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Discrete & Continuous Dynamical Systems

July 2021 , Volume 41 , Issue 7

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Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem
Isabeau Birindelli, Françoise Demengel and Fabiana Leoni
2021, 41(7): 3021-3029 doi: 10.3934/dcds.2020395 +[Abstract](407) +[HTML](144) +[PDF](315.39KB)

We prove gradient boundary blow up rates for ergodic functions in bounded domains related to fully nonlinear degenerate/singular elliptic operators. As a consequence, we deduce the uniqueness, up to constants, of the ergodic functions. The results are obtained by means of a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators.

Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source
Lu Xu, Chunlai Mu and Qiao Xin
2021, 41(7): 3031-3043 doi: 10.3934/dcds.2020396 +[Abstract](380) +[HTML](149) +[PDF](342.51KB)

This paper deals with the global boundedness of solutions to the forager-exploiter model with logistic sources

under homogeneous Neumann boundary conditions in a smoothly bounded domain \begin{document}$ \Omega \subset R^2 $\end{document}, where the constants \begin{document}$ \mu $\end{document}, \begin{document}$ \mu_1 $\end{document}, \begin{document}$ \mu_2 $\end{document}, \begin{document}$ \lambda $\end{document}, \begin{document}$ m $\end{document} and \begin{document}$ l $\end{document} are positive. We prove that the corresponding initial-boundary value problem possesses a global classical solution that is uniformly bounded under conditions \begin{document}$ 2\leq m < 3 $\end{document}, \begin{document}$ l \geq 3 $\end{document}, \begin{document}$ r(x,t) \in C^1(\overline{\Omega}\times[0,\infty))\cup L^{\infty}(\Omega\times(0,\infty)) $\end{document} and the smooth nonnegative initial functions, which improves the results obtained by Wang and Wang (MMMAS 2020).

The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence
Yao Nie and Jia Yuan
2021, 41(7): 3045-3062 doi: 10.3934/dcds.2020397 +[Abstract](336) +[HTML](153) +[PDF](407.24KB)

In this paper, we consider the Littlewood-Paley \begin{document}$ p $\end{document}th-order (\begin{document}$ 1\le p<\infty $\end{document}) moments of the three-dimensional MHD periodic equations, which are defined by the infinite-time and space average of \begin{document}$ L^p $\end{document}-norm of velocity and magnetic fields involved in the spectral cut-off operator \begin{document}$ \dot\Delta_m $\end{document}. Our results imply that in some cases, \begin{document}$ k^{-\frac{1}{3}} $\end{document} is an upper bound at length scale \begin{document}$ 1/k $\end{document}. This coincides with the scaling law of many observations on astrophysical systems and simulations in terms of 3D MHD turbulence.

Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions
Amru Hussein, Martin Saal and Marc Wrona
2021, 41(7): 3063-3092 doi: 10.3934/dcds.2020398 +[Abstract](398) +[HTML](158) +[PDF](462.24KB)

The \begin{document}$ 3D $\end{document}-primitive equations with only horizontal viscosity are considered on a cylindrical domain \begin{document}$ \Omega = (-h,h) \times G $\end{document}, \begin{document}$ G\subset \mathbb{R}^2 $\end{document} smooth, with the physical Dirichlet boundary conditions on the sides. Instead of considering a vanishing vertical viscosity limit, we apply a direct approach which in particular avoids unnecessary boundary conditions on top and bottom. For the initial value problem, we obtain existence and uniqueness of local \begin{document}$ z $\end{document}-weak solutions for initial data in \begin{document}$ H^1((-h,h),L^2(G)) $\end{document} and local strong solutions for initial data in \begin{document}$ H^1(\Omega) $\end{document}. If \begin{document}$ v_0\in H^1((-h,h),L^2(G)) $\end{document}, \begin{document}$ \partial_z v_0\in L^q(\Omega) $\end{document} for \begin{document}$ q>2 $\end{document}, then the \begin{document}$ z $\end{document}-weak solution regularizes instantaneously and thus extends to a global strong solution. This goes beyond the global well-posedness result by Cao, Li and Titi (J. Func. Anal. 272(11): 4606-4641, 2017) for initial data near \begin{document}$ H^1 $\end{document} in the periodic setting. For the time-periodic problem, existence and uniqueness of \begin{document}$ z $\end{document}-weak and strong time periodic solutions is proven for small forces. Since this is a model with hyperbolic and parabolic features for which classical results are not directly applicable, such results for the time-periodic problem even for small forces are not self-evident.

