Regularity and integrability of rotating shallow-water equations

We consider classical shallow-water equations for a rapidly rotating fluid layer. The Poincaré/Kelvin linear propagator describes fast oscillating waves for the linearized system. We show that solutions of the full nonlinear shallow-water equations can be decomposed as U(t,x₁,x₂) + Ũ(t,x₁,x₂) + W’(t,x₁,x₂) + r, where Ũ is a solution of the quasigeostrophic (QG) equation. Here r is a remainder, which is uniformly estimated from above by a majorant of order 1/f₀. The vector field W’(t,x₁,x₂) describes the rapidly oscillating ageostrophic (AG) component. This component is exactly solved in terms of Poincaré/Kelvin waves with phase shifts explicitly determined from the nonlinear quasigeostrophic equations. The mathematically rigorous control of the error r, based on estimates of small divisors, is used to prove the existence, on a long time interval T^*, of regular solutions to classical shallow-water equations with general initial data (T^* → +∞, as 1/f₀ → 0).     

Periodic solutions of non-smooth friction oscillators

We study the differential equation x"+g(x￠)+m(x) sgn x￠+f(x)=j(t)x''+g(x')+\mu(x)\,{\rm sgn}\, x'+f(x)=\varphi(t) with T-periodic right-hand side, which models e.g. a mechanical system with one degree of freedom subjected to dry friction and periodic external force. If, in particular, the damping term g is present and acts, up to a bounded difference, like a linear damping, we get existence of a T-periodic solution.¶In the more difficult case g = 0, we concentrate on the model equation x"+m(x) sgn x￠+x=j(t)x''+\mu(x)\,{\rm sgn}\,x'+x=\varphi(t) and obtain sufficient conditions for the existence of a T-periodic solution by application of Brouwer's fixed point theorem. For this purpose we show that a certain associated autonomous differential equation admits a periodic orbit such that the surrounded set (minus some neighborhood of the equilibria) is forward invariant for the equation above. Under additional assumptions on 7 we prove boundedness of all solutions.¶Finally, we provide a principle of linearized stability for periodic solutions without deadzones, where the "linearized" differential equation is an impulsive Hill equation.     

A geometric approach to periodically forced dynamical systems in presence of a separatrix

We consider the periodic boundary value problem for the non-autonomous scalar second-order equation , with e(·) a continuous and T-periodic forcing term. Using a continuation theorem adapted from Capietto et al. (Trans. Amer. Math. Soc. 329 (1992) 41-72), we propose some new conditions for the existence of T-periodic solutions to the forced equation in terms of the dynamical properties of the trajectories of the associated autonomous equation . Special emphasis will be addressed to the study of the case in which the presence of an unbounded separatrix for the autonomous system in the phase-plane allows to obtain a priori bounds for the T-periodic solutions of the homotopic equation .     

On the Response of Autonomous Sweeping Processes to Periodic Perturbations

If x ₀ is an equilibrium of an autonomous differential equation \(\dot x=f(x)\) and det∥f ^′(x ₀)∥≠0, then x ₀ persists under autonomous perturbations and x ₀ transforms into a T-periodic solution under non-autonomous T-periodic perturbations. In this paper we discover a similar structural stability for Moreau sweeping processes of the form \(-\dot u\in N_{B}(u)+f_{0}(u)\), \(u\in \mathbb {R}^{2}, \)i. e. we consider the simplest case where the derivative is taken with respect to the Lebesgue measure and where the convex set B of the reduced system is a non-moving unit ball of \(\mathbb {R}^{2}\). We show that an equilibrium ∥u ₀∥=1 persists under periodic perturbations, if the projection \(\overline {f}:\partial B\to \mathbb {R}^{2}\) of f ₀ on the tangent to the boundary ? B is nonsingular at u ₀.     

Global branches of periodic solutions for forced delay differential equations on compact manifolds

We prove a global bifurcation result for T-periodic solutions of the T-periodic delay differential equation x^′(t)=λf(t,x(t),x(t−1)) depending on a real parameter λ?0. The approach is based on the fixed point index theory for maps on ANRs.     

