Emergent behaviors of discrete Lohe aggregation flows

<p style='text-indent:20px;'>The Lohe sphere model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group, respectively. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the continuous Lohe type aggregation models and study their emergent behaviors using the Lyapunov function method. For suitable discretization of the Lohe sphere model, we employ a scheme consisting of two steps. In the first step, we solve the first-order forward Euler scheme, and in the second step, we project the intermediate state onto the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical frameworks leading to complete state aggregation and asymptotic state-locking.</p>

On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation

<p style='text-indent:20px;'>We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data <inline-formula><tex-math id="M1">\begin{document}$ u_{01} \in L^2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ u_{02} \in H^{-1 + \eta} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M3">\begin{document}$ \eta &gt; 0 $\end{document}</tex-math></inline-formula>.</p>

Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

<p style='text-indent:20px;'>In this article, we are concerned with statistical solutions for the nonautonomous coupled Schrödinger-Boussinesq equations on infinite lattices. Firstly, we verify the existence of a pullback-<inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor and establish the existence of a unique family of invariant Borel probability measures carried by the pullback-<inline-formula><tex-math id="M2">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor for this lattice system. Then, it will be shown that the family of invariant Borel probability measures is a statistical solution and satisfies a Liouville type theorem. Finally, we illustrate that the invariant property of the statistical solution is indeed a particular case of the Liouville type theorem.</p>

Multi-valued random dynamics of stochastic wave equations with infinite delays

<p style='text-indent:20px;'>This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays and additive white noise. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor.</p>

Boundedness in a two species attraction-repulsion chemotaxis system with two chemicals

<p style='text-indent:20px;'>This paper deals with a class of attraction-repulsion chemotaxis systems in a smoothly bounded domain. When the system is parabolic-elliptic-parabolic-elliptic and the domain is <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional, if the repulsion effect is strong enough then the solutions of the system are globally bounded. Meanwhile, when the system is fully parabolic and the domain is either one-dimensional or two-dimensional, the system also possesses a globally bounded classical solution.</p>

Exponential ergodicity for regime-switching diffusion processes in total variation norm

<p style='text-indent:20px;'>We investigate the long time behavior for a class of regime-switching diffusion processes. Based on direct evaluation of moments and exponential functionals of hitting time of the underlying process, we adopt coupling method to obtain existence and uniqueness of the invariant probability measure and establish explicit exponential bounds for the rate of convergence to the invariant probability measure in total variation norm. In addition, we provide some concrete examples to illustrate our main results which reveal impact of random switching on stochastic stability and convergence rate of the system.</p>

On weak (measure-valued)-strong uniqueness for compressible MHD system with non-monotone pressure law

<p style='text-indent:20px;'>In this paper, we define a renormalized dissipative measure-valued (rDMV) solution of the compressible magnetohydrodynamics (MHD) equations with non-monotone pressure law. We prove the existence of the rDMV solutions and establish a suitable relative energy inequality. And we obtain the weak (measure-valued)-strong uniqueness property of this rDMV solution with the help of the relative energy inequality.</p>

Asymptotic behavior of supercritical wave equations driven by colored noise on unbounded domains

<p style='text-indent:20px;'>This paper deals with the asymptotic behavior of the non-autonomous random dynamical systems generated by the wave equations with supercritical nonlinearity driven by colored noise defined on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ n\le 6 $\end{document}</tex-math></inline-formula>. Based on the uniform Strichartz estimates, we prove the well-posedness of the equation in the natural energy space and define a continuous cocycle associated with the solution operator. We also establish the existence and uniqueness of tempered random attractors of the equation by showing the uniform smallness of the tails of the solutions outside a bounded domain in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.</p>

Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4

<p style='text-indent:20px;'>We characterize the meromorphic Liouville integrability of the Hamiltonian systems with Hamiltonian <inline-formula><tex-math id="M2">\begin{document}$ H = \left(p_1^2+p_2^2\right)/2+1/P(q_1, q_2) $\end{document}</tex-math></inline-formula>, being <inline-formula><tex-math id="M3">\begin{document}$ P(q_1, q_2) $\end{document}</tex-math></inline-formula> a homogeneous polynomial of degree <inline-formula><tex-math id="M4">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> of one of the following forms <inline-formula><tex-math id="M5">\begin{document}$ \pm q_1^4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ 4q_1^3q_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \pm 6q_1^2q_2^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \pm \left(q_1^2+q_2^2\right)^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ \pm q_2^2\left(6q_1^2-q_2^2\right) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \pm q_2^2\left(6q_1^2+q_2^2\right) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ q_1^4+6\mu q_1^2q_2^2-q_2^4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ -q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M13">\begin{document}$ \mu&gt;-1/3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ \mu\neq 1/3 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M15">\begin{document}$ q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M16">\begin{document}$ \mu \neq \pm 1/3 $\end{document}</tex-math></inline-formula>. We note that any homogeneous polynomial of degree <inline-formula><tex-math id="M17">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> after a linear change of variables and a rescaling can be written as one of the previous polynomials. We remark that for the polynomial <inline-formula><tex-math id="M18">\begin{document}$ q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M19">\begin{document}$ \mu\in\left\{-5/3, -2/3\right\} $\end{document}</tex-math></inline-formula> we only can prove that it has no a polynomial first integral.</p>