Discrete Contin Research Articles

Morse inequalities at infinity for a resonant mean field equation

In this paper, we study the following mean field type equation: [Formula: see text] where [Formula: see text] is a closed oriented surface of unit volume [Formula: see text] = 1, [Formula: see text] positive smooth function and [Formula: see text], [Formula: see text]. Building on the critical points at infinity approach initiated in [M. Ahmedou, M. Ben Ayed and M. Lucia, On a resonant mean field type equation: A “critical point at infinity” approach, Discrete Contin. Dyn. Syst. 37(4) (2017) 1789–1818] we develop, under generic condition on the function K and the metric g, a full Morse theory by proving Morse inequalities relating the Morse indices of the critical points, the indices of the critical points at infinity, and the Betti numbers of the space of formal barycenters [Formula: see text]. We derive from these Morse inequalities at infinity various new existence as well as multiplicity results of the mean field equation in the resonant case, i.e. [Formula: see text].

Generalized Analytic Integrability of a Class of Polynomial Differential Systems in $\mathbb{C}^{2}$

This paper study the type of integrability of differential systems with separable variables \(\dot{x}=h\left (x\right )f\left (y\right )\), \(\dot{y}= g\left (y\right )\), where \(h\), \(f\) and \(g\) are polynomials. We provide a criterion for the existence of generalized analytic first integrals of such differential systems. Moreover we characterize the polynomial integrability of all such systems.In the particular case \(h\left (x\right )=\left (ax+b\right )^{m}\) we provide necessary and sufficient conditions in order that this subclass of systems has a generalized analytic first integral. These results extend known results from Giné et al. (Discrete Contin. Dyn. Syst. 33:4531–4547, 2013) and Llibre and Valls (Discrete Contin. Dyn. Syst., Ser. B 20:2657–2661, 2015). Such differential systems of separable variables are important due to the fact that after a blow-up change of variables any planar quasi-homogeneous polynomial differential system can be transformed into a special differential system of separable variables \(\dot{x}=xf\left (y\right )\), \(\dot{y}=g\left (y\right )\), with \(f\) and \(g\) polynomials.

Continuous-time stochastic mutual fund management game between active and passive funds

This paper investigates a Stackelberg game between a mutual fund manager and an investor. Throughout the paper, we follow the literature (see, e.g. Li and Sethi. A review of dynamic stackelberg game models. Discrete Contin. Dyn. Syst. B, 2016, 22(1), 125) on Stackelberg games, i.e. leader-follower games, and call the leader and follower, respectively, her and him. In our model, the leader (she) refers to the mutual fund manager, while the follower (he) is the investor. The individual investor (he), manages an active mutual fund and can only allocate his wealth among a risk-free asset, the active mutual fund, and a passive index fund. The passive index fund is composed of an exogenously given portfolio of all stocks in the market. The investor aims to maximize the expected constant relative risk aversion utility of his terminal wealth, while the mutual fund manager's objective is to maximize the expected value of the accumulative discounted management fees from the investor. Assume that the mutual fund manager faces portfolio constraints, which reflects the investment objective and style of the fund. We first focus on a special case in which the mutual fund is a sector fund, investing only in a small subset of available stocks in the market, namely a specific sector or industry. Then, this special case is extended to the general case in which the portfolio constraint is given by a nonempty, closed convex set. By applying the dynamic programming principle approach, we solve two associated Hamilton-Jacobi-Bellman (HJB) equations and obtain Stackelberg equilibrium strategies for both the mutual fund manager and the investor in closed-form expressions. Finally, in the special case we provide numerical examples to analyze the effects of some parameters on the equilibrium strategies.

