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Discrete and Continuous Dynamical Systems

June 2022 , Volume 42 , Issue 6

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Hydrodynamic limits of the nonlinear Schrödinger equation with the Chern-Simons gauge fields
Jeongho Kim and Bora Moon
2022, 42(6): 2541-2561 doi: 10.3934/dcds.2021202 +[Abstract](504) +[HTML](161) +[PDF](410.25KB)
Abstract:

We present two types of the hydrodynamic limit of the nonlinear Schrödinger-Chern-Simons (SCS) system. We consider two different scalings of the SCS system and show that each SCS system asymptotically converges towards the compressible and incompressible Euler system, coupled with the Chern-Simons equations and Poisson equation respectively, as the scaled Planck constant converges to 0. Our method is based on the modulated energy estimate. In the case of compressible limit, we observe that the classical theory of relative entropy method can be applied to show the hydrodynamic limit, with the additional quantum correction term. On the other hand, for the incompressible limit, we directly estimate the modulated energy to derive the desired asymptotic convergence.

On the one dimensional cubic NLS in a critical space
Marco Bravin and Luis Vega
2022, 42(6): 2563-2584 doi: 10.3934/dcds.2021203 +[Abstract](400) +[HTML](143) +[PDF](386.87KB)
Abstract:

In this note we study the initial value problem in a critical space for the one dimensional Schrödinger equation with a cubic non-linearity and under some smallness conditions. In particular the initial data is given by a sequence of Dirac deltas with different amplitudes but equispaced. This choice is motivated by a related geometrical problem; the one describing the flow of curves in three dimensions moving in the direction of the binormal with a velocity that is given by the curvature.

Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems
Héctor Barge and José M. R. Sanjurjo
2022, 42(6): 2585-2601 doi: 10.3934/dcds.2021204 +[Abstract](461) +[HTML](178) +[PDF](329.39KB)
Abstract:

In this paper we study generalized Poincaré-Andronov-Hopf bifurcations of discrete dynamical systems. We prove a general result for attractors in \begin{document}$ n $\end{document}-dimensional manifolds satisfying some suitable conditions. This result allows us to obtain sharper Hopf bifurcation theorems for fixed points in the general case and other attractors in low dimensional manifolds. Topological techniques based on the notion of concentricity of manifolds play a substantial role in the paper.

Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space
Hailiang Li, Houzhi Tang and Haitao Wang
2022, 42(6): 2603-2636 doi: 10.3934/dcds.2021205 +[Abstract](610) +[HTML](146) +[PDF](421.66KB)
Abstract:

In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate \begin{document}$ (1+t)^{-\frac{1}{2}} $\end{document} in \begin{document}$ L^\infty $\end{document} norm.

Global well-posedness for fractional Sobolev-Galpern type equations
Huy Tuan Nguyen, Nguyen Anh Tuan and Chao Yang
2022, 42(6): 2637-2665 doi: 10.3934/dcds.2021206 +[Abstract](528) +[HTML](155) +[PDF](481.43KB)
Abstract:

This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontrivial limit technique for singular integral and some appropriate choices of weighted Banach space to prove the global existence result. For the gradient nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to show that the solution either exists globally in time or blows up at finite time. For the polynomial form nonlinearity, by assuming the smallness of the initial data we derive the global well-posed results. And for the case of exponential nonlinearity in two-dimensional space, we derive the global well-posedness by additionally using an Orlicz space.

Higher order parabolic boundary Harnack inequality in C1 and Ck, α domains
Teo Kukuljan
2022, 42(6): 2667-2698 doi: 10.3934/dcds.2021207 +[Abstract](432) +[HTML](146) +[PDF](470.3KB)
Abstract:

We study the boundary behaviour of solutions to second order parabolic linear equations in moving domains. Our main result is a higher order boundary Harnack inequality in C1 and Ck, α domains, providing that the quotient of two solutions vanishing on the boundary of the domain is as smooth as the boundary.

As a consequence of our result, we provide a new proof of higher order regularity of the free boundary in the parabolic obstacle problem.

On involution kernels and large deviations principles on $ \beta $-shifts
Victor Vargas
2022, 42(6): 2699-2718 doi: 10.3934/dcds.2021208 +[Abstract](409) +[HTML](140) +[PDF](445.69KB)
Abstract:

Consider \begin{document}$ \beta > 1 $\end{document} and \begin{document}$ \lfloor \beta \rfloor $\end{document} its integer part. It is widely known that any real number \begin{document}$ \alpha \in \Bigl[0, \frac{\lfloor \beta \rfloor}{\beta - 1}\Bigr] $\end{document} can be represented in base \begin{document}$ \beta $\end{document} using a development in series of the form \begin{document}$ \alpha = \sum_{n = 1}^\infty x_n\beta^{-n} $\end{document}, where \begin{document}$ x = (x_n)_{n \geq 1} $\end{document} is a sequence taking values into the alphabet \begin{document}$ \{0,\; ...\; ,\; \lfloor \beta \rfloor\} $\end{document}. The so called \begin{document}$ \beta $\end{document}-shift, denoted by \begin{document}$ \Sigma_\beta $\end{document}, is given as the set of sequences such that all their iterates by the shift map are less than or equal to the quasi-greedy \begin{document}$ \beta $\end{document}-expansion of \begin{document}$ 1 $\end{document}. Fixing a Hölder continuous potential \begin{document}$ A $\end{document}, we show an explicit expression for the main eigenfunction of the Ruelle operator \begin{document}$ \psi_A $\end{document}, in order to obtain a natural extension to the bilateral \begin{document}$ \beta $\end{document}-shift of its corresponding Gibbs state \begin{document}$ \mu_A $\end{document}. Our main goal here is to prove a first level large deviations principle for the family \begin{document}$ (\mu_{tA})_{t>1} $\end{document} with a rate function \begin{document}$ I $\end{document} attaining its maximum value on the union of the supports of all the maximizing measures of \begin{document}$ A $\end{document}. The above is proved through a technique using the representation of \begin{document}$ \Sigma_\beta $\end{document} and its bilateral extension \begin{document}$ \widehat{\Sigma_\beta} $\end{document} in terms of the quasi-greedy \begin{document}$ \beta $\end{document}-expansion of \begin{document}$ 1 $\end{document} and the so called involution kernel associated to the potential \begin{document}$ A $\end{document}.

A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior
Yuki Kaneko, Hiroshi Matsuzawa and Yoshio Yamada
2022, 42(6): 2719-2745 doi: 10.3934/dcds.2021209 +[Abstract](580) +[HTML](158) +[PDF](418.97KB)
Abstract:

We study a free boundary problem of a reaction-diffusion equation \begin{document}$ u_t = \Delta u+f(u) $\end{document} for \begin{document}$ t>0,\ |x|<h(t) $\end{document} under a radially symmetric environment in \begin{document}$ \mathbb{R}^N $\end{document}. The reaction term \begin{document}$ f $\end{document} has positive bistable nonlinearity, which satisfies \begin{document}$ f(0) = 0 $\end{document} and allows two positive stable equilibrium states and a positive unstable equilibrium state. The problem models the spread of a biological species, where the free boundary represents the spreading front and is governed by a one-phase Stefan condition. We show multiple spreading phenomena in high space dimensions. More precisely the asymptotic behaviors of solutions are classified into four cases: big spreading, small spreading, transition and vanishing, and sufficient conditions for each dynamical behavior are also given. We determine the spreading speed of the spherical surface \begin{document}$ \{x\in \mathbb{R}^N:\ |x| = h(t)\} $\end{document}, which expands to infinity as \begin{document}$ t\to\infty $\end{document}, even when the corresponding semi-wave problem does not admit solutions.

Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system
Andrea Bondesan and Marc Briant
2022, 42(6): 2747-2773 doi: 10.3934/dcds.2021210 +[Abstract](375) +[HTML](126) +[PDF](487.73KB)
Abstract:

Recently, the authors proved [2] that the Maxwell-Stefan system with an incompressibility-like condition on the total flux can be rigorously derived from the multi-species Boltzmann equation. Similar cross-diffusion models have been widely investigated, but the particular case of a perturbative incompressible setting around a non constant equilibrium state of the mixture (needed in [2]) seems absent of the literature. We thus establish a quantitative perturbative Cauchy theory in Sobolev spaces for it. More precisely, by reducing the analysis of the Maxwell-Stefan system to the study of a quasilinear parabolic equation on the sole concentrations and with the use of a suitable anisotropic norm, we prove global existence and uniqueness of strong solutions and their exponential trend to equilibrium in a perturbative regime around any macroscopic equilibrium state of the mixture. As a by-product, we show that the equimolar diffusion condition naturally appears from this perturbative incompressible setting.

On $ n $-tuplewise IP-sensitivity and thick sensitivity
Jian Li and Yini Yang
2022, 42(6): 2775-2793 doi: 10.3934/dcds.2021211 +[Abstract](454) +[HTML](147) +[PDF](371.22KB)
Abstract:

Let \begin{document}$ (X,T) $\end{document} be a topological dynamical system and \begin{document}$ n\geq 2 $\end{document}. We say that \begin{document}$ (X,T) $\end{document} is \begin{document}$ n $\end{document}-tuplewise IP-sensitive (resp. \begin{document}$ n $\end{document}-tuplewise thickly sensitive) if there exists a constant \begin{document}$ \delta>0 $\end{document} with the property that for each non-empty open subset \begin{document}$ U $\end{document} of \begin{document}$ X $\end{document}, there exist \begin{document}$ x_1,x_2,\dotsc,x_n\in U $\end{document} such that

is an IP-set (resp. a thick set).

