On Singular Limits of Mean-Field Equations

Mean-field equations arise as steady state versions of convection-diffusion systems where the convective field is determined by solution of a Poisson equation whose right-hand side is affine in the solutions of the convection-diffusion equations. In this paper we consider the repulsive coupling case for a system of two convection-diffusion equations. For general diffusivities we prove the existence of a unique solution of the mean-field equation by a variational analysis of a saddle point problem (usually without coercivity). Also we analyze the small-Debye-length limit and prove convergence to either the so-called charge-neutral case or to a double obstacle problem for the limiting potential depending on the data.     

Role of Prehistories in the Initial Value Problems of Fractional Viscoelastic Equations

The fractional viscoelastic equation (FVE), which is a second-order differential equation with fractional derivatives describing the dynamical behavior of a single-degree-of-freedom viscoelastic oscillator, is considered. Some viscoelastic damped mechanical systems may be described by FVEs. However, FVEs with conventional nonzero initial values cannot generally be solved. In this paper, the prehistories of the unknown functions before the initial times, referred to as the initial functions, are taken into account to solve FVEs. Mathematically, appropriate initial functions are essential for unique solutions of FVEs. Physically, the initial functions reflect the processes of giving the initial values. FVEs are solved for some initial functions both by analytical and numerical methods. The initial functions affect the solutions of FVEs. It is discussed how the solutions depend on the initial functions. Implication of the solutions to viscoelastic materials will be discussed.     

Role of Prehistories in the Initial Value Problems of Fractional Viscoelastic Equations

The fractional viscoelastic equation (FVE), which is a second-order differential equation with fractional derivatives describing the dynamical behavior of a single-degree-of-freedom viscoelastic oscillator, is considered. Some viscoelastic damped mechanical systems may be described by FVEs. However, FVEs with conventional nonzero initial values cannot generally be solved. In this paper, the prehistories of the unknown functions before the initial times, referred to as the initial functions, are taken into account to solve FVEs. Mathematically, appropriate initial functions are essential for unique solutions of FVEs. Physically, the initial functions reflect the processes of giving the initial values. FVEs are solved for some initial functions both by analytical and numerical methods. The initial functions affect the solutions of FVEs. It is discussed how the solutions depend on the initial functions. Implication of the solutions to viscoelastic materials will be discussed.     

Existence,Uniqueness, and Stability of Generalized Traveling Waves in Time Dependent Monostable Equations

The current paper is devoted to the study of traveling wave solutions of spatially homogeneous monostable reaction diffusion equations with ergodic or recurrent time dependence, which includes periodic and almost periodic time dependence as special cases. Such an equation has two spatially homogeneous and time recurrent solutions with one of them being stable and the other being unstable. Traveling wave solutions are a type of entire solutions connecting the two spatially homogeneous and time recurrent solutions. Recently, the author of the current paper proved that a spatially homogeneous time almost periodic monostable equation has a spreading speed in any given direction. This result can be easily extended to monostable equations with recurrent time dependence. In this paper, we introduce generalized traveling wave solutions for time recurrent monostable equations and show the existence of such solutions in any given direction with average propagating speed greater than or equal to the spreading speed in that direction and non-existence of such solutions of slower average propagating speed. We also show the uniqueness and stability of generalized traveling wave solutions in any given direction with average propagating speed greater than the spreading speed in that direction. Moreover, we show that a generalized traveling wave solution in a given direction with average propagating speed greater than the spreading speed in that direction is unique ergodic in the sense that its wave profile and wave speed are unique ergodic, and if the time dependence of the monostable equation is almost periodic, it is almost periodic in the sense that its wave profile and wave speed are almost periodic.     

Integration of Special Linear Equations of the Second Order

Under certain conditions on the coefficients, the Chazy equation with constant coefficients reduces to a second-order linear differential equation with six singular points. Investigating this equation with the use of the Schwarz derivative, we obtain linear equations in which some of these six singular points coincide. An integration procedure for these equations is considered. Their general solutions are obtained in explicit form.     

Relaxation of Hamilton-Jacobi Equations

We study the relaxation of Hamilton-Jacobi equations. The relaxation in our terminology is the following phenomenon: the pointwise supremum over a certain collection of subsolutions, in the almost everywhere sense, of a Hamilton-Jacobi equation yields a viscosity solution of the ``convexified' Hamilton-Jacobi equation. This phenomenon has recently been observed in [13] in eikonal equations. We show in this paper that this relaxation is a common phenomenon for a wide range of Hamilton-Jacobi equations.     

Exact Averaging of Stochastic Equations for Flow in Porous Media

It is well-known that at present, exact averaging of the equations for flow and transport in random porous media have been proposed for limited special fields. Moreover, approximate averaging methods—for example, the convergence behavior and the accuracy of truncated perturbation series—are not well-studied, and in addition, calculation of high-order perturbations is very complicated. These problems have for a long time stimulated attempts to find the answer to the question: Are there in existence some, exact, and sufficiently general forms of averaged equations? Here, we present an approach for finding the general exactly averaged system of basic equations for steady flow with sources in unbounded stochastically homogeneous fields. We do this by using (1) the existence and some general properties of Green’s functions for the appropriate stochastic problem, and (2) some information about the random field of conductivity. This approach enables us to find the form of the averaged equations without directly solving the stochastic equations or using the usual assumption regarding any small parameters. In the common case of a stochastically homogeneous conductivity field we present the exactly averaged new basic non-local equation with a unique kernel-vector. We show that in the case of some type of global symmetry (isotropy, transversal isotropy, or orthotropy), we can for three-dimensional and two-dimensional flow in the same way derive the exact averaged non-local equations with a unique kernel-tensor. When global symmetry does not exist, the non-local equation with a kernel-tensor involves complications and leads to an ill-posed problem.     