Relaxation Oscillations in Predator–Prey Systems

We characterize a criterion for the existence of relaxation oscillations in planar systems of the form $$\begin{aligned} \frac{du}{dt}= u^{k+1} g(u,v,\varepsilon ), \qquad \frac{dv}{dt}=\varepsilon f(u,v,\varepsilon ) + u^{k+1} h(u,v,\varepsilon ), \end{aligned}$$ where $$k\ge 0$$ is an arbitrary constant and $$\varepsilon >0$$ is a sufficiently small parameter. Taking into account of possible degeneracy of the “discriminant” function occurred when $$k\ge 0$$ , this criterion generalizes and strengthens those for the case $$k=0$$ obtained by Hsu (SIAM J Appl Dyn Syst 18:33–67, 2019) and Hsu and Wolkowicz (Discrete Contin Dyn Syst Ser B 25:1257–1277, 2020). Differing from the case of $$k=0$$ , our proof of the criterion is based on the construction of an invariant, thin annular region in an arbitrarily prescribed small neighborhood of a singular closed orbit and the establishment of an asymptotic formula for solutions near the v-axis. As applications of this criterion, we will give concrete conditions ensuring the existence of relaxation oscillations in general predator–prey systems, as well as spatially homogeneous relaxation oscillations and relaxed periodic traveling waves in a class of diffusive predator–prey systems.

Hölder Continuous Regularity of Stochastic Convolutions with Distributed Delay

In this work, we consider the Hölder continuous regularity of stochastic convolutions for a class of linear stochastic retarded functional differential equations with distributed delay in Hilbert spaces. By focusing on distributed delays, we first establish some more delicate estimates for fundamental solutions than those given in Liu (Discrete Contin. Dyn. Syst. Ser. B 25(4), 1279–1298, 2020). Then we apply these estimates to stochastic convolutions incurred by distributed delay to study their regularity property. Last, we present some easily-verified results by considering the regularity of a class of systems whose delay operators have the same order derivatives as those in instantaneous ones.

Consistency, convergence and error estimates for a mixed finite element–finite volume scheme to compressible Navier–Stokes equations with general inflow/outflow boundary data

Abstract We study convergence of a mixed finite element-finite volume scheme for the compressible Navier–Stokes equations in the isentropic regime under the full range $1<\gamma <\infty $ of the adiabatic coefficients $\gamma $ for the problem with general nonzero inflow–outflow boundary conditions. We propose a modification of Karper’s scheme (2013, A convergent FEM-DG method for the compressible Navier–Stokes equations. Numer. Math., 125, 441–510) in order to accommodate the nonzero boundary data, prove existence of its solutions, establish the stability and uniform estimates, derive a convenient consistency formulation of the balance laws and use it to show the weak convergence of the numerical solutions to a dissipative solution with the Reynolds defect introduced in Abbatiello et al. (2021, Generalized solutions to models of compressible viscous fluids. Discrete Contin. Dyn. Syst., 41, 1--28). If the target system admits a strong solution then the convergence is strong towards the strong solution. Moreover, we establish the convergence rate of the strong convergence in terms of the size of the space discretization $h$ (which is supposed to be comparable with the time step $\varDelta t$). In the case of the nonzero inflow–outflow boundary data all results are new. The latter result is new also for the no-slip boundary conditions and adiabatic coefficients $\gamma $ less than the threshold $3/2$.

A regularity criterion for a 2D tropical climate model with fractional dissipation

Tropical climate model derived by Frierson et al. (Commun Math Sci 2:591–626, 2004) and its modified versions have been investigated in a number of papers [see, e.g., Li and Titi (Discrete Contin Dyn Syst Series A 36(8):4495–4516, 2016), Wan (J Math Phys 57(2):021507, 2016), Ye (J Math Anal Appl 446:307–321, 2017) and more recently Dong et al. (Discrete Contin Dyn Syst Ser B 24(1):211–229, 2019)]. Here, we deal with the 2D tropical climate model with fractional dissipative terms in the equation of the barotropic mode u and in the equation of the first baroclinic mode v of the velocity, but without diffusion in the temperature equation, and we establish a regularity criterion for this system.

