 2671-2685 doi: 10.3934/dcds.2020145 +[Abstract](702) +[HTML](57) +[PDF](359.32KB)
Abstract:

This article concerns with the existence of multiple positive solutions for the following logarithmic Schrödinger equation

where \begin{document}$\epsilon >0$\end{document}, \begin{document}$N \geq 1$\end{document} and \begin{document}$V$\end{document} is a continuous function with a global minimum. Using variational method, we prove that for small enough \begin{document}$\epsilon>0$\end{document}, the "shape" of the graph of the function \begin{document}$V$\end{document} affects the number of nontrivial solutions.

2020, 40(5): 2687-2703 doi: 10.3934/dcds.2020146 +[Abstract](658) +[HTML](63) +[PDF](374.87KB)
Abstract:

In this paper, we establish a variational principle for topological pressure on compact subsets in the context of amenable group actions. To be precise, for a countable amenable group action on a compact metric space, say \begin{document}$G\curvearrowright X$\end{document}, for any potential \begin{document}$f\in C(X)$\end{document}, we define and study topological pressure on an arbitrary subset and measure theoretic pressure for any Borel probability measure on \begin{document}$X$\end{document} (not necessarily invariant); moreover, we prove a variational principle for this topological pressure on a given nonempty compact subset \begin{document}$K\subseteq X$\end{document}.

2020, 40(5): 2705-2738 doi: 10.3934/dcds.2020147 +[Abstract](453) +[HTML](82) +[PDF](466.56KB)
Abstract:

We consider the Cauchy problem for the damped wave equation under the initial state that the sum of an initial position and an initial velocity vanishes. When the initial position is non-zero, non-negative and compactly supported, we study the large time behavior of the spatial null, critical, maximum and minimum sets of the solution. The behavior of each set is totally different from that of the corresponding set under the initial state that the sum of an initial position and an initial velocity is non-zero and non-negative.

The spatial null set includes a smooth hypersurface homeomorphic to a sphere after a large enough time. The spatial critical set has at least three points after a large enough time. The set of spatial maximum points escapes from the convex hull of the support of the initial position. The set of spatial minimum points consists of one point after a large enough time, and the unique spatial minimum point converges to the centroid of the initial position at time infinity.

2020, 40(5): 2739-2766 doi: 10.3934/dcds.2020148 +[Abstract](716) +[HTML](80) +[PDF](445.02KB)
Abstract:

In this article, we deal with the existence, qualitative and symmetry properties of normalized solutions to the following nonlinear Schrödinger system

where \begin{document}$\mu_{i}>0$\end{document} (\begin{document}$i$\end{document} = 1, 2), \begin{document}$\beta>0$\end{document}, and the frequencies \begin{document}$\lambda_{1}$\end{document}, \begin{document}$\lambda_{2}$\end{document} are unknown and appear as Lagrange multipliers. In addition, we study the stability of the corresponding standing waves for the related time-dependent Schrödinger systems. We mainly extend the results in J. Bellazzini et al. (Commun. Math. Phys. 2017), which dealt with mass-supercritical nonlinear Schrödinger equation with partial confinement, to cubic nonlinear Schrödinger systems with partial confinement.

2020, 40(5): 2767-2789 doi: 10.3934/dcds.2020149 +[Abstract](599) +[HTML](62) +[PDF](402.46KB)
Abstract:

This paper studies the measure theoretic pressure of measures that are not necessarily ergodic. We define the measure theoretic pressure of an invariant measure (not necessarily ergodic) via the Carathéodory-Pesin structure described in [13], and show that this quantity is equal to the essential supremum of the absolute value of free energy of the measures in an ergodic decomposition. Meanwhile, we define the measure theoretic pressure in another way by using separated sets, it is showed that this quantity is exactly the absolute value of free energy if the measure is ergodic. Particularly, if the dynamical system satisfies the uniform separation condition and the ergodic measures are entropy dense, this quantity is still equal to the the absolute value of free energy even if the measure is non-ergodic. As an application of the main results, we find that the Hausdorff dimension of an invariant measure supported on an average conformal repeller is given by the zero of the measure theoretic pressure of this measure. Furthermore, if a hyperbolic diffeomorphism is average conformal and volume-preserving, the Hausdorff dimension of any invariant measure on the hyperbolic set is equal to the sum of the zeros of measure theoretic pressure restricted to stable and unstable directions.

2020, 40(5): 2791-2826 doi: 10.3934/dcds.2020150 +[Abstract](602) +[HTML](67) +[PDF](470.32KB)
Abstract:

We define the notions of impulsive evolution processes and their pullback attractors, and exhibit conditions under which a given impulsive evolution process has a pullback attractor. We apply our results to a nonautonomous ordinary differential equation describing an integrate-and-fire model of neuron membrane, as well as to a heat equation with nonautonomous impulse and a nonautonomous 2D Navier-Stokes equation. Finally, we introduce the notion of tube conditions to impulsive evolution processes, and use them as an alternative way to obtain pullback attractors.

