Time Periodic Solutions to the One-Dimensional Nonlinear Wave Equation

This paper is concerned with the time periodic solutions to the one-dimensional nonlinear wave equation with either variable or constant coefficients. By adjusting the basis of L ² function space, we can circumvent the difficulties caused by η _u  = 0 and obtain the existence of a weak periodic solution, which was posed as an open problem by Baubu and Pavel in (Trans Am Math Soc 349:2035–2048, 1997). Finally, an application to the forced Sine-Gordon equation is presented to illustrate the utility of this technique.     

Removing Collision Singularities from Action Minimizers for the N-Body Problem with Free Boundaries

For the planar and spatial N-body problems, it has been proved by Marchal and Chenciner that solutions for the minimizing problem with fixed ends are free from interior collisions. This important result has been extended by Ferrario & Terracini to Newtonian-type problems and equivariant problems. It has also been used to construct many symmetric solutions for the N-body problem. In this paper we are interested in action minimizing solutions in function spaces with free boundaries. The function spaces are imposed with boundary conditions, such that every mass point starts and ends on two transversal proper subspaces of ℝ^d, d≥2. We will prove that solutions for this minimizing problem with free boundaries are always free from collisions, including boundary collisions. This result can be used to construct certain classes of relative periodic solutions of the N-body problem.     

Orbits Homoclinic to Exponentially Small Periodic Orbits for a Class of Reversible Systems. Application to Water Waves

In this paper a class of reversible analytic vector fields is investigated near an equilibrium. For these vector fields, the part of the spectrum of the differential at the equilibrium which lies near the imaginary axis comes from the perturbation of a double eigenvalue 0 and two simple eigenvalues , . In the first part of this paper, we study the 4-dimensional problem. The existence of a family of solutions homoclinic to periodic orbits of size less than μ^N for any fixed N, where μ is the bifurcation parameter, is known for vector fields. Using the analyticity of the vector field, we prove here the existence of solutions homoclinic to a periodic orbit the size of which is exponentially small ( of order . This result receives its significance from the still unsolved question of whether there exist solutions that are homoclinic to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum. In the second part of this paper we prove that the exponential estimates still hold in infinite dimensions. This result cannot be simply obtained from the study of the 4-dimensional analysis by a center-manifold reduction since this result is based on analyticity of the vector field. One example of such a vector field in infinite dimensions occurs when describing the irrotational flow of an inviscid fluid layer under the influence of gravity and small surface tension (Bond number ) for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalized solitary wave. Our work shows that there exist generalized solitary waves with exponentially small oscillations at infinity. More precisely, we prove that for each F close enough to 1, there exist two reversible solutions homoclinic to a periodic orbit, the size of which is less than , l being any number between 0 and π and satisfying . (Accepted October 2, 1995)     

Minimizing Periodic Orbits with Regularizable Collisions in the <Emphasis Type="Italic">n</Emphasis>-Body Problem

In the Newtonian n-body problem, there are various subsystems with two degrees of freedom, such as the collinear three-body problem and the isosceles three-body problem. After we determine a normal form of the Lagrangians of these subsystems, we prove the existence of periodic solutions with regularizable collisions for these systems. Our result includes several examples, such as Schubart’s orbit with or without equal masses, among others.     

Symmetric Trajectories for the 2N-Body Problem with Equal Masses

We consider the problem of 2N bodies of equal masses in for the Newtonian-like weak-force potential r ^−σ, and we prove the existence of a family of collision-free nonplanar and nonhomographic symmetric solutions that are periodic modulo rotations. In addition, the rotation number with respect to the vertical axis ranges in a suitable interval. These solutions have the hip-hop symmetry, a generalization of that introduced in [19], for the case of many bodies and taking account of a topological constraint. The argument exploits the variational structure of the problem, and is based on the minimization of Lagrangian action on a given class of paths.     

Application of simple,periodic homogenization techniques to non-linear heat conduction problems in non-periodic,porous media

Often, detailed simulations of heat conduction in complicated, porous media have large runtimes. Then homogenization is a powerful tool to speed up the calculations by preserving accurate solutions at the same time. Unfortunately real structures are generally non-periodic, which requires unpractical, complicated homogenization techniques. We demonstrate in this paper, that the application of simple, periodic techniques to realistic media, that are just close to periodic, gives accurate, approximative solutions. In order to obtain effective parameters for the homogenized heat equation, we have to solve a so called “cell problem”. In contrast to periodic structures it is not trivial to determine a suitable unit cell, which represents a non-periodic media. To overcome this problem, we give a rule of thumb on how to choose a good cell. Finally we demonstrate the efficiency of our method for virtually generated foams as well as real foams and compare these results to periodic structures.     