Periodic solutions for a nonautonomous ordinary differential equation

We consider the nonautonomous differential equation of second order x^″+a(t)x−b(t)x²+c(t)x³=0, where a(t),b(t),c(t) are T-periodic functions. This is a biomathematical model of an aneurysm in the circle of Willis. We prove the existence of at least two positive T-periodic solutions for this equation, using coincidence degree theories.     

Homogenization of the acoustic equations for a porous long-memory viscoelastic material filled with a viscous fluid

We consider the system of linear differential and integro-differential equations describing small vibrations in an ?-periodic combined medium consisting of a porous long-memory viscoelastic material and a viscous fluid filling the pores. By using the two-scale convergence method, we construct the system of homogenized equations and prove the convergence of solutions of the original problems to the solution of the homogenized problem as ? ?? 0.     

Periodic solutions with prescribed minimal period for the second order Hamiltonian systems with even potentials

In this paper, under a similar but stronger condition than that of Ambrosetti and Rabinowitz we find a T-periodic solution of the autonomous superquadratic second order Hamiltonian system with even potential for any T 〉 0; moreover, such a solution has T as its minimal period.     

Conditional Fredholm determinant for the S-periodic orbits in Hamiltonian systems

For S being a symplectic orthogonal matrix on R2n, the S-periodic orbits in Hamiltonian systems are a solution which satisfies x(0)=Sx(T) for some period T. This paper is devoted to establishing the theory of conditional Fredholm determinant in studying the S-periodic orbits in Hamiltonian systems. First, we study the property of the conditional Fredholm determinant, such as the Fréchet differentiability, the splittingness for the cyclic type symmetric solutions. Also, we generalize the Hill formula originally gotten by Hill and Poincaré. More precisely, let M be the monodromy matrix of the S-periodic orbits, then we get the formula relating the characteristic polynomial of the matrix SM and the conditional Fredhom determinant. Moreover, we study the relation of the conditional Fredholm determinant and the relative Morse index. Applications to the problem of linear stability for the S-periodic orbits are given.     

Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach

We are concerned with non-autonomous radially symmetric systems with a singularity, which are T-periodic in time. By the use of topological degree theory, we prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T. Precise estimates are then given in the case of Keplerian-like systems, showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system.     

A remark on the genericity of multiplicity results for forced oscillations on manifolds

We discuss the genericity of some multiplicity results for periodically perturbed autonomous first- and second-order ODEs on manifolds.?In particular, the genericity of the following property is investigated: if the differentiable manifold M is compact, then the equation _π=h(x,)+f(t,x,) on M has |χ(M)| geometrically distinct T-periodic solutions for any small enough T-periodic perturbing function f. Received: January 24, 2000; in final form: January 16, 2001?Published online: March 19, 2002     

On the determination of the number of periodic (or closed) solutions of a scalar differential equation with convexity

It is well known that a scalar differential equation , where f(t,x) is continuous, T-periodic in t and weakly convex or concave in x has no, one or two T-periodic solutions or a connected band of T-periodic solutions. The last possibility can be excluded if f(t,x) is strictly convex or concave for some t in the period interval. In this paper we investigate how the actual number of T-periodic solutions for a given equation of this type in principle can be determined, if f(t,x) is also assumed to have a continuous derivative . It turns out that there are three cases. In each of these cases we indicate the monotonicity properties and the domain of values for the function P(ξ)=S(ξ)−ξ, where S(ξ) is the Poincaré successor function. From these informations the actual number of periodic solutions can be determined, since a zero of P(ξ) represents a periodic solution.     