Second Order Abstract Neutral Functional Differential Equations

In this paper we are concerned with a class of second order abstract neutral functional differential equations with finite delay in a Banach space. We establish the existence of mild and classical solutions for the nonlinear equation, and we show that the map defined by the mild solutions of the linear equation is a strongly continuous semigroup of bounded linear operators on an appropriate space. We use this semigroup to establish a variation of constants formula to solve the inhomogeneous linear equation.     

Existence of Multi-Pulses of the Regularized Short-Pulse and Ostrovsky Equations

The existence of multi-pulse solutions near orbit-flip bifurcations of a primary single-humped pulse is shown in reversible, conservative, singularly perturbed vector fields. Similar to the non-singular case, the sign of a geometric condition that involves the first integral decides whether multi-pulses exist or not. The proof utilizes a combination of geometric singular perturbation theory and Lyapunov–Schmidt reduction through Lin’s method. The motivation for considering orbit flips in singularly perturbed systems comes from the regularized short-pulse equation and the Ostrovsky equation, which both fit into this framework and are shown here to support multi-pulses.     

From the Boltzmann Equations¶to the Equations of¶Incompressible Fluid Mechanics, I

We consider here the problem of deriving rigorously from Boltzmann's equation, globally in time and for general initial conditions, fluid mechanics equations such as the Navier-Stokes or Euler equations.     

Long-time Solvability of the Navier–Stokes Equations in a Rotating Frame with Spatially Almost Periodic Large Data

We consider existence of solutions, for large times, to the Navier–Stokes equations in a rotating frame with spatially almost periodic large data provided by a sufficiently large Coriolis force. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. To show existence of solutions for large times, we use the ℓ ¹-norm of amplitudes. Existence for large times is proven by means of techniques of fast singular oscillating limits and bootstrapping from a global-in-time unique solution to the limit equation.     

Lie Symmetry Analysis of Differential Equations in Finance

Lie group theory is applied to differential equations occurring as mathematical models in financial problems. We begin with the complete symmetry analysis of the one-dimensional Black–Scholes model and show that this equation is included in Sophus Lie's classification of linear second-order partial differential equations with two independent variables. Consequently, the Black–Scholes transformation of this model into the heat transfer equation follows directly from Lie's equivalence transformation formulas. Then we carry out the classification of the two-dimensional Jacobs–Jones model equations according to their symmetry groups. The classification provides a theoretical background for constructing exact (invariant) solutions, examples of which are presented.     

Attractors of Parabolic Equations Without Uniqueness

In this paper we study the existence of global compact attractors for nonlinear parabolic equations of the reaction-diffusion type and variational inequalities. The studied equations are generated by a difference of subdifferential maps and are not assumed to have a unique solution for each initial state. Applications are given to inclusions modeling combustion in porous media and processes of transmission of electrical impulses in nerve axons.     

FPK方程的近似闭合解

讨论了FPK方程的近似闭合解问题。假设FPK方程的解具有指数多项式的形式，利用比较系数的方法确定其中待定的常数。计算表明，本方法适用于强非线性系统，在特殊情形下还能求出原方程的精确解。     

Asymptotic Variational Wave Equations

We investigate the equation (u _t +(f(u)) _x ) _x =f ^{′ ′}(u) (u _x )²/2 where f(u) is a given smooth function. Typically f(u)=u ²/2 or u ³/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u _tt − c(u) (c(u)u _x ) _x =0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data.     

Fractional Diffusion Limit for Collisional Kinetic Equations

This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation.     

Equations of nonlinear anisotropic flow through a porous medium

A general law of nonlinear anisotropic flow through a porous medium is proposed. A corresponding equation for the pressure of the fluid is obtained in velocity hodograph variables. The conditions of ellipticity of this equation are expressed in terms of the dissipative function.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 158–160, September–October, 1980.I thank V. M. Entob for discussing the work.     

Asymptotic Stability of Differential Equations on Convex Sets

We show a sufficient condition for the asymptotic stability of solutions of an abstract evolutionary equation with the right-hand side defined on a convex set. This condition is applied to a generalized version of the Tjon-Wu equation and to the Boltzmann-Kac equation.     

Exact Solutions of the Boltzmann Equations with a Source

Exact solutions of a nonlinear Boltzmann kinetic equation with a source are constructed in the case of an isotropic distribution function and Maxwell model of isotropic scattering. These solutions are constructed with the use of an equivalence group such that one of its transformations uniquely identifies the class of the source functions that are linear in terms of the distribution function; moreover, the transformed equation has a zero right side. As a result, invariant solutions of the type of the Bobylev–Krook–Wu solutions can be explicitly found, in particular, those that admit physical interpretation.     

Renormalization Group Method. Applications to Partial Differential Equations

Our aim in this article is to present a simplified form of the renormalization group (RG) method introduced by Chen, Goldenfeld, and Oono and to derive a rigorous study of the validity in time of the asymptotic solutions furnished by the RG method. We apply the renormalization group method to a slightly compressible fluid equation and to the Swift–Hohenberg equation.