On the local measures of mean rotation in continuum mechanics

We implement Cauchy's concept of a rotation-angle function on an oriented plane, and characterize situations when a rotation-angle function exists, and hence when measuring mean rotations in the manner of Cauchy or Novozhilov makes sense. We also discuss in passing the role of the skew part of the deformation gradient in measuring the mean deformation.     

Asymptotics of the Solutions of the Stochastic Lattice Wave Equation

We consider the long time limit for the solutions of a discrete wave equation with weak stochastic forcing. The multiplicative noise conserves energy, and in the unpinned case also conserves momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds for both square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sense in the former case, and in the weak sense in the latter. On the other hand, the weak limit for square integrable initial data is deterministic.     

Well-posedness of characteristic symmetric hyperbolic systems

We consider the initial-boundary-value problem for quasi-linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity. We show the well-posedness in Hadamard's sense (i.e., existence, uniqueness and continuous dependence of solutions on the data) of regular solutions in suitable functions spaces which take into account the loss of regularity in the normal direction to the characteristic boundary.     

The numerical solution of a singularly perturbed problem for quasilinear parabolic differential equation

We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L~2 norm is proved and numerical examples are presented.     

Asymptotic representations of solutions of one class of second-order ordinary differential equations

We establish asymptotic representations for some classes of solutions of nonautonomous second-order differential equations close, in a certain sense, to linear equations.     

Weak Attractor for a Dissipative Euler Equation

A two-dimensional dissipative Euler equation is considered. We proved the existence of a global attractor in a weak sense, for the corresponding shift dynamical system in path space.     

Displacement Convexity and Minimal Fronts at Phase Boundaries

We show that certain free energy functionals that are not convex with respect to the usual convex structure on their domain of definition are strictly convex in the sense of displacement convexity under a natural change of variables.We use this to show that, in certain cases, the only critical points of these functionals are minimizers. This approach based on displacement convexity permits us to treat multicomponent systems as well as single component systems. The developments produce new examples of displacement convex functionals and, in the multi-component setting, jointly displacement convex functionals.     

Initial Layers and Uniqueness of¶Weak Entropy Solutions to¶Hyperbolic Conservation Laws

We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weak-star in L ^∞ as t→0₊ and satisfy the entropy inequality in the sense of distributions for t>0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solutions are always unique and the initial layers do not appear. We also discuss applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation. Accepted: October 26, 1999     

Monotonicity theorems for two-phase solids

Certain alloys, such as gold-copper, have two solid phases. We establish a general mathematical framework in which we show that the fraction in one phase and the compositions within each phase are in some sense decreasing in the overall composition. The tools used include useful new lemmas on minima of functions of several variables and parameters.     

Secondary flow of Rivlin-Ericksen fluids between two concentric spheres oscillating about a fixed diameter

Summary The paper deals with the study of the nature of secondary flow of aRivlin-Ericksen fluid, contained between two concentric spheres, which perform oscillations about a fixed diameter. The steady part of the secondary flow is discussed in detail in the following three cases (i) the outer sphere at rest, the inner oscillating, (ii) the two spheres oscillating with the same angular velocity in the same sense and (iii) the spheres oscillating with the same angular velocity in opposite sense. In a previous paper, a similar problem was discussed for theOldroyd fluids. We find that the secondary flow is strongly dependent on the common frequency of oscillation of the two spheres and on the rotational nature of the motion for the present investigation also. Certain contrasting features of interest between the secondary flow field of the two fluids are also noted.     

Stability of Subsonic Planar Phase Boundaries in a van der Waals Fluid

We are concerned with the structural stability of dynamic phase changes occurring across sharp interfaces in a multidimensional van der Waals fluid. Such phase transitions can be viewed as propagating discontinuities. However, they are usually subsonic, and thus undercompressive. The lacking information lies in an additional jump condition, which may be derived from the viscosity-capillarity criterion. This condition is rather simple in the case of reversible phase transitions, since it reduces to a generalized equal area rule. In a previous work, I proved that reversible planar phase boundaries are weakly linearly stable, in the sense introduced by Majda for shock fronts. This means that they satisfy a generalized Lopatinsky condition but not a uniform one. The aim of this paper is to point out the influence of viscosity on the stability analysis, in order to deal with the more realistic case of dissipative phase transitions. The main difficulty lies in the additional jump condition, which is no longer explicit and depends on the (unknown) internal structure of the interface. We overcome it by using bifurcation arguments on the nondimensional parameter measuring the competition between viscosity and capillarity. We show by perturbation that the positivity of this parameter stabilizes the phase transitions. As a conclusion, we find that dissipative planar phase boundaries are uniformly linearly stable, in the sense of the uniform Lopatinsky condition. Accepted December 14, 1998     

Uniform difference scheme for hyperbolic-hyperbolic singular perturbation mixed problems

A difference scheme is established in this paper for second-order hyperbolic-hyperbolic singular perturbation mixed problems. We give an energy inequality of the numerical solution and prove that the numerical solution converges to the solution of the singular perturbation problems uniformly with respect to a small parameter and in the sense of a discrete norm.This paper was read out in the Second National Symposium on Computing Hydromechanics which was held in May of 1984 in Wuxi, Jiangsu, China.     

Inhomogeneous deformations of a cone of a neo-Hookean solid

We study the inhomogeneous deformations of a cone due to squeezing or expanding the cone. Reminiscent of solutions to Jefferey-Hamel flows between intersecting planes and the deformation of elastic wedges, we find that depending on the cone angle we can have solutions that correspond to radial filaments being purely stretched or contracted, or solutions in which the displacement does not have the same sense, certain subparts being stretched, while others are contracted.     

Optimal design of an optical lens

We wish to construct an optimal optical lens, in the sense that it focuses all rays from a given source onto a given focal point. The construction can be reduced mathematically to a global study of a system of two nonlinear functional differential equations. We prove that this system has precisely two analytic solutions.     

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.     

On Second-Order Differentiability with Respect to Parameters for Differential Equations with State-Dependent Delays

In this paper we consider a class of differential equations with state-dependent delays. We show first and second-order differentiability of the solution with respect to parameters in a pointwise sense and also using the $C$ -norm on the state-space, assuming that the state-dependent time lag function is piecewise strictly monotone.     

Helicity and other conservation laws in perfect fluid motion

In this review paper, we discuss helicity from a geometrical point of view and see how it applies to the motion of a perfect fluid. We discuss its relation with the Hamiltonian structure, and then its extension to arbitrary space dimensions. We also comment about the existence of additional conservation laws for the Euler equation, and its unlikely integrability in Liouville's sense.     

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.     

Theory of Complex Systems and Economic Dynamics

The purpose of this paper is to discuss the significance of the theory of complex systems for analyzing economic evolution. We emphasize the necessity of the structural dynamic approach and discuss possible implications of the theory of complex systems for studying economic processes with different speeds of change. We illustrate a way to construct a general economic theory that includes the main economic theories of competition with government intervention as special cases in the structural sense.     

Hamiltonian finite-dimensional oscillator-type reductions of Lax-integrable superconformal hierarchies

For bi-Hamiltonian superconformal hierarchies of nonlinear Benny-Kaup and Kaup-Broer dynamical systems, we develop a method for reduction to nonlocal finite-dimensional invariant subspaces of Neumann and Bargmann types, respectively. We prove that there exist even supersymplectic structures on these spaces and that the reduced commuting vector fields generated by the hierarchies are integrable in the Lax-Liouville sense. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 15–30, January–March, 2006.