From the Boltzmann Equation to an Incompressible Navier–Stokes–Fourier System

We establish a Navier–Stokes–Fourier limit for solutions of the Boltzmann equation considered over any periodic spatial domain of dimension two or more. We do this for a broad class of collision kernels that relaxes the Grad small deflection cutoff condition for hard potentials and includes for the first time the case of soft potentials. Appropriately scaled families of DiPerna–Lions renormalized solutions are shown to have fluctuations that are compact. Every limit point is governed by a weak solution of a Navier–Stokes–Fourier system for all time.     

Time-Periodic Solutions to the Full Navier–Stokes–Fourier System

We consider the full Navier–Stokes–Fourier system describing the motion of a compressible viscous and heat conducting fluid driven by a time-periodic external force. We show the existence of at least one weak time periodic solution to the problem under the basic hypothesis that the system is allowed to dissipate the thermal energy through the boundary. Such a condition is in fact necessary, as energetically closed fluid systems do not possess non-trivial (changing in time) periodic solutions as a direct consequence of the Second law of thermodynamics.     

Homogenization of scalar wave equations with hysteresis

The paper deals with a scalar wave equation of the form where is a Prandtl–Ishlinskii operator and are given functions. This equation describes longitudinal vibrations of an elastoplastic rod. The mass density and the Prandtl–Ishlinskii distribution function are allowed to depend on the space variable x. We prove existence, uniqueness and regularity of solution to a corresponding initial-boundary value problem. The system is then homogenized by considering a sequence of equations of the above type with spatially periodic data and , where the spatial period tends to 0. We identify the homogenized limits and and prove the convergence of solutions to the solution of the homogenized equation. Received June 17, 1999     

On the permanence of periodic predator–prey systems with stage structure and pulse action

We establish permanence conditions for a periodic predator–prey system with stage structure, pulse action, and Beddington–DeAngelis functional response.     

On Periodic Solutions of a System of Nonlinear Functional Partial Differential Equations

We obtain sufficient conditions for the existence of periodic solutions of a system of nonlinear functional partial differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 154–158, April–June, 2005.     

Bifurcations of rotating structures in a parabolic functional differential equation

We investigate the Andronov-Hopf bifurcation of the birth of a periodic solution from a space-homogeneous stationary solution of the Neumann problem on a disk for a parabolic equation with a transformation of space variables in the case where this transformation is the composition of a rotation by a constant angle and a radial contraction. Under general assumptions, we prove a theorem on the existence of a rotating structure, deduce conditions for its orbital stability, and construct its asymptotic form. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 155–169, April–June, 2006.     

The effect of pulsed harvesting policy on the inshore–offshore fishery model with the impulsive diffusion

In this paper, we propose the inshore–offshore fishing model with impulsive diffusion and pulsed harvesting at the different fixed time. The existence and globally asymptotical stability of both the trivial periodic solution and the positive periodic solution are obtained. We show that the pulsed harvesting has a strong impact on the persistence of the fish population. By the numerical simulation, we obtain that the best time of fishing is at the end of the period τ.     

Periodic Step-Like Contrast Structures for a Quasilinear Parabolic Equation with Small Diffusion Coefficient

Using the method of boundary functions, for a quasilinear parabolic equation with small diffusion coefficient we construct an asymptotic expansion of a periodic solution with internal transition layer. Sufficient conditions for the existence of this solution are obtained. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 3, pp. 329–350, July–September, 2005.     

New approach to the investigation of the existence of periodic solutions of systems of differential equations and their construction

Methods developed for the solution of general equations with restrictions are applied to the construction of periodic solutions of systems of differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 55–70, January–March, 2007.     

Large Eddy Simulations Using the Subgrid-Scale Estimation Model and Truncated Navier–Stokes Dynamics

We describe a procedure for large eddy simulations of turbulence which uses the subgrid-scale estimation model and truncated Navier–Stokes dynamics. In the procedure the large eddy simulation equations are advanced in time with the subgrid-scale stress tensor calculated from the parallel solution of the truncated Navier–Stokes equations on a mesh two times smaller in each Cartesian direction than the mesh employed for a discretization of the resolved quantities. The truncated Navier–Stokes equations are solved through a sequence of runs, each initialized using the subgrid-scale estimation model. The modeling procedure is evaluated by comparing results of large eddy simulations for isotropic turbulence and turbulent channel flow with the corresponding results of experiments, theory, direct numerical simulations, and other large eddy simulations. Subsequently, simplifications of the general procedure are discussed and evaluated. In particular, it is possible to formulate the procedure entirely in terms of the truncated Navier–Stokes equation and a periodic processing of the small-scale component of its solution. Received 27 April 2001 and accepted 16 December 2001     

Finite Dimensional Global Attractor for 3D MHD-α Models: A Comparison

We consider two magnetohydrodynamic-α (MHDα) models with kinematic viscosity and magnetic diffusivity for an incompressible fluid in a three-dimensional periodic box (torus). More precisely, we consider the Navier–Stokes-α-MHD and the Modified Leray-α-MHD models. Similar models are useful to study the turbulent behavior of fluids in presence of a magnetic field because of the current impossibility to handle non-regularized systems neither analytically nor via numerical simulations. In both cases, the global existence of the solution and of a global attractor can be shown. We provide an upper bound for the Hausdorff and the fractal dimension of the attractor. This bound can be interpreted in terms of degrees of freedom of the long-time dynamics of the involved system and gives information about the numerical stability of the model. We get the same bound that holds for the Simplified Bardina-MHD model, considered in a previous paper (this result provides, in some sense, an intermediate bound between the number of degrees of freedom for the Simplified Bardina model and the Navier–Stokes-α equation in the nonmagnetic case). However, the Navier–Stokes-α-MHD system is preferable since, in the ideal case, it conserves more quadratic invariants derived from the standard MHD model.     