2020, 40(5): 2827-2873 doi: 10.3934/dcds.2020151 +[Abstract](472) +[HTML](69) +[PDF](559.08KB)
Abstract:

In this paper we will prove a functional central limit theorem (CLT) for random functions of the form

where the \begin{document}$q_i$\end{document}'s are certain type of bivariate polynomials, \begin{document}$F = F(x_1, ..., x_\ell)$\end{document} is a locally Hölder continuous function and the sequence of random variables \begin{document}$\{\xi_n\}$\end{document} satisfies some mixing and moment conditions. This paper continues the line of research started in [15] and [17], and it is a generalization of the results in [9] and Chapter 3 of [11]. We will also prove a strong law of large numbers (SLLN) for the averages \begin{document}$N^{-\frac12} {\mathcal S}_N(1)$\end{document} which extends the results from the beginning of Chapter 3 of [11] to general bivariate polynomial functions \begin{document}$q_i$\end{document}. Our results hold true for sequences \begin{document}$\{\xi_n\}$\end{document} generated by a wide class of Markov chains and dynamical systems. As an application we obtain functional CLT's for expressions of the form \begin{document}$N^{-\frac12}M([Nt])$\end{document}, where \begin{document}$M(N)$\end{document} counts the number of multiple recurrence of the sequence \begin{document}$\{\xi_n\}$\end{document} to certain sets \begin{document}$A_1, ..., A_\ell$\end{document} which occur at the times \begin{document}$q_1(n, N), ..., q_\ell(n, N)$\end{document}, as well as SLLN's for these \begin{document}$M(N)$\end{document}'s. One of the simplest examples is when \begin{document}$\xi_n$\end{document} is \begin{document}$n$\end{document}-the digit of a random \begin{document}$m$\end{document}-base or continued fraction expansion, and each \begin{document}$A_i$\end{document} is singleton (i.e. it represent one possible value of a digit).

2020, 40(5): 2875-2889 doi: 10.3934/dcds.2020152 +[Abstract](671) +[HTML](71) +[PDF](349.06KB)
Abstract:

In a bounded domain, we consider a thermoelastic plate with rotational forces. The rotational forces involve the spectral fractional Laplacian, with power parameter \begin{document}$0\le\theta\le 1$\end{document}. The model includes both the Euler-Bernoulli (\begin{document}$\theta = 0$\end{document}) and Kirchhoff (\begin{document}$\theta = 1$\end{document}) models for thermoelastic plate as special cases. First, we show that the underlying semigroup is of Gevrey class \begin{document}$\delta$\end{document} for every \begin{document}$\delta>(2-\theta)/(2-4\theta)$\end{document} for both the clamped and hinged boundary conditions when the parameter \begin{document}$\theta$\end{document} lies in the interval \begin{document}$(0, 1/2)$\end{document}. Then, we show that the semigroup is exponentially stable for hinged boundary conditions, for all values of \begin{document}$\theta$\end{document} in \begin{document}$[0, 1]$\end{document}. Finally, we prove, by constructing a counterexample, that, under hinged boundary conditions, the semigroup is not analytic, for all \begin{document}$\theta$\end{document} in the interval \begin{document}$(0, 1]$\end{document}. The main features of our Gevrey class proof are: the frequency domain method, appropriate decompositions of the components of the system and the use of Lions' interpolation inequalities.

2020, 40(5): 2891-2901 doi: 10.3934/dcds.2020153 +[Abstract](485) +[HTML](73) +[PDF](324.09KB)
Abstract:

In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal \begin{document}$\mathbb Z$\end{document} action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group \begin{document}$G$\end{document}, the centralizer of any minimal aperiodic \begin{document}$G$\end{document}-action on the Cantor set is a subgroup of the centralizer of a minimal \begin{document}$\mathbb Z$\end{document}-action.

2020, 40(5): 2903-2915 doi: 10.3934/dcds.2020154 +[Abstract](506) +[HTML](76) +[PDF](363.01KB)
Abstract:

Let \begin{document}$\{f_t\}_{t\in(1,2]}$\end{document} be the family of core tent maps of slopes \begin{document}$t$\end{document}. The parameterized Barge-Martin construction yields a family of disk homeomorphisms \begin{document}$\Phi_t\colon D^2\to D^2$\end{document}, having transitive global attractors \begin{document}$\Lambda_t$\end{document} on which \begin{document}$\Phi_t$\end{document} is topologically conjugate to the natural extension of \begin{document}$f_t$\end{document}. The unique family of absolutely continuous invariant measures for \begin{document}$f_t$\end{document} induces a family of ergodic \begin{document}$\Phi_t$\end{document}-invariant measures \begin{document}$\nu_t$\end{document}, supported on the attractors \begin{document}$\Lambda_t$\end{document}.

