52,791 research outputs found
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented
Nonlinear Regularizing Effect for Conservation Laws
20 pagesInternational audienceCompactness of families of solutions --- or of approximate solutions --- is a feature that distinguishes certain classes of nonlinear hyperbolic equations from the case of linear hyperbolic equations, in space dimension one. This paper shows that some classical compactness results in the context of hyperbolic conservation laws, such as the Lax compactness theorem for the entropy solution semigroup associated with a nonlinear scalar conservation laws with convex flux, or the Tartar-DiPerna compensated compactness method, can be turned into quantitative compactness estimates --- in terms of epsilon-entropy, for instance --- or even nonlinear regularization estimates. This regularizing effect caused by the nonlinearity is discussed in detail on two examples: a) the case of a scalar conservation law with convex flux, and b) the case of isentropic gas dynamics, in space dimension one
Two-scale convergence for locally-periodic microstructures and homogenization of plywood structures
The introduced notion of locally-periodic two-scale convergence allows to
average a wider range of microstructures, compared to the periodic one. The
compactness theorem for the locally-periodic two-scale convergence and the
characterisation of the limit for a sequence bounded in are
proven. The underlying analysis comprises the approximation of functions, which
periodicity with respect to the fast variable depends on the slow variable, by
locally-periodic functions, periodic in subdomains smaller than the considered
domain, but larger than the size of microscopic structures. The developed
theory is applied to derive macroscopic equations for a linear elasticity
problem defined in domains with plywood structures.Comment: 22 pages, 4 figure
Alternative results and robustness for fractional evolution equations with periodic boundary conditions
In this paper, we study periodic boundary value problems for a class of linear fractional evolution equations involving the Caputo fractional derivative. Utilizing compactness of the constructed evolution operators and Fredholm alternative theorem, some interesting alternative results for the mild solutions are presented. Periodic motion controllers that are robust to parameter drift are also designed for given a periodic motion. An example is given to illustrate the results
A note on the fractional Cauchy problems with nonlocal initial conditions
AbstractOf concern is the Cauchy problems for fractional integro-differential equations with nonlocal initial conditions. Using a new strategy in terms of the compactness of the semigroup generated by the operator in the linear part and approximating technique, a new existence theorem for mild solutions is established. An application to a fractional partial integro-differential equation with a nonlocal initial condition is also considered
Symmetries and global solvability of the isothermal gas dynamics equations
We study the Cauchy problem associated with the system of two conservation
laws arising in isothermal gas dynamics, in which the pressure and the density
are related by the -law equation with
. Our results complete those obtained earlier for . We
prove the global existence and compactness of entropy solutions generated by
the vanishing viscosity method. The proof relies on compensated compactness
arguments and symmetry group analysis. Interestingly, we make use here of the
fact that the isothermal gas dynamics system is invariant modulo a linear
scaling of the density. This property enables us to reduce our problem to that
with a small initial density. One symmetry group associated with the linear
hyperbolic equations describing all entropies of the Euler equations gives rise
to a fundamental solution with initial data imposed to the line . This
is in contrast to the common approach (when ) which prescribes
initial data on the vacuum line . The entropies we construct here are
weak entropies, i.e. they vanish when the density vanishes. Another feature of
our proof lies in the reduction theorem which makes use of the family of weak
entropies to show that a Young measure must reduce to a Dirac mass. This step
is based on new convergence results for regularized products of measures and
functions of bounded variation.Comment: 29 page
Solitary waves for linearly coupled nonlinear Schrodinger equations with inhomogeneous coefficients
Motivated by the study of matter waves in Bose-Einstein condensates and
coupled nonlinear optical systems, we study a system of two coupled nonlinear
Schrodinger equations with inhomogeneous parameters, including a linear
coupling. For that system we prove the existence of two different kinds of
homoclinic solutions to the origin describing solitary waves of physical
relevance. We use a Krasnoselskii fixed point theorem together with a suitable
compactness criterion.Comment: 16 page
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