The cyclicity of the period annulus of a reversible quadratic system

We prove that perturbing the periodic annulus of the reversible quadratic polynomial differential system $\dot x=y+ax^2$, $\dot y=-x$ with a ≠ 0 inside the class of all quadratic polynomial differential systems we can obtain at most two limit cycles, including their multiplicities. Since the first integral of the unperturbed system contains an exponential function, the traditional methods cannot be applied, except in Figuerasa, Tucker and Villadelprat (2013, J. Diff. Equ., 254, 3647–3663) a computer-assisted method was used. In this paper, we provide a method for studying the problem. This is also the first purely mathematical proof of the conjecture formulated by Dumortier and Roussarie (2009, Discrete Contin. Dyn. Syst., 2, 723–781) for q ⩽ 2. The method may be used in other problems.

Long time behavior of solutions to the damped forced generalized Ostrovsky equation below the energy space

In this paper, we investigate the long time behavior of the damped forced generalized Ostrovsky equation below the energy space. First, by using Fourier restriction norm method and Tao’s [ k , Z ] [k,Z] - multiplier method, we establish the multi-linear estimates, including the bilinear and trilinear estimates on the Bourgain space X s , b . X_{s,b}. Then, combining the multi-linear estimates with the contraction mapping principle as well as L ~ 2 \widetilde {L}^{2} energy method, we establish the global well-posedness and existence of the bounded absorbing sets in L ~ 2 . \widetilde {L}^{2}. Finally, we show the existence of global attractor in L ~ 2 \widetilde {L}^{2} and its compactness in H ~ 5 \widetilde {H}^{5} by means of the high-low frequency decomposition method, cut-off function, tail estimate together with Kuratowski α \alpha -measure in order to overcome the non-compactness of the classical Sobolev embedding. This result improves earlier ones in the literatures, such as Goubet and Rosa [J. Differential Equations 185 (2002), no. 1, 25–53], Moise and Rosa [Adv. Differential Equations 2 (1997), no. 2, 251–296], Wang et al. [J. Math. Anal. Appl. 390 (2012), no. 1, 136–150], Wang [Discrete Contin. Dyn. Syst. 35 (2015), no. 8, 3799–3825], and Guo and Huo [J. Math. Anal. App. 329 (2007), no. 1, 392–407].

Regularity and stability of finite energy weak solutions for the Camassa–Holm equations with nonlocal viscosity

We consider the n-dimensional ( $$n=2,3$$ ) Camassa–Holm equations with nonlocal diffusion of type $$(-\,\Delta )^{s}, \ \frac{n}{4}\le s<1$$ . In Gan et al. (Discrete Contin Dyn Syst 40(6):3427–3450, 2020), the global-in-time existence and uniqueness of finite energy weak solutions is established. In this paper, we show that with regular initial data, the finite energy weak solutions are indeed regular for all time. Moreover, the weak solutions are stable with respect to the initial data. The main difficulty lies in establishing higher order uniform estimates with the presence of the fractional Laplacian diffusion. To achieve this, we need to explore suitable fractional Sobolev type inequalities and bilinear estimates for fractional derivatives. The critical case $$s=\frac{n}{4}$$ contains extra difficulties and a smallness assumption on the initial data is imposed.

Standing waves with prescribed mass for the coupled Hartree–Fock system with partial confinement

