Integral equations as evolution equations

We study a class of time-dependent linear integrodifferential equations (VE) with the evolution equation approach. We determine the generators of a time-dependent evolution equation (DE) which is equivalent to the given integrodifferential equation. Under very general assumptions we prove the well-posedness and continuity of (VE) from the stability of (DE). The related question of convergence of a family of approximate solutions is examined. As an application, we include an example of hyperbolic integro-partial-differential equation to illustrate the theory.     

Convolution equations and nonlinear functional equations

The survey is devoted to applications of nonlinear integral equations to linear convolution equations, their discrete analogues, and also the connection of these equations with problems of radiative transfer, in particular, with the Ambartsumyan equations.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 22, pp. 175–244, 1984.     

Integrability of Riccati equations and stationary Korteweg-de vries equations

By using the Lie infinitesimal method, we establish the correspondence between the integrability of a one-parameter family of Riccati equations and the hierarchy of the higher Korteweg-de Vries equations. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal. Vol. 51, No. 6, pp. 856–860, June, 1999.     

Dissipativity of differential equations and the corresponding difference equations

We establish conditions under which the existence of a bounded solution of a difference equation yields the existence of a bounded solution of the corresponding differential equation. We investigate the relationship between the dissipativities of differential and difference equations in terms of Lyapunov functions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1249–1256, September, 2006.     

Some nonlinear equations reducible to diffusion-type equations

The problem of the reducibility of a system of second-order quasi-linear parabolic differential equations to diffusion-type equations is considered. An effective solution algorithm is suggested for this problem in the nondegenerate case. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 26–37, April, 2000.     

Generating equations approach for quadratic matrix equations

We show how Van Loan's method for annulling the (2,1) block of skew‐Hamiltonian matrices by symplectic‐orthogonal similarity transformation generalizes to general matrices and provides a numerical algorithm for solving the general quadratic matrix equation: For skew‐Hamiltonian matrices we find their canonical form under a similarity transformation and find the class of all symplectic‐orthogonal similarity transformations for annulling the (2,1) block and simultaneously bringing the (1,1) block to Hessenberg form. We present a structure‐preserving algorithm for the solution of continuous‐time algebraic Riccati equation. Unlike other methods in the literature, the final transformed Hamiltonian matrix is not in Hamiltonian–Schur form. Three applications are presented: (a) for a special system of partial differential equations of second order for a single unknown function, we obtain the matrix of partial derivatives of second order of the unknown function by only algebraic operations and differentiation of functions; (b) for a similar transformation of a complex matrix into a symmetric (and three‐diagonal) one by applying only finite algebraic transformations; and (c) for finite‐step reduction of the eigenvalues–eigenvectors problem of a Hermitian matrix to the eigenvalues– eigenvectors problem of a real symmetric matrix of the same dimension. Copyright © 1999 John Wiley & Sons, Ltd.     

Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations

The combined quasi-neutral and non-relativistic limit of compressible Navier-Stokes-Maxwell equations for plasmas is studied. For well-prepared initial data, it is shown that the smooth solution of compressible Navier-Stokes-Maxwell equations converges to the smooth solution of incompressible Navier-Stokes equations by introducing new modulated energy functional.     

A class of integral equations and approximation of p-Laplace equations

Let ${\Omega\subset\mathbb{R}^n}$ be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equation $$ M[u](x) = f_0(x)\quad {\rm in}\,\Omega$$ with the boundary condition u = g ₀ on ?Ω, where ${f_0\in C(\overline\Omega)}$ and ${g_0\in C(\partial\Omega)}$ are given functions and M is the singular integral operator given by $$M[u](x)={\rm p.v.} \int\limits_{B(0,\rho(x))} \frac{p-\sigma}{|z|^{n+\sigma}}|u(x+z)-u(x)|^{p-2} (u(x+z)-u(x))\,{\rm dz},$$ with some choice of ${\rho\in C(\overline\Omega)}$ having the property, 0 < ρ(x) ≤ dist (x, ?Ω). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on ${\overline\Omega}$ , as σ → p, of the solution u _σ of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation νΔ _p u = f ₀ in Ω with the Dirichlet condition u = g ₀ on ?Ω, where the factor ν is a positive constant (see (7.2)).     

Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations

In this paper, we study the incompressible limit of the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. Based on the convergence-stability principle, we show that, when the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient are sufficiently small, the initial-value problem of the model has a unique smooth solution in the time interval where the ideal incompressible magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient go to zero. Moreover, we obtain the convergence of smooth solutions for the model forwards those for the ideal incompressible magnetohydrodynamic equations with a sharp convergence rate.     

Systems of differential equations of higher dimension and delay equations

We establish connections between the solutions to a class of systems of ordinary differential equations of higher dimension and delay equations.     

A link of stochastic differential equations to nonlinear parabolic equations

Using Girsanov transformation,we derive a new link from stochastic differential equations of Markovian type to nonlinear parabolic equations of Burgers-KPZ type,in such a manner that the obtained BurgersKPZ equation characterizes the path-independence property of the density process of Girsanov transformation for the stochastic differential equation.Our assertion also holds for SDEs on a connected differential manifold.     

Harmonic analysis of stochastic equations and backward stochastic differential equations

The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations in ${\mathcal{R}^p}$ ( ${p\in [1,\infty)}$ ) and backward stochastic differential equations (BSDEs) in ${\mathcal{R}^p\times \mathcal{H}^p}$ ( ${p\in (1, \infty)}$ ) and in ${\mathcal{R}^\infty\times\overline{L^\infty}^{\rm BMO}}$ , with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Fefferman’s inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multi-dimensional linear cases for stochastic differential equations (SDEs) and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse Hölder inequality for some suitable exponent p ≥ 1. Finally, some relations are established between Kazamaki’s quadratic critical exponent b(M) of a BMO martingale M and the spectral radius of the stochastic integral operator with respect to M, which lead to a characterization of Kazamaki’s quadratic critical exponent of BMO martingales being infinite.     

Second-order type-changing evolution equations with first-order intermediate equations

This paper presents a partial classification for C^∞ type-changing symplectic Monge-Ampère partial differential equations (PDEs) that possess an infinite set of first-order intermediate PDEs. The normal forms will be quasi-linear evolution equations whose types change from hyperbolic to either parabolic or to zero. The zero points can be viewed as analogous to singular points in ordinary differential equations. In some cases, intermediate PDEs can be used to establish existence of solutions for ill-posed initial value problems.     

Some differences between difference equations and differential equations

In this expository paper we discuss some instances in which analogues which might be expected between the behavior of solutions of differential equations and difference equations fail to hold, focussing particularly on questions related to boundedness and oscillation.