We show that this family \begin{document}$\nu_t$\end{document} varies weakly continuously, and that the measures \begin{document}$\nu_t$\end{document} are physical with respect to a weakly continuously varying family of background Oxtoby-Ulam measures \begin{document}$\rho_t$\end{document}.

Similar results are obtained for the family \begin{document}$\chi_t\colon S^2\to S^2$\end{document} of transitive sphere homeomorphisms, constructed in a previous paper of the authors as factors of the natural extensions of \begin{document}$f_t$\end{document}.

2020, 40(5): 2917-2944 doi: 10.3934/dcds.2020155 +[Abstract](650) +[HTML](59) +[PDF](419.36KB)
Abstract:

In the present paper, we deal with a new continuous and compact embedding theorems for the fractional Orlicz-Sobolev spaces, also, we study the existence of infinitely many nontrivial solutions for a class of non-local fractional Orlicz-Sobolev Schrödinger equations whose simplest prototype is

where \begin{document}$0<s<1$\end{document}, \begin{document}$d\geq2$\end{document} and \begin{document}$(-\triangle)^{s}_{m}$\end{document} is the fractional \begin{document}$M$\end{document}-Laplace operator. The proof is based on the variant Fountain theorem established by Zou.

2020, 40(5): 2945-2962 doi: 10.3934/dcds.2020156 +[Abstract](477) +[HTML](113) +[PDF](534.02KB)
Abstract:

We prove various pointwise properties of \begin{document}$L^p$\end{document}-viscosity solutions of fully nonlinear uniformly elliptic equations with unbounded measurable terms and possibly quadratically growing gradient terms. These include differentiability properties and "pointwise maximum principle". In particular we discuss an equivalent pointwise definition of \begin{document}$L^p$\end{document}-viscosity solution which, together with pointwise properties, allows to operate on \begin{document}$L^p$\end{document}-viscosity solutions almost as if they were strong solutions and move them freely from one equation to another. Such results were proved before in [6,8,30] for equations with linearly growing gradient terms, however it appears that they have not been widely used. Here we generalize pointwise properties of \begin{document}$L^p$\end{document}-viscosity solutions to a larger class of equations and show that they create a very powerful set of tools which can be used for instance to prove regularity results for equations and differential inequalities.

2020, 40(5): 2963-2986 doi: 10.3934/dcds.2020157 +[Abstract](564) +[HTML](74) +[PDF](396.97KB)
Abstract:

The Cauchy problem for a class of non-uniformly parabolic equations including (4) is studied for initial data with less regularity. When \begin{document}$m\in(1,2]$\end{document}, it is shown that there exists a smooth solution for \begin{document}$t>0$\end{document} when the initial data belongs to \begin{document}$L_{\text{loc}}^p, p>1$\end{document}. When \begin{document}$m>2$\end{document}, the same results holds when the initial data belongs to \begin{document}$W_{\text{loc}}^{1,p}, p\geq m-1$\end{document}. An example is displayed to show that a smooth solution may not exist when the initial data is merely in \begin{document}$L^p_{\text{loc}}, p>1$\end{document}. Solvability of the weak solution is also studied.

2020, 40(5): 2987-3011 doi: 10.3934/dcds.2020158 +[Abstract](579) +[HTML](71) +[PDF](407.24KB)
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In this paper, we prove that for the ill-prepared initial data of the form

the Cauchy problem of the incompressible Navier-Stokes equations on \begin{document}$\mathbb{R}^3$\end{document} is locally well-posed for all \begin{document}$\epsilon > 0$\end{document}, provided that the initial velocity profile \begin{document}$v_0$\end{document} is analytic in \begin{document}$x_3$\end{document} but independent of \begin{document}$\epsilon$\end{document}.

2020, 40(5): 3013-3030 doi: 10.3934/dcds.2020159 +[Abstract](624) +[HTML](127) +[PDF](358.85KB)
Abstract:

Given an arbitrary fixed closed subset \begin{document}$\mathcal{C}\subset\mathbb{R}^n$\end{document}, we provide an explicit method to construct a dynamical system which admits the regular part of \begin{document}$\mathcal{C}$\end{document} as globally bp-attracting set, i.e. a closed and invariant set which attracts every bounded positive orbit of the dynamical system. As application, we provide an explicit method of leafwise asymptotic bp-stabilization of the regular part of an a-priori given invariant set of a conservative system. The theoretical results are illustrated for the completely integrable case of the Rössler dynamical system.

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