In this paper, we focus on the standing waves with prescribed mass for the coupled Hartree–Fock system, which is the basic quantum chemistry model of small number of electrons interacting with static nuclei. This leads to study the normalized solutions to the following nonlocal elliptic system $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+(x_{1}^{2}+x_{2}^{2}+\cdot \cdot \cdot +x_{N-1}^{2})u=\lambda _{1}u+\mu _{1}\phi _{u}|u|^{p-2}u+\beta \phi _{v}|u|^{p-2}u, &{}\quad x\in \mathbb {R}^N,\\ \displaystyle -\Delta v+(x_{1}^{2}+x_{2}^{2}+\cdot \cdot \cdot +x_{N-1}^{2})v=\lambda _{2}v+\mu _{2}\phi _{v}|v|^{p-2}v+\beta \phi _{u}|v|^{p-2}v, &{}\quad x\in \mathbb {R}^N, \end{array}\right. } \end{aligned}$$ where $$N\ge 3$$ , $$\mu _{i}>0$$ (i=1,2), $$\beta >0$$ and the frequencies $$\lambda _{1}$$ , $$\lambda _{2}$$ are unknown and appear as Lagrange multipliers. For this purpose, we firstly give the existence of normalized solutions to the related single equation, which extends the main results in Bellazzini et al. (Commun Math Phys 1:229–251, 2017) to the nonlocal case. Then, combining constrained minimization method and coupled rearrangement technique, we prove the existence of normalized solutions to the above nonlocal elliptic system together with some qualitative properties. In addition, the stability of the corresponding standing waves for the related time-dependent coupled Hartree–Fock system is discussed. These results can be viewed as an extension of the main results in Wang et al. (Discrete Contin Dyn Syst 56, 2019) from mass-subcritical case $$\frac{N+\alpha }{N}<p<\frac{N+\alpha +2}{N}$$ , to mass-supercritical setting $$\frac{N+\alpha +2}{N}<p<\min \{4,\frac{N+\alpha }{N-2}\}$$ .

Strichartz estimates and Strauss conjecture on non-trapping asymptotically hyperbolic manifolds

We prove global-in-time Strichartz estimates for the shifted wave equations on non-trapping asymptotically hyperbolic manifolds. The key tools are the spectral measure estimates from [Ann. Inst. Fourier, Grenoble 68 (2018), pp. 1011–1075] and arguments borrowed from [Analysis PDE 9 (2016), pp. 151–192], [Adv. Math. 271 (2015), pp. 91–111]. As an application, we prove the small data global existence for any power p ∈ ( 1 , 1 + 4 n − 1 ) p\in (1, 1+\frac {4}{n-1}) for the shifted wave equation in this setting, involving nonlinearities of the form ± | u | p \pm |u|^p or ± | u | p − 1 u \pm |u|^{p-1}u , which answers partially an open question raised in [Discrete Contin. Dyn. Syst. 39 (2019), pp. 7081–7099].

Global regularity to the 2D non-isothermal inhomogeneous nematic liquid crystal flows

ABSTRACT In this paper, we prove the global strong solutions for the Cauchy problem of two-dimensional (2D) incompressible non-isothermal nematic liquid crystal flows, if the initial orientation satisfies a geometric condition. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states. When d is a constant vector and , we also extend the corresponding result in Wang Y. [Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier–Stokes flows with vacuum. Discrete Contin Dyn Syst B. doi:10.3934/dcdsb.2020099.] to the whole space , where the global strong solution of 2D inhomogeneous incompressible heat conducting Navier–Stokes flows is established on bounded domain.

The $${\mathcal {F}}$$-transitivity and recurrence of translation semigroups on complex sectors

In this paper, we study the $${\mathcal {F}}$$ -transitive behaviour of the translation semigroups on complex sectors, where $${\mathcal {F}}$$ is a Furstenberg family of the semigroup. We present the characterizations for $${\mathcal {F}}$$ -transitive translation semigroups on complex sectors, which implies that the $${\mathcal {F}}$$ -transitivity equals the thickly $${\mathcal {F}}$$ -transitivity for the translation semigroups on complex sectors. The relationship between the recurrence and the $${\mathcal {F}}$$ -transitivity for the translation semigroup is illustrated. The obtained characterizations generalize the recent results of Conejero and Peris (Discrete Contin Dyn Syst 25(4):1195–1208, 2009).