Complex order van der Pol oscillator

In this paper a complex-order van der Pol oscillator is considered. The complex derivative D^a±jbD^{\alpha\pm\jmath\beta}, with α,β∈R ⁺ is a generalization of the concept of integer derivative, where α=1, β=0. By applying the concept of complex derivative, we obtain a high-dimensional parameter space. Amplitude and period values of the periodic solutions of the two versions of the complex-order van der Pol oscillator are studied for variation of these parameters. Fourier transforms of the periodic solutions of the two oscillators are also analyzed.     

On the motion of a gyrostat similar to Lagrange’s gyroscope under the influence of a gyrostatic moment vector

In this paper, the problem of the motion of a gyrostat fixed at one point under the action of a gyrostatic moment vector whose components are ℓ _i (i=1,2,3) about the axes of rotation, similar to a Lagrange gyroscope is investigated. We assume that the center of mass G of this gyrostat is displaced by a small quantity relative to the axis of symmetry, and that quantity is used to obtain the small parameter ε (Elfimov in PMM, 42(2):251–258, [1978]). The equations of motion will be studied under certain initial conditions of motion. The Poincaré small parameter method (Malkin in USAEC, Technical Information Service, ABC. Tr-3766, [1959]; Nayfeh in Perturbation methods, Wiley-Interscience, New York, [1973]) is applied to obtain the periodic solutions of motion. The periodic solutions for the case of irrational frequencies ratio are given. The periodic solutions are analyzed geometrically using Euler’s angles to describe the orientation of the body at any instant t of time. These solutions are performed by our computer programs to get their graphical representations.     

A proof of the Benjamin-Feir instability

The existence and linear stability problem for the Stokes periodic wavetrain on fluids of finite depth is formulated in terms of the spatial and temporal Hamiltonian structure of the water-wave problem. A proof, within the Hamiltonian framework, of instability of the Stokes periodic wavetrain is presented. A Hamiltonian center-manifold analysis reduces the linear stability problem to an ordinary differential eigenvalue problem on ℝ⁴. A projection of the reduced stability problem onto the tangent space of the 2-manifold of periodic Stokes waves is used to prove the existence of a dispersion relation Λ(λ,σ, I ₁, I ₂)=0 where λ ε ℂ is the stability exponent for the Stokes wave with amplitude I ₁ and mass flux I ₂ and σ is the “sideband’ or spatial exponent. A rigorous analysis of the dispersion relation proves the result, first discovered in the 1960's, that the Stokes gravity wavetrain of sufficiently small amplitude is unstable for F ε (0,F₀) where F ₀ ≈ 0.8 and F is the Froude number.     

Viscosity Solutions of a Degenerate Parabolic-Elliptic System Arising in the Mean-Field Theory of Superconductivity

In a Type‐II superconductor the magnetic field penetrates the superconducting body through the formation of vortices. In an extreme Type‐II superconductor these vortices reduce to line singularities. Because the number of vortices is large it seems feasible to model their evolution by an averaged problem, known as the mean-field model of superconductivity. We assume that the evolution law of an individual vortex, which underlies the averaging process, involves the current of the generated magnetic field as well as the curvature vector. In the present paper we study a two‐dimensional reduction, assuming all vortices to be perpendicular to a given direction. Since both the magnetic field H and the averaged vorticity ω are curl‐free, we may represent them via a scalar magnetic potential q and a scalar stream function ψ, respectively. We study existence, uniqueness and asymptotic behaviour of solutions (ψ, q) of the resulting degenerate elliptic‐parabolic system (with curvature taken into account or not) by means of viscosity and weak solutions. In addition we relate (ψ, q) to solutions (ω, H) of the mean‐field equations without curvature. Finally we construct special solutions of the corresponding stationary equations with two or more superconducting phases. (Accepted August 8, 1997)     

Rigorous Results in Steady Finger Selection in Viscous Fingering

This paper concerns the existence of a steadily translating finger solution in a Hele-Shaw cell for small but non-zero surface tension (ɛ²). Though there are numerous numerical and formal asymptotic results for this problem, we know of no mathematically rigorous results that address the selection problem. We rigorously conclude that for relative finger width λ in the range , with small, analytic symmetric finger solutions exist in the asymptotic limit of surface tension if and only if the Stokes constant for a relatively simple nonlinear differential equation is zero. This Stokes constant S depends on the parameter and earlier calculations by a number of authors have shown it to be zero for a discrete set of values of a. The methodology consists of proving the existence and uniqueness of analytic solutions for a weak half-strip problem for any λ in a compact subset of (0, 1). The weak problem is shown to be equivalent to the original finger problem in the function space considered, provided we invoke a symmetry condition. Next, we consider the behavior of the solution in a neighborhood of an appropriate complex turning point for the restricted case , for some . This turning point accounts for exponentially small terms in ɛ, as ɛ→0⁺ that generally violate the symmetry condition. We prove that the symmetry condition is satisfied for small ɛ when the parameter a is constrained appropriately. (Accepted July 4, 2002 Published online January 15, 2003) Communicated by F. OTTO     

Second-order nonlinear differential equations with infinite set of periodic solutions

For the differential equation u″ = f(t, u, u′), where the function f: R × R ² → R is periodic in the first variable and f (t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic solutions are found. Published in Neliniini Kolyvannya, Vol. 11, No. 4, pp. 495–500, October–December, 2008.     

