Stochastic functional differential equations on manifolds

In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters. Received: 6 July 1999 / Revised version: 19 April 2000 /?Published online: 14 June 2001     

Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds

We study the dependence on initial data for solutions of diffusion equations with globally non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may not be everywhere twice differentiable, we show that, under certain monotonicity conditions on the coefficients and curvature of the manifold, there are estimates exponential in time for the continuity of a diffusion process with respect to initial data. These estimates are combined with methods of the theory of absolutely continuous functions to achieve the first-order regularity of solutions with respect to initial data. The suggested approach neither appeals to the local stopping time arguments, nor applies the exponential mappings on the tangent space, nor uses imbeddings of a manifold to linear spaces of higher dimensions. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1299–1316, October, 2008.     

Solitary waves in a cold plasma

We study the existence of soliton-like solutions (solitary waves) to the equations describing the one-dimensional motion of a cold quasi-neutral plasma. It is shown that in some range of the angle between the norperturbed magnetic field and the wave propagation direction there exists a branch of solitary hydromagnetic waves that is a bifurcation of the zero wave number. These solutions lie on a two-dimensional center manifold.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 719–728, May, 1996.     

Asymptotics of solutions of partial differential equations with higher degenerations and the Laplace equation on a manifold with a cuspidal singularity

We study the asymptotics of solutions of partial differential equations with higher degenerations. Such equations arise, for example, when studying solutions of elliptic equations on manifolds with cuspidal singular points. We construct the asymptotics of a solution of the Laplace equation defined on a manifold with a cuspidal singularity of order k.     

On the Maxwell problem

We study the large-time behavior of global smooth solutions to the Cauchy problem for hyperbolic regularization of conservation laws. An attracting manifold of special smooth global solutions is determined by the Chapman–Enskog projection onto the phase space of consolidated variables. For small initial data we construct the Chapman–Enskog projection and describe its properties in the case of the Cauchy problem for moment approximations of kinetic equations. The existence conditions for the Chapman–Enskog projection are expressed in terms of the solvability of the Riccati matrix equations with parameter. Bibliography: 21 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 27–63.     

Geodesic vector fields and Eikonal equation on a Riemannian manifold

In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.     

Removable Singularities for Anisotropic Elliptic Equations

We study a class of quasi-linear elliptic equations with model representative \(\sum _{i=1}^{n}(|u_{x_{i}}|^{p_{i}-2}u_{x_{i}})_{x_{i}}=0\) , which solutions have singularities on a smooth manifold. We establish the condition for removability of singularity on a manifold for solutions of such equations.     

Higher dimensional vortex standing waves for nonlinear Schrödinger equations

We study standing wave solutions to nonlinear Schrödinger equations, on a manifold with a rotational symmetry, which transform in a natural fashion under the group of rotations. We call these vortex solutions. They are higher dimensional versions of vortex standing waves that have been studied on the Euclidean plane. We focus on two types of vortex solutions, which we call spherical vortices and axial vortices.     

Long-time limit for a class of quadratic infinite-dimensional dynamical systems inspired by models of viscoelastic fluids

We study a class of quadratic, infinite-dimensional dynamical systems, inspired by models for viscoelastic fluids. We prove that these equations define a semi-flow on the cone of positive, essentially bounded functions. As time tends to infinity, the solutions tend to an equilibrium manifold in the L²-norm. Convergence to a particular function on the equilibrium manifold is only proved under additional assumptions. We discuss several possible generalizations.     

Long time behaviour of solutions of abstract inequalities: Applications to thermo-hydraulic and magnetohydrodynamic equations

We study some scalar inequalities of parabolic type and we give the leading term of an asymptotic expansion as t → ∞ for solutions of thermo-hydraulic equations without external excitation. A phenomenon of resonance is pointed out. We also treat M. H. D. equations and Navier-Stokes equations on a Riemannian manifold.     

Normal forms for semilinear equations with non-dense domain with applications to age structured models

Normal form theory is very important and useful in simplifying the forms of equations restricted on the center manifolds in studying nonlinear dynamical problems. In this paper, using the center manifold theorem associated with the integrated semigroup theory, we develop a normal form theory for semilinear Cauchy problems in which the linear operator is not densely defined and is not a Hille–Yosida operator and present procedures to compute the Taylor expansion and normal form of the reduced system restricted on the center manifold. We then apply the main results and computation procedures to determine the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions in a structured evolutionary epidemiological model of influenza A drift and an age structured population model.     

Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?

It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. The appearance of curvature in the structure of high-order variational equations is discussed and a family of a priori nonlinear estimates of regularity of any order is obtained. By using the relationship between the differential equations on manifolds and semigroups, we study C ^∞-regular properties of solutions of the parabolic Cauchy problems with coefficients increasing at infinity. The obtained conditions of regularity generalize the classical coercivity and dissipation conditions to the case of a manifold and correlate (in a unified way) the behavior of diffusion and drift coefficients with the geometric properties of the manifold without traditional separation of curvature. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1011–1034, August, 2006.     

