In this paper, we prove the existence of non-negative solutions for a non-local higher order degenerate parabolic equation arising in the modelling of hydraulic fractures. The equation is similar to the well-known thin-film equation, but the Laplace operator is replaced by a Dirichlet-to-Neumann operator, corresponding to the square root of the Laplace operator on a bounded domain with Neumann boundary conditions (which can also be defined using the periodic Hilbert transform). In our study, we have to deal with the usual difficulties associated with higher order equations (e. g. a lack of maximum principle). However, there are important differences to, for instance, the thin-film equation. Firstly, our equation is non-local. Secondly the natural energy estimate is not as good as that of the thin-film equation, and does not yield, for instance, boundedness and continuity of the solutions (our equation is critical in dimension 1 in that respect).
To decompose the (2 + I)-dimensional Gardner equation, an isospectral problem and a corresponding hierarchy of (I + 1)-dimensional soliton equations are proposed. The (2 + 1)-dimensional Gardner equation is separated into the first two non-trivial (I + I)-dimensional soliton systems in the hierarchy, and in turn into two new compatible Hamiltonian systems of ordinary differential equations. Using the generating function flow method, the involutivity and the functional independence of the integrals are proved. The Abel-Jacobi coordinates are introduced to straighten out the associated flows. The Riemann-Jacobi inversion problem is discussed, from which quasi-periodic solutions of the (2 + I)-dimensional Gardner equation are obtained by resorting to the Riemann theta functions.
Under the influence of long-range attractive and short-range repulsive forces, thin liquid films rupture and form complex dewetting patterns. This paper studies this phenomenon in one space dimension within the framework of fourth-order degenerate parabolic equations of lubrication type. We derive the global structure of the bifurcation diagram for steady-state solutions. A stability analysis of the solution branches and numerical simulations suggest coarsening occurs. Furthermore, we study the behaviour of solutions in the limit that short-range repulsive forces are neglected. Both asymptotic analysis and numerical experiments show that this limit can concentrate mass in delta -distributions.
This paper is devoted to the study of limit laws of entrance times to cylinder sets for Cantor minimal systems of zero entropy using their representation by means of ordered Bratteli diagrams. We study in detail substitution subshifts and we prove these limit laws are piecewise linear functions. The same kind of results is obtained for classical low complexity systems given by non stationary ordered Bratteli diagrams.
We describe a numerical method for computing the linearized normal behaviour of an invariant curve of a diffeomorphism of R-n, n greater than or equal to 2. In the reducible case, the method computes not only the normal eigenvalues-either elliptic or hyperbolic-but also the corresponding eigendirections, that are the first-order approximation to the invariant manifolds (stable, unstable and central) around the curve. Moreover, the method seems to be able to detect the nonreducibility-if this is the case-of the linearized system. The input of the method is the invariant curve-including its rotation number-as well as a numerical procedure for computing the map and its differential. Hence, this method can be easily used on Poincare sections of ordinary differential equations. Due to the spectral character of the approximations used, the convergence of the process is very fast for sufficiently smooth cases. We note that the method is also valid for computing the normal behaviour of tori of higher dimensions. Finally, as examples, we study the stability of the invariant curves that appear in some concrete problems. In particular, we compute the unstable manifold for a given invariant curve of a six-dimensional symplectic map.
Piecewise linear differential equations in R-2 with a line of discontinuity and Z(2)-symmetry are investigated. Using the theory of differential inclusions and the method of point transformation a complete analysis is provided including uniqueness and non-uniqueness for the initial-value problem, stability and geometrical properties of equilibrium points, existence of sliding motion solutions, number and stability of periodic trajectories and existence of homoclinic solutions.
Grazing bifurcations are local bifurcations that can occur in dynamical models of impacting mechanical systems. The motion resulting from a grazing bifurcation can be complex. In this paper we discuss the creation of periodic orbits associated with grazing bifurcations, and we give sufficient conditions for the existence of a such a family of orbits. We also give a numerical example for an impacting system with one degree of freedom.
We consider the dynamics of singularly perturbed differential equations near points where the critical manifold has a transcritical or a pitchfork singularity. Our main tool is the recently developed blow-up method, which allows a detailed geometric analysis of such problems. A version of the Melnikov method to study transversality properties in this and related problems is developed.
Impacting systems are found in a great variety of mechanical constructions and they are intrinsically nonlinear. In this paper it is shown how near-grazing systems, i.e. systems in which the impacts take place at low speed, can be described by discrete mappings. The derivation of this mapping for a harmonic oscillator with a stop is dealt with in detail. It is found that the resulting mapping for rigid obstacles is somewhat different from those presented earlier in the literature. The derivations are extended to systems with a compliant obstacle. We find that the map for impacts with a compliant obstacle is very similar to the one describing collisions with a rigid obstacle. A notable difference is a change of scale of the bifurcation parameter. We illustrate our findings in the limit of large damping, where the mechanism of period adding can be analysed exactly. The relevance of our results to experiments on practical impact systems is indicated.