Well-Posedness of the Generalized Proudman–Johnson Equation Without Viscosity

The generalized Proudman–Johnson equation, which was derived from the Navier–Stokes equations by Jinghui Zhu and the author, are considered in the case where the viscosity is neglected and the periodic boundary condition is imposed. The equation possesses two nonlinear terms: the convection and stretching terms. We prove that the solution exists globally in time if the stretching term is weak in the sense to be specified below. We also discuss on blow-up solutions when the stretching term is strong. Partly supported by the Grant-in-Aid for Scientific Research from JSPS No. 14204007.     

Rotationally symmetric spontaneous swirling in MHD flows

The stability of steady axisymmetricMHD flows of an inviscid, incompressible, perfectly conducting fluid with respect to swirling—perturbations of the azimuthal components of the velocity field—is studied in a linear approximation. It is shown that for flows similar to a magnetohydrodynamic Hill-Shafranov vortex, the problem reduces to a one-dimensional problem on a closed streamline of the unperturbed flow (the arc length of the streamline is the spatial coordinate). A spectral boundary-value eigenvalue problem is formulated for a system of two ordinary differential equations with periodic coefficients and periodic boundary conditions. Sufficient conditions under which swirling is impossible are obtained. Numerical solution of the characteristic equation shows that, under certain conditions, for each streamline there is a real eigenvalue that yields monotonic exponential growth of the initial perturbations. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 120–129, September–October, 2000.     

Conditions for the existence of a solution of a Noether boundary-value problem for a second-order system

We consider a weakly nonlinear boundary-value problem for a system of second-order ordinary differential equations. We find a sufficient condition for the existence of at least one solution of this problem and propose a convergent iterative algorithm for the determination of its solution. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 368–375, July–September, 2006.     

Hopf Bifurcation of the Generalized Lorenz Canonical Form

Hopf bifurcation of a unified chaotic system – the generalized Lorenz canonical form (GLCF) – is investigated. Based on rigorous mathematical analysis and symbolic computations, some conditions for stability and direction of the periodic obits from the Hopf bifurcation are derived.     

Vlasov–Maxwell–Boltzmann Diffusive Limit

Inspired by the work (Bastea et al. in J Stat Phys 1011087–1136, 2000) for binary fluids, we study the diffusive expansion for solutions around Maxwellian equilibrium and in a periodic box to the Vlasov–Maxwell–Boltzmann system, the most fundamental model for an ensemble of charged particles. Such an expansion yields a set of dissipative new macroscopic PDEs, the incompressible Vlasov–Navier–Stokes–Fourier system and its higher order corrections for describing a charged fluid, where the self-consistent electromagnetic field is present. The uniform estimate on the remainders is established via a unified nonlinear energy method and it guarantees the global in time validity of such an expansion up to any order.     

Approximation of periodic solutions of linear differential-difference equations of neutral type by spline-collocation method

We construct a method for the approximation of periodic solutions of linear differential-difference equations of the neutral type by using cubic splines and investigate conditions for the convergence of the proposed approximation scheme. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 147–154, April–June, 2006.     

Infinite-Dimensional Hyperbolic Sets and Spatio-Temporal Chaos in Reaction Diffusion Systems in $${\mathbb{R}^{n}}$$

The paper is devoted to a rigorous construction of a parabolic system of partial differential equations which displays space–time chaotic behavior in its global attractor. The construction starts from a periodic array of identical copies of a temporally chaotic reaction-diffusion system (RDS) on a bounded domain with Dirichlet boundary conditions. We start with the case without coupling where space–time chaos, defined via embedding of multi- dimensional Bernoulli schemes, is easily obtained. We introduce small coupling by replacing the Dirichlet boundary conditions by strong absorption between the active islands. Using hyperbolicity and delicate PDE estimates we prove persistence of the embedded Bernoulli scheme. Furthermore we smoothen the nonlinearity and obtain a RDS which has polynomial interaction terms with space and time-periodic coefficients and which has a hyperbolic invariant set on which the dynamics displays spatio-temporal chaos. Finally we show that such a system can be embedded in a bigger system which is autonomous and homogeneous and still contains space–time chaos. Obviously, hyperbolicity is lost in this step. Research partially supported by the INTAS project Attractors for Equations of Mathematical Physics, by CRDF and by the Alexander von Humboldt–Stiftung.     

Construction of asymptotic formulas for solutions of singularly perturbed degenerate systems of differential equations

We obtain an asymptotic solution of the Cauchy problem for a singularly perturbed degenerate system of differential equations in the case of a singular limit pencil of matrices. Translated from Neliniini Kolyvannya, Vol. 11, No. 3, pp. 408–420, July–September, 2008.     

On periodic solutions of nonlinear autonomous differential systems in the critical case

We propose a new numerical-analytic algorithm for the investigation of periodic solutions of nonlinear autonomous systems of ordinary differential equations in the critical case. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 73–82, January–March, 2009.