Stability of the geometric Ekeland variational principle: Convex case

In geometric terms, the Ekeland variational principle says that a lower-bounded proper lower-semicontinuous functionf defined on a Banach spaceX has a point (x ₀,f(x ₀)) in its graph that is maximal in the epigraph off with respect to the cone order determined by the convex coneK _λ = {(x, α) ∈X × ?:λ ∥x∥ ≤ ? α}, where λ is a fixed positive scalar. In this case, we write (x ₀,f(x ₀))∈λ-extf. Here, we investigate the following question: if (x ₀,f(x ₀))∈λ-extf, wheref is a convex function, and if 〈f _n 〉 is a sequence of convex functions convergent tof in some sense, can (x ₀,f(x ₀)) be recovered as a limit of a sequence of points taken from λ-extf _n ? The convergence notions that we consider are the bounded Hausdorff convergence, Mosco convergence, and slice convergence, a new convergence notion that agrees with the Mosco convergence in the reflexive setting, but which, unlike the Mosco convergence, behaves well without reflexivity.     

Periodic and almost periodic solutions of volterra integral differential equations with infinite memory

Conditions are given which guarantee that if T > 0 is sufficiently small, then x(t) = ∝₀^∞ [dE(s)] x(t — s)+ f(t) has a unique T-periodic solution x for each continuous T-periodic function f. The vectors x and f are n-dimensional; the matrix function E(s) is n × n with bounded total variation. The proof adapts readily to provide an analogous result when x and f are almost periodic functions whose non-zero Fourier frequencies are bounded away from zero. The results are applied to study certain perturbations of the above equation.     

Positive periodic solutions for a system of anisotropic parabolic equations

We consider a system of two degenerate parabolic equations with nonlocal terms and Dirichlet boundary conditions. More precisely, the degeneracy in each equation of the system is of the type r(x)-Laplacian where r(x) is a function depending on x∈Ω, where Ω is a bounded smooth domain of Rn. The system models the diffusion and the interaction between two different biological species sharing the same territory Ω. The paper provides conditions on the parameters of the problem that guarantee the coexistence of a T-periodic non-negative solution (u,v) with both non-trivial u,v.     

A generalized normal form and formal equivalence of two-dimensional systems with quadratic zero approximation: IV

We continue the study of invertible formal transformations of two-dimensional autonomous systems of differential equations with zero approximation represented by homogeneous polynomials of degree 2 and with perturbations in the form of power series without terms of order < 3. In the regular case, we consider systems that have the canonical form (αx ₁ ² ? sgnα x ₂ ² , x ₁ x ₂) with α ≠ 0 as the zero approximation. For such systems, we obtain resonance equations in closed form and use them to prove the theorem on the formal equivalence of systems and establish a generalized normal form to which any original system can be reduced by an invertible change of variables.     

Plane structures in thermal runaway

We consider the problem

1. u _t=u _xx+e ^u whenx ∈ ?,t > 0,
2. u(x, 0) =u ₀(x) whenx ∈ ?,

whereu ₀(x) is continuous, nonnegative and bounded. Equation (1) appears as a limit case in the analysis of combustion of a one-dimensional solid fuel. It is known that solutions of (1), (2) blow-up in a finite timeT, a phenomenon often referred to as thermal runaway. In this paper we prove the existence of blow-up profiles which are flatter than those previously observed. We also derive the asymptotic profile ofu(x, T) near its blow-up points, which are shown to be isolated.     

Functional differential equations and nonlinear semigroups in Lp-spaces

The autonomous nonlinear functional differential equation x(t) = F(x_t), t ? 0, x₀ = φ is studied as a semigroup of nonlinear operators in L_p function spaces. The method employed is to construct a semigroup of nonlinear operators which may be associated with the solutions of this equation. New existence and stability results are obtained for this equation by means of the semigroup approach.     

An inertial proximal scheme for nonmonotone mappings

We present an inertial proximal method for solving an inclusion involving a nonmonotone set-valued mapping enjoying some regularity properties. More precisely, we investigate the local convergence of an implicit scheme for solving inclusions of the type T(x)∋0 where T is a set-valued mapping acting from a Banach space into itself. We consider subsequently the case when T is strongly metrically subregular, metrically regular and strongly regular around a solution to the inclusion. Finally, we study the convergence of our algorithm under variational perturbations.     

Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for the first order neutral functional differential equation of the form

(x(t)+Bx(t−δ)^)′=g₁(t,x(t))+g₂(t,x(t−τ))+p(t).