The hierarchy of multi-point probability density functions and characteristic functions in compressible turbulence

The hierarchies of equations for a general multi-point probability density function (PDF) and its characteristic function (CF) are derived for compressible turbulent flows, obeying the ideal gas law. The closure problem of turbulence is clearly exhibited in each of the approaches, with n-point statistics being dependent on the (n + 1)-point statistics and, for some cases, even the (n + 2)-point statistics. When dynamic viscosity and heat conductivity are dependent on temperature as a power-law, the CF hierarchy could contain fractional derivatives if the exponent is a non-integer. The additional conditions satisfied by all the PDFs and CFs in both the hierarchies are also prescribed. The PDF and CF equations derived in this paper, with the unclosed terms explicitly written in terms of higher order PDF/CF, act as a starting point in constructing symmetry-based invariant solutions of compressible turbulence, analogous to the works of Wacławczyk et al. [“Statistical symmetries of the Lundgren–Monin–Novikov hierarchy,” Phys. Rev. E 90, 013022 (2014)] and Oberlack and Rosteck [“New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws,” Discrete Contin. Dyn. Syst. 3, 451–471 (2010)] for incompressible turbulence.

Brualdi-type inclusion sets of Z-eigenvalues and lk,s-singular values for tensors

We give a Brualdi-type Z-eigenvalue inclusion set of tensors, and prove that it is tighter than the inclusion set given by G. Wang, G. L. Zhou, and L. Caccetta [Discrete Contin. Dyn. Syst. Ser. B, 2017, 22: 187–198] in a special case. We also give an inclusion set for lk,s-singular values of rectangular tensors.

Pointwise persistence and shadowing

We show that every pointwise persistent homeomorphism with the shadowing property is persistent. The proof relies on a new tracing property, the persistent shadowing property, which has its own interest. We shown for instance that expansive homeomorphisms with this property are s-topologically stable in the sense of Kawaguchi (Discrete Contin Dyn Syst 39(5):2743–2761, 2019). We also present orbital versions of our results.

Global Existence and Exponential Decay of Strong Solutions to the 2D Density-dependent Nematic Liquid Crystal Flows with Vacuum

This paper deals with the 2D incompressible nematic liquid crystal flows with density-dependent viscosity in bounded domain. The global well-posedness of strong solutions are established in the vacuum cases, provided the assumption that $\overline{\rho} + \|\nabla d_0\|_{L^2}$ is suitably small with large velocity, which extends the recent work [Discrete Contin. Dyn. Syst. 37 (2017), 4907--4922] and [Methods Appl. Anal. 22 (2015), 201--220] to the case of variable viscosity. Furthermore, the exponential decay of the solution is also obtained.

Lyapunov exponents for expansive homeomorphisms

AbstractWe address the problem of defining Lyapunov exponents for an expansive homeomorphism f on a compact metric space (X, dist) using similar techniques as those developed in Barreira and Silva [Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dynam. Sys.13 (2005), 469–490]; Kifer [Characteristic exponents of dynamical systems in metric spaces, Ergod. Th. Dynam. Sys.3 (1983), 119–127]. Under certain conditions on the topology of the space X where f acts we obtain that there is a metric D defining the topology of X such that the Lyapunov exponents of f are different from zero with respect to D for every point x ∈ X. We give an example showing that this may not be true with respect to the original metric dist. But expansiveness of f ensures that Lyapunov exponents do not vanish on a Gδ subset of X with respect to any metric defining the topology of X. We define Lyapunov exponents on compact invariant sets of Peano spaces and prove that if the maximal exponent on the compact set is negative then the compact is an attractor.

Corrigendum to "The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension" [Discrete Contin. Dyn. Syst., 36 (2016), 5743–5761

We correct an error which has occurred in the proof of Lemma 4.1 in the paper The lifespan of small solutions to cubic derivative nonlinear Schrodinger equations in one space dimension [Discrete Contin. Dyn. Syst., 36 (2016), 5743-5761].