Extensions of expansive dynamical systems
Mauricio Achigar
2021, 41(7): 3093-3108 doi: 10.3934/dcds.2020399 +[Abstract](377) +[HTML](147) +[PDF](359.37KB)

We characterize and describe the extensions of expansive and Ano- sov homeomorphisms on compact spaces. As an application we obtain a stability result for extensions of Anosov systems, and show a construction that embeds any expansive system inside an expansive system having the shadowing property for the pseudo orbits of the original space.

Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media
Izumi Takagi and Conghui Zhang
2021, 41(7): 3109-3140 doi: 10.3934/dcds.2020400 +[Abstract](423) +[HTML](168) +[PDF](748.22KB)

This paper is concerned with the existence and stability of steady states of a reaction-diffusion-ODE system arising from the theory of biological pattern formation. We are interested in spontaneous emergence of patterns from spatially heterogeneous environments, hence assume that all coefficients in the equations can depend on the spatial variable. We give some sufficient conditions on the coefficients which guarantee the existence of far-from-the-equilibrium patterns with jump discontinuity and then verify their stability in a weak sense. Our conditions cover the case where the number of equilibria of the kinetic system (i.e., without diffusion) changes from one to three in the spatial interval, which is not obtained by a small perturbation of constant coefficients. Moreover, we consider the asymptotic behavior of steady states as the diffusion coefficient tends to infinity. Some examples and numerical simulations are given to illustrate the theoretical results.

On some model problem for the propagation of interacting species in a special environment
Michel Chipot and Mingmin Zhang
2021, 41(7): 3141-3161 doi: 10.3934/dcds.2020401 +[Abstract](404) +[HTML](155) +[PDF](385.92KB)

The purpose of this note is to study the existence of a nontrivial solution for an elliptic system which comes from a newly introduced mathematical problem so called Field-Road model. Specifically, it consists of coupled equations set in domains of different dimensions together with some interaction of non classical type. We consider a truncated problem by imposing Dirichlet boundary conditions and an unbounded setting as well.

Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice
Mia Jukić and Hermen Jan Hupkes
2021, 41(7): 3163-3209 doi: 10.3934/dcds.2020402 +[Abstract](372) +[HTML](140) +[PDF](1017.18KB)

In this paper we consider the discrete Allen-Cahn equation posed on a two-dimensional rectangular lattice. We analyze the large-time behaviour of solutions that start as bounded perturbations to the well-known planar front solution that travels in the horizontal direction. In particular, we construct an asymptotic phase function \begin{document}$ \gamma_j(t) $\end{document} and show that for each vertical coordinate \begin{document}$ j $\end{document} the corresponding horizontal slice of the solution converges to the planar front shifted by \begin{document}$ \gamma_j(t) $\end{document}. We exploit the comparison principle to show that the evolution of these phase variables can be approximated by an appropriate discretization of the mean curvature flow with a direction-dependent drift term. This generalizes the results obtained in [47] for the spatially continuous setting. Finally, we prove that the horizontal planar wave is nonlinearly stable with respect to perturbations that are asymptotically periodic in the vertical direction.

Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms
Raffaele Folino, Ramón G. Plaza and Marta Strani
2021, 41(7): 3211-3240 doi: 10.3934/dcds.2020403 +[Abstract](586) +[HTML](149) +[PDF](821.47KB)

This paper establishes the emergence of slowly moving transition layer solutions for the \begin{document}$ p $\end{document}-Laplacian (nonlinear) evolution equation,

where \begin{document}$ \varepsilon>0 $\end{document} and \begin{document}$ p>1 $\end{document} are constants, driven by the action of a family of double-well potentials of the form

indexed by \begin{document}$ \theta>1 $\end{document}, \begin{document}$ \theta\in \mathbb{R} $\end{document} with minima at two pure phases \begin{document}$ u = \pm1 $\end{document}. The equation is endowed with initial conditions and boundary conditions of Neumann type. It is shown that interface layers, or solutions which initially are equal to \begin{document}$ \pm 1 $\end{document} except at a finite number of thin transitions of width \begin{document}$ \varepsilon $\end{document}, persist for an exponentially long time in the critical case with \begin{document}$ \theta = p $\end{document}, and for an algebraically long time in the supercritical (or degenerate) case with \begin{document}$ \theta>p $\end{document}. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg–Landau type are established. In contrast, in the subcritical case with \begin{document}$ \theta<p $\end{document}, the transition layer solutions are stationary.