Center manifolds for periodic functional differential equations of mixed type

We study the behaviour of solutions to nonlinear functional differential equations of mixed type (MFDEs), that remain sufficiently close to a prescribed periodic solution. Under a discreteness condition on the Floquet spectrum, we show that all such solutions can be captured on a finite dimensional invariant center manifold, that inherits the smoothness of the nonlinearity. This generalizes the results that were obtained previously in [H.J. Hupkes, S.M. Verduyn Lunel, Center manifold theory for functional differential equations of mixed type, J. Dynam. Differential Equations 19 (2007) 497-560] for bifurcations around equilibrium solutions to MFDEs.     

Tronquée solutions of the painlevé II equation

We study special solutions of the Painlevé II (PII) equation called tronquée solutions, i.e., those having no poles along one or more critical rays in the complex plane. They are parameterized by special monodromy data of the Lax pair equations. The manifold of the monodromy data for a general solution is a twodimensional complex manifold with one- and zero-dimensional singularities, which arise because there is no global parameterization of the manifold. We show that these and only these singularities (together with zeros of the parameterization) are related to the tronquée solutions of the PII equation. As an illustration, we consider the known Hastings-McLeod and Ablowitz-Segur solutions and some other solutions to show that they belong to the class of tronquée solutions and correspond to one or another type of singularity of the monodromy data.     

The profile of bubbling solutions of a class of fourth order geometric equations on 4-manifolds

We study a class of fourth order geometric equations defined on a 4-dimensional compact Riemannian manifold which includes the Q-curvature equation. We obtain sharp estimates on the difference near the blow-up points between a bubbling sequence of solutions and the standard bubble.     

The continuous dependence of solutions to algebraic differential systems on the initial data

We consider a nonlinear system of ordinary differential equations which is unsolved with respect to the derivative of the desired vector function and identically degenerate in the definition domain. We study the consistency manifold under assumptions that guarantee the existence of a solution. We prove an analog of the theorem on the continuous dependence of solutions on the initial data, assuming that the latter belong to the consistency manifold.     

The effect of nonlocal interactions on the dynamics of the Ginzburg-Landau equation

Nonlocal amplitude equations of the complex Ginzburg-Landau type arise in a few physical contexts, such as in ferromagnetic systems. In this paper, we study the effect of the nonlocal term on the global dynamics by considering a model nonlocal complex amplitude equation. First, we discuss the global existence, uniqueness and regularity of solutions to this equation. Then we prove the existence of the global attractor, and of a finite dimensional inertial manifold. We provide upper and lower bounds to their dimensions, and compare them with those of the cubic complex Ginzburg-Landau equation. It is observed that the nonlocal term plays a stabilizing or destabilizing role depending on the sing of the real part of its coefficient. Moreover, the nonlocal term affects not only the diameter of the attractor but also its dimension.     

The onset of resonance processes in the Couette–Taylor problem close to the point of codimension-2 bifurcation

The bifurcations on passing around the point of intersection of two neutral curves (points of codimension-2 bifurcation) are considered in the Couette–Taylor problem of the fluid motion between rotating cylinders. The secondary modes in a small neighbourhood of a point of codimension-2 bifurcation are studied using a system of non-linear amplitude equations in a central manifold. The steady-state solutions of the amplitude systems, to which secondary periodic modes of the travelling-wave type, non-linear mixtures of travelling waves and unsteady two-, three- and four-frequency quasiperiodic solutions of the system of Navier–Stokes equations correspond, are analysed. A numerical analysis of the conditions for the existence and stability of irrotationally symmetric steady-state fluid flows between unidirectionally rotating cylinders is carried out.     

The unique solvability of a problem without initial conditions for linear and nonlinear elliptic-parabolic equations

The existence and the uniqueness of generalized solutions of a problem without initial conditions are established for linear and nonlinear anisotropic elliptic-parabolic second-order equations in domains unbounded in spatial variables. We put the restrictions on the behavior of solutions of the problem and the growth of its initial data at infinity. The equations have the nonlinearity exponents depending on points of the domain of definition and the direction of differentiation. Their weak solutions are taken from generalized Lebesgue–Sobolev spaces.     

A Numerical Study of the Stability of Solitary Waves of the Bona–Smith Family of Boussinesq Systems

In this paper we study, from a numerical point of view, some aspects of stability of solitary-wave solutions of the Bona–Smith systems of equations. These systems are a family of Boussinesq-type equations and were originally proposed for modelling the two-way propagation of one-dimensional long waves of small amplitude in an open channel of water of constant depth. We study numerically the behavior of solitary waves of these systems under small and large perturbations with the aim of illuminating their long-time asymptotic stability properties and, in the case of large perturbations, examining, among other, phenomena of possible blow-up of the perturbed solutions in finite time.