We introduce a discretization of the Lagrange-d'Alembert principle for Lagrangian systems with non-holonomic constraints, which allows us to construct numerical integrators that approximate the continuous flow. We study the geometric invariance properties of the discrete flow which provide an explanation for the good performance of the proposed method. This is tested on two examples: a non-holonomic particle with a quadratic potential and a mobile robot with fixed orientation.
A haptotaxis-dominated model of cell invasion is considered for small cell diffusion and fast protease adjustment to the cell-collagen matrix interaction. A simplified limit model has travelling wave cell invasion profiles that are blunt, that is end with a 'shock-like' step, and that evolve stably from initial data that lie to one side of some initial plane. In common with diffusion-dominated systems, the travelling wave which evolves from such initial data has the minimum wavespeed permissible in the model. This minimum wavespeed is not, however, determined by the local stability of the steady states in the travelling wave phase plane, but by a novel combination of singular behaviour within the phase plane and hyperbolic shock conditions. It is shown that more accurate models including the detailed fast dynamics of the protease require small amounts of diffusion (of the same order as the fast dynamics timescale) in order to remain stable. However, small diffusion and fast protease adjustment then give physically relevant and interesting solutions that evolve from semicompact initial data and stably invade at speeds well predicted by the simple model.
We study dynamical systems generated by skew products T : S-1 x R > S-1 x R T (x, y) = (lx, lambday + f (x)) where l greater than or equal to 2, 1/l < lambda < 1 and f is a C-2 function on S-1. We show that the SBR measure for T is absolutely continuous for almost every f.
Let f be a diffeomorphism of a compact finite-dimensional boundaryless manifold M exhibiting infinitely many coexisting attractors. Assume that each attractor supports a stochastically stable probability measure and that the union of the basins of attraction of each attractor covers Lebesgue almost all points of M. We prove that the time averages of almost all orbits under random perturbations are given by a finite number of probability measures. Moreover, these probability measures are close to the probability measures supported by the attractors when the perturbations are close to the original map f.
We establish local balance equations for smooth functions of the vorticity in the DiPerna-Majda weak solutions of two-dimensional (2D) incompressible Euler equations, analogous to the balance proved by Duchon and Robert for kinetic energy in three dimensions. The anomalous term or defect distribution therein corresponds to the 'enstrophy cascade' of 2D turbulence. It is used to define a rather natural notion of a 'dissipative Euler solution' in 2D. However, we show that the DiPerna-Majda solutions with vorticity in L-p for p > 2 are conservative and have zero defect. Instead, we must seek an alternative approach to dissipative solutions in 2D. If we assume an upper bound on the energy spectrum of 2D incompressible Navier-Stokes solutions by the Kraichnan-Batchelor k(-3) spectrum, uniformly for high Reynolds number, then we show that the zero viscosity limits of the Navier-Stokes solutions-exist, with vorticities in the zero-index Besov space B-2(0,infinity), and that these give a weak solution of the 2D incompressible Euler equations. We conjecture that for this class of weak solutions enstrophy dissipation may indeed occur, in a sense which is made precise.
In this paper we report on numerical studies of the Cauchy problem for equivariant wave maps from (2 + 1)-dimensional Minkowski spacetime into the 2-sphere. Our results provide strong evidence for the conjecture that large-energy initial data develop singularities in finite time and that singularity formation has the universal form of adiabatic shrinking of the degree-one harmonic map from R-2 into S-2.
We provide a general framework to construct integrable mappings of the plane that preserve a one-parameter family B(x, y, K) of biquadratic invariant curves where parametrization by K is very general. These mappings are reversible by construction (i.e. they are the composition of two involutions) and can be shown to be measure preserving. They generalize integrable maps previously given by McMillan and Quispel, Roberts and Thompson. By considering a transformation of the case of the symmetric biquadratic to a canonical form, we provide a normal form for the symmetric integrable map acting on each invariant curve. We give a Lax pair for a large subclass of our symmetric integrable maps, including at least a 10-parameter subfamily of the 12-parameter symmetric Quispel-Roberts-Thompson maps.
We show how to link topological tools with a local hyperbolic behaviour to prove the existence of homoclinic and heteroclinic trajectories for a map. We apply this technique for the Henon map h with classical parameter values (a = 1.4, b = 0.3). For this map we give a computer-assisted proof of the existence of an infinite number of homoclinic and heteroclinic trajectories. We also introduce the method for computation of the lower bound of the topological entropy of a map based on the covering relations involving different iterations of the map and we prove that the topological entropy of h is larger than 0.3381.
We construct the soliton solutions for N = 1 supersymmetric KdV equations. Our starting point is the bilinear transcription of the latter using the super-bilinear operators. We show explicitly the form of the two- and three-soliton solution and give the procedure for constructing the higher ones. The main difference of those solitons with the classical case is that their fermionic part becomes dressed through the interaction. Our approach allows us to compute this dressing in an explicit way.