Nonlinear Stability of Homogeneous Models in Newtonian Cosmology

We consider the Vlasov‐Poisson system in a cosmological setting as studied in [18] and prove nonlinear stability of homogeneous solutions against small, spatially periodic perturbations in the L ^∞‐norm of the spatial mass density. This result is connected with the question of how large scale structures such as galaxies have evolved out of the homogeneous state of the early universe. (Accepted June 28, 1996)     

Sobolev Periodic Solutions of Nonlinear Wave Equations in Higher Spatial Dimensions

We prove the existence of Cantor families of periodic solutions for nonlinear wave equations in higher spatial dimensions with periodic boundary conditions. We study both forced and autonomous PDEs. In the latter case our theorems generalize previous results of Bourgain to more general nonlinearities of class C ^k and assuming weaker non-resonance conditions. Our solutions have Sobolev regularity both in time and space. The proofs are based on a differentiable Nash–Moser iteration scheme, where it is sufficient to get estimates of interpolation-type for the inverse linearized operators. Our approach works also in presence of very large “clusters of small divisors”.     

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.     

Homogenization and Enhancement for the G—Equation

We consider the so-called G-equation, a level set Hamilton–Jacobi equation used as a sharp interface model for flame propagation, perturbed by an oscillatory advection in a spatio-temporal periodic environment. Assuming that the advection has suitably small spatial divergence, we prove that, as the size of the oscillations diminishes, the solutions homogenize (average out) and converge to the solution of an effective anisotropic first-order (spatio-temporal homogeneous) level set equation. Moreover, we obtain a rate of convergence and show that, under certain conditions, the averaging enhances the velocity of the underlying front. We also prove that, at scale one, the level sets of the solutions of the oscillatory problem converge, at long times, to the Wulff shape associated with the effective Hamiltonian. Finally, we also consider advection depending on position at the integral scale.     

Global stabilization of periodic orbits using a proportional feedback control with pulses

We investigate the stabilization of periodic orbits of one-dimensional discrete maps by using a proportional feedback method applied in the form of pulses. We determine a range of the parameter μ values representing the strength of the feedback for which all positive solutions of the controlled equation converge to a periodic orbit.     

Univalent σ-Harmonic Mappings

We study mappings from ℝ² into ℝ² whose components are weak solutions to the elliptic equation in divergence form, div (σ∇u)= 0, which we call σ-harmonic mappings. We prove sufficient conditions for the univalence, i.e., injectivity, of such mappings. Moreover we prove local bounds in BMO on the logarithm of the Jacobian determinant of such univalent mappings, thus obtaining the a.e. nonvanishing of their Jacobian. In particular, our results apply to σ-harmonic mapping associated with any periodic structure and therefore they play an important role in homogenization. Accepted October 30, 2000?Published online April 23, 2001     

Existence,uniqueness and asymptotics of steady jets

We consider the three-dimensional flow through an aperture in a plane either with a prescribed flux or pressure drop condition. We discuss the existence and uniqueness of solutions for small data in weighted spaces and derive their complete asymptotic behaviour at infinity. Moreover, we show that each solution with a bounded Dirichlet integral, which has a certain weak additional decay, behaves like O(r ⁻²) as r=|x|→∞ and admits a wide jet region. These investigations are based on the solvability properties of the linear Stokes system in a half space ℝ ₊ ³ . To investigate the Stokes problem in ℝ ₊ ³ , we apply the Mellin transform technique and reduce the Stokes problem to the determination of the spectrum of the corresponding invariant Stokes-Beltrami operator on the hemisphere.     

Concerning the W k,p -Inviscid Limit for 3-D Flows Under a Slip Boundary Condition

We consider the 3-D evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, see (1.2), and study the problem of the strong convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We prove here, in the flat boundary case, convergence in Sobolev spaces W ^k, p (Ω), for arbitrarily large k and p (for previous results see Xiao and Xin in Comm Pure Appl Math 60:1027–1055, 2007 and Beir?o da Veiga and Crispo in J Math Fluid Mech, 2009, doi:). However this problem is still open for non-flat, arbitrarily smooth, boundaries. The main obstacle consists in some boundary integrals, which vanish on flat portions of the boundary. However, if we drop the convective terms (Stokes problem), the inviscid, strong limit result holds, as shown below. The cause of this different behavior is quite subtle. As a by-product, we set up a very elementary approach to the regularity theory, in L ^p -spaces, for solutions to the Navier–Stokes equations under slip type boundary conditions.