We obtain several sufficient and necessary conditions of a dynamical system to be \begin{document}$ n $\end{document}-tuplewise IP-sensitive or \begin{document}$ n $\end{document}-tuplewise thickly sensitive and show that any non-trivial weakly mixing system is \begin{document}$ n $\end{document}-tuplewise IP-sensitive for all \begin{document}$ n\geq 2 $\end{document}, while it is \begin{document}$ n $\end{document}-tuplewise thickly sensitive if and only if it has at least \begin{document}$ n $\end{document} minimal points. We characterize two kinds of sensitivity by considering some kind of factor maps. We introduce the opposite side of pairwise IP-sensitivity and pairwise thick sensitivity, named (almost) pairwise IP\begin{document}$ ^* $\end{document}-equicontinuity and (almost) pairwise syndetic equicontinuity, and obtain dichotomies results for them. In particular, we show that a minimal system is distal if and only if it is pairwise IP\begin{document}$ ^* $\end{document}-equicontinuous. We show that every minimal system admits a maximal almost pairwise IP\begin{document}$ ^* $\end{document}-equicontinuous factor and admits a maximal pairwise syndetic equicontinuous factor, and characterize them by the factor maps to their maximal distal factors.

Stability of hyperbolic Oseledets splittings for quasi-compact operator cocycles
Harry Crimmins
2022, 42(6): 2795-2857 doi: 10.3934/dcds.2022001 +[Abstract](349) +[HTML](128) +[PDF](737.62KB)
Abstract:

We develop a random version of the perturbation theory of Gouëzel, Keller, and Liverani, and consequently obtain results on the stability of Oseledets splittings and Lyapunov exponents for operator cocycles. By applying the theory to the Perron-Frobenius operator cocycles associated to random \begin{document}$ \mathcal{C}^k $\end{document} expanding maps on \begin{document}$ S^1 $\end{document} (\begin{document}$ k \ge 2 $\end{document}) we provide conditions for the stability of Lyapunov exponents and Oseledets splitting of the cocycle under (ⅰ) uniformly small fiber-wise \begin{document}$ \mathcal{C}^{k-1} $\end{document}-perturbations to the random dynamics, and (ⅱ) numerical approximation via a Fejér kernel method. A notable addition to our approach is the use of Saks spaces, which allow us to weaken the hypotheses of Gouëzel-Keller-Liverani perturbation theory, provides a unifying framework for key concepts in the so-called 'functional analytic' approach to studying dynamical systems, and has applications to the construction of anisotropic norms adapted to dynamical systems.

The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates
Donatella Donatelli and Nóra Juhász
2022, 42(6): 2859-2892 doi: 10.3934/dcds.2022002 +[Abstract](371) +[HTML](136) +[PDF](502.18KB)
Abstract:

A widely used approach to mathematically describe the atmosphere is to consider it as a geophysical fluid in a shallow domain and thus to model it using classical fluid dynamical equations combined with the explicit inclusion of an \begin{document}$ \epsilon $\end{document} parameter representing the small aspect ratio of the physical domain. In our previous paper [14] we proved a weak convergence theorem for the polluted atmosphere described by the Navier-Stokes equations extended by an advection-diffusion equation. We obtained a justification of the generalised hydrostatic limit model including the pollution effect described for the case of classical, east-north-upwards oriented local Cartesian coordinates. Here we give a two-fold improvement of this statement. Firstly, we consider a meteorologically more meaningful coordinate system, incorporate the analytical consequences of this coordinate change into the governing equations, and verify that the weak convergence still holds for this altered system. Secondly, still considering this new, so-called downwind-matching coordinate system, we prove an analogous strong convergence result, which we make complete by providing a closely related existence theorem as well.

Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $ \mathbb{R}^{N} $
Wenxian Shen and Shuwen Xue
2022, 42(6): 2893-2925 doi: 10.3934/dcds.2022003 +[Abstract](436) +[HTML](160) +[PDF](500.17KB)
Abstract:

In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on \begin{document}$ \mathbb{R}^{N} $\end{document},

where \begin{document}$ \chi, \ a,\ b,\ \lambda,\ \mu $\end{document} are positive constants and \begin{document}$ N $\end{document} is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption \begin{document}$ b>\frac{N\chi\mu}{4} $\end{document}, the global existence of a unique classical solution \begin{document}$ (u(x,t;u_0, v_0),v(x,t;u_0, v_0)) $\end{document} of (1) with \begin{document}$ u(x,0;u_0, v_0) = u_0(x) $\end{document} and \begin{document}$ v(x,0;u_0, v_0) = v_0(x) $\end{document} for every nonnegative, bounded, and uniformly continuous function \begin{document}$ u_0(x) $\end{document}, and every nonnegative, bounded, uniformly continuous, and differentiable function \begin{document}$ v_0(x) $\end{document}. Next, under the same assumption \begin{document}$ b>\frac{N\chi\mu}{4} $\end{document}, we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function \begin{document}$ u_0 $\end{document} is bounded below by a positive constant independent of \begin{document}$ (u_0, v_0) $\end{document} when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function \begin{document}$ u_0 $\end{document}. We show that there is \begin{document}$ K = K(a,\lambda,N)>\frac{N}{4} $\end{document} such that if \begin{document}$ b>K \chi\mu $\end{document} and \begin{document}$ \lambda\geq \frac{a}{2} $\end{document}, then for every strictly positive initial function \begin{document}$ u_0(\cdot) $\end{document}, it holds that

Global regularity for the 3D compressible magnetohydrodynamics with general pressure
Anthony Suen
2022, 42(6): 2927-2943 doi: 10.3934/dcds.2022004 +[Abstract](361) +[HTML](126) +[PDF](357.72KB)
Abstract:

We address the compressible magnetohydrodynamics (MHD) equations in \begin{document}$ \mathbb R^3 $\end{document} and establish a blow-up criterion for the local-in-time smooth solutions in terms of the density only. Namely, if the density is away from vacuum (\begin{document}$ \rho = 0 $\end{document}) and the concentration of mass (\begin{document}$ \rho = \infty $\end{document}), then a local-in-time smooth solution can be continued globally in time. The results generalise and strengthen the previous ones in the sense that there is no magnetic field present in the criterion and the assumption on the pressure is significantly relaxed. The proof is based on some new a priori estimates for three-dimensional compressible MHD equations.

On the analyticity of the trajectories of the particles in the planar patch problem for some active scalar equations
Josep M. Burgués and Joan Mateu
2022, 42(6): 2945-3003 doi: 10.3934/dcds.2022005 +[Abstract](351) +[HTML](121) +[PDF](526.76KB)
Abstract:

We prove analyticity in time of the particle trajectories associated with the solutions of some transport equations when the initial condition is the characteristic function of a regular bounded domain. These results are obtained from a detailed study of the Beurling transform, that represents a derivative of the velocity field. The precise estimates obtained for the solutions of an equation satisfied by the Lagrangian flow, are a key point in the development.

The mean-field limit of the Lieb-Liniger model
Matthew Rosenzweig
2022, 42(6): 3005-3037 doi: 10.3934/dcds.2022006 +[Abstract](364) +[HTML](136) +[PDF](562.4KB)
Abstract:

We consider the well-known Lieb-Liniger (LL) model for \begin{document}$ N $\end{document} bosons interacting pairwise on the line via the \begin{document}$ \delta $\end{document} potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [3] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65,66,67] and Knowles and Pickl [44]. To overcome difficulties stemming from the singularity of the \begin{document}$ \delta $\end{document} potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the \begin{document}$ N $\end{document}-body wave function in a single particle variable. By further exploiting the \begin{document}$ L^2 $\end{document}-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finite mass, but only for a very special class of \begin{document}$ N $\end{document}-body initial states.

A Cantor dynamical system is slow if and only if all its finite orbits are attracting
Silvère Gangloff and Piotr Oprocha
2022, 42(6): 3039-3064 doi: 10.3934/dcds.2022007 +[Abstract](376) +[HTML](123) +[PDF](452.63KB)
Abstract:

In this paper we completely solve the problem of when a Cantor dynamical system \begin{document}$ (X, f) $\end{document} can be embedded in \begin{document}$ \mathbb{R} $\end{document} with vanishing derivative. For this purpose we construct a refining sequence of marked clopen partitions of \begin{document}$ X $\end{document} which is adapted to a dynamical system of this kind. It turns out that there is a huge class of such systems.

Local well-posedness for the Maxwell-Dirac system in temporal gauge
Hartmut Pecher
2022, 42(6): 3065-3076 doi: 10.3934/dcds.2022008 +[Abstract](281) +[HTML](96) +[PDF](330.13KB)
Abstract:

We consider the low regularity well-posedness problem for the Maxwell-Dirac system in 3+1 dimensions:

where \begin{document}$ F_{\mu \nu} = \partial^{\mu} A_{\nu} - \partial^{\nu} A_{\mu} $\end{document}, and \begin{document}$ \alpha^{\mu} $\end{document} are the 4x4 Dirac matrices. We assume the temporal gauge \begin{document}$ A_0 = 0 $\end{document} and make use of the fact that some of the nonlinearities fulfill a null condition. Because we work in the temporal gauge we also apply a method, which was used by Tao for the Yang-Mills system.

2021 Impact Factor: 1.588
5 Year Impact Factor: 1.568
2021 CiteScore: 2.4

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