Another point of view on Kusuoka's measure
Ugo Bessi
2021, 41(7): 3241-3271 doi: 10.3934/dcds.2020404 +[Abstract](336) +[HTML](144) +[PDF](402.05KB)

Kusuoka's measure on fractals is a Gibbs measure of a very special kind, since its potential is discontinuous while the standard theory of Gibbs measures requires continuous (in its simplest version, Hölder) potentials. In this paper we shall see that for many fractals it is possible to build a class of matrix-valued Gibbs measures completely within the scope of the standard theory; there are naturally some minor modifications, but they are only due to the fact that we are dealing with matrix-valued functions and measures. We shall use these matrix-valued Gibbs measures to build self-similar bilinear forms on fractals. Moreover, we shall see that Kusuoka's measure and bilinear form can be recovered in a simple way from the matrix-valued Gibbs measure.

$ BV $ solution for a non-linear Hamilton-Jacobi system
Ahmad El Hajj, Hassan Ibrahim and Vivian Rizik
2021, 41(7): 3273-3293 doi: 10.3934/dcds.2020405 +[Abstract](401) +[HTML](130) +[PDF](636.34KB)

In this work, we are dealing with a non-linear eikonal system in one dimensional space that describes the evolution of interfaces moving with non-signed strongly coupled velocities. For such kind of systems, previous results on the existence and uniqueness are available for quasi-monotone systems and other special systems in Lipschitz continuous space. It is worth mentioning that our system includes, in particular, the case of non-decreasing solution where some existence and uniqueness results arose for strictly hyperbolic systems with a small total variation. In the present paper, we consider initial data with unnecessarily small \begin{document}$ BV $\end{document} seminorm, and we use some \begin{document}$ BV $\end{document} bounds to prove a global-in-time existence result of this system in the framework of discontinuous viscosity solution.

Complex planar Hamiltonian systems: Linearization and dynamics
Hua Shi, Xiang Zhang and Yuyan Zhang
2021, 41(7): 3295-3317 doi: 10.3934/dcds.2020406 +[Abstract](318) +[HTML](152) +[PDF](367.56KB)

Global dynamics of complex planar Hamiltonian polynomial systems is difficult to be characterized. In this paper, for general complex quadratic Hamiltonian systems of one degree of freedom, we obtain some sufficient conditions on the existence of family of invariant tori. We also complete characterization on locally analytic linearizability of complex planar Hamiltonian systems with homogeneous nonlinearity of degrees either 2 or 3 at a nondegenerate singularity, and present their global dynamics. For these classes of systems we also prove existence of families of invariant tori, together with isochronous periodic orbits.

Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems
Adrian Viorel, Cristian D. Alecsa and Titus O. Pinţa
2021, 41(7): 3319-3341 doi: 10.3934/dcds.2020407 +[Abstract](425) +[HTML](145) +[PDF](446.96KB)

The present work deals with the numerical long-time integration of damped Hamiltonian systems. The method that we analyze combines a specific Strang splitting, that separates linear dissipative effects from conservative ones, with an energy-preserving averaged vector field (AVF) integrator for the Hamiltonian subproblem. This construction faithfully reproduces the energy-dissipation structure of the continuous model, its equilibrium points and its natural Lyapunov function. As a consequence of these structural similarities, both the convergence to equilibrium and, more interestingly, the energy decay rate of the continuous dynamical system are recovered at a discrete level. The possibility of replacing the implicit AVF integrator by an explicit Störmer-Verlet one is also discussed, while numerical experiments illustrate and support the theoretical findings.

Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains
Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan and Alain Miranville
2021, 41(7): 3343-3366 doi: 10.3934/dcds.2020408 +[Abstract](330) +[HTML](133) +[PDF](411.35KB)

This paper is concerned with the tempered pullback dynamics of the 2D Navier-Stokes equations with sublinear time delay operators subject to non-homogeneous boundary conditions in Lipschitz-like domains. By virtue of the estimates of background flow in Lipschitz-like domain and a new retarded Gronwall inequality, we establish the existence of pullback attractors in a general setting involving tempered universes.

A constructive approach to robust chaos using invariant manifolds and expanding cones
Paul A. Glendinning and David J. W. Simpson
2021, 41(7): 3367-3387 doi: 10.3934/dcds.2020409 +[Abstract](310) +[HTML](146) +[PDF](1228.98KB)

Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously in some parameter regimes. Here we provide formal results for robust chaos in the original parameter regime of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049-3052, 1998]. We first construct a trapping region in phase space to prove the existence of a topological attractor. We then construct an invariant expanding cone in tangent space to prove that tangent vectors expand and so no invariant set can have only negative Lyapunov exponents. Under additional assumptions we characterise an attractor as the closure of the unstable manifold of a fixed point and prove that it satisfies Devaney's definition of chaos.

Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance
Mario Bukal
2021, 41(7): 3389-3414 doi: 10.3934/dcds.2021001 +[Abstract](289) +[HTML](117) +[PDF](460.97KB)

In this paper we construct a unique global in time weak nonnegative solution to the corrected Derrida-Lebowitz-Speer-Spohn equation, which statistically describes the interface fluctuations between two phases in a certain spin system. The construction of the weak solution is based on the dissipation of a Lyapunov functional which equals to the square of the Hellinger distance between the solution and the constant steady state. Furthermore, it is shown that the weak solution converges at an exponential rate to the constant steady state in the Hellinger distance and thus also in the \begin{document}$ L^1 $\end{document}-norm. Numerical scheme which preserves the variational structure of the equation is devised and its convergence in terms of a discrete Hellinger distance is demonstrated.

A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains
Beom-Seok Han, Kyeong-Hun Kim and Daehan Park
2021, 41(7): 3415-3445 doi: 10.3934/dcds.2021002 +[Abstract](257) +[HTML](116) +[PDF](462.25KB)

We introduce a weighted \begin{document}$ L_{p} $\end{document}-theory (\begin{document}$ p>1 $\end{document}) for the time-fractional diffusion-wave equation of the type

where \begin{document}$ \alpha\in (0,2) $\end{document}, \begin{document}$ \partial_{t}^{\alpha} $\end{document} denotes the Caputo fractional derivative of order \begin{document}$ \alpha $\end{document}, and \begin{document}$ \Omega $\end{document} is a \begin{document}$ C^1 $\end{document} domain in \begin{document}$ \mathbb{R}^d $\end{document}. We prove existence and uniqueness results in Sobolev spaces with weights which allow derivatives of solutions to blow up near the boundary. The order of derivatives of solutions can be any real number, and in particular it can be fractional or negative.

Constant-speed ramps for a central force field
Rafael López and Óscar Perdomo
2021, 41(7): 3447-3464 doi: 10.3934/dcds.2021003 +[Abstract](227) +[HTML](103) +[PDF](561.02KB)

We investigate the problem of determining the planar curves that describe ramps where a particle of mass \begin{document}$ m $\end{document} moves with constant-speed when is subject to the action of the friction force and a force whose magnitude \begin{document}$ F(r) $\end{document} depends only on the distance \begin{document}$ r $\end{document} from the origin. In this paper we describe all the constant-speed ramps for the case \begin{document}$ F(r) = -m/r $\end{document}. We show the circles and the logarithmic spirals play an important role. Not only they are solutions but every other solution approaches either a circle or a logarithmic spiral.

Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators
Françoise Demengel
2021, 41(7): 3465-3488 doi: 10.3934/dcds.2021004 +[Abstract](231) +[HTML](109) +[PDF](439.22KB)

We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when at least one of the components of the gradient vanishes. We extend here the results in [16], [10], [24].

On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow
Ling-Bing He and Li Xu
2021, 41(7): 3489-3530 doi: 10.3934/dcds.2021005 +[Abstract](266) +[HTML](114) +[PDF](574.21KB)

The present work aims at the mathematical derivation of the equations for the isentropic flow from those for the non-isentropic flow for perfect gases in the whole space. Suppose that the following things hold for the entropy equation: (1). both conduction of heat and its generation by dissipation of mechanical energy are sufficiently weak(with the order of \begin{document}$ \varepsilon $\end{document}); (2). initially the entropy \begin{document}$ S^{N}_ \varepsilon $\end{document} is around a constant \begin{document}$ c_S $\end{document}, that is, \begin{document}$ S^{N}_ \varepsilon |_{t = 0} = c_S+O( \varepsilon ) $\end{document}. Then the non-isentropic compressible Navier-Stokes equations admit a unique and global solution \begin{document}$ ( \rho^{N}_ \varepsilon , \, \boldsymbol{u}^{N}_ \varepsilon , \, S^{N}_ \varepsilon ) $\end{document} with the initial data \begin{document}$ ( \rho_0, \, \boldsymbol{u}_0, \, c_S+ \varepsilon S_0) $\end{document}, which is a perturbation of the equilibrium \begin{document}$ (1, \boldsymbol{0}, c_S) $\end{document}. Moreover, \begin{document}$ ( \rho^{N}_ \varepsilon , u^{N}_ \varepsilon ) $\end{document} can be approximated by \begin{document}$ ( \rho^{I}, \, u^{I}) $\end{document}, the solution to the associated isentropic compressible Navier-Stokes equations equipped with the initial data \begin{document}$ ( \rho_0, \, \boldsymbol{u}_0) $\end{document}, in the sense that

which holds globally in the so-called critical Besov spaces for the compressible Navier-Stokes equations.

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