Cette thèse a été préparée à l'Institut de Mécanique Céleste et de Calcul des Ephémérides de l'Observatoire de Paris, de septembre 2001 à avril 2005, sous la direction de MM. Alain Chenciner et Alain Albouy. Elle traite du problème des N corps, qui consiste en l'étude des solutions des équations de Newton. Celles-ci décrivent le mouvement de N particules ponctuelles en interaction gravitationnelle. Cette thèse a plus précisément pour objet l'étude des solutions homographiques (les rapports entre les distances mutuelles sont constants) associées à des masses de signe quelconque. On étudie le cas des mouvements rigides (les distances mutuelles sont constantes). Ce problème est plus difficile que le problème à masses positives, car il n'est plus possible d'associer un produit scalaire aux masses.<br /><br />On s'intéresse au cas où la somme des masses est nulle. Le centre d'inertie devient alors un vecteur, invariant par translation. Ceci rend les équations de Newton "plus intégrables". Ainsi, sous une hypothèse sur les vitesses initiales, le problème colinéaire des trois corps devient intégrable. Cette propriété permet de calculer les configurations centrales (configurations qui engendrent un effondrement homothétique sur un centre) pour des masses x, -x, y, -y.<br /><br />On applique une propriété des équilibres absolus à somme des masses nulle au problème des chorégraphies. Une chorégraphie est une solution dans laquelle les corps se suivent sur la même courbe avec des intervalles de temps égaux. On montre que, pour le potentiel logarithmique, les masses d'une chorégraphie sont nécessairement égales.

This dissertation deals with a class of nonlinear wave equations of the type discovered by R. Camassa and D. D. Holm which includes the Camassa-Holm, the Degasperis-Procesi, and the two component Camassa-Holm equations. All these equations admit certain non-smooth soliton-like solutions, called peakons as well as other non-smooth solutions like cuspons. We apply the techniques of the theory of distributions of L. Schwartz to study these solutions. In particular, every non-smooth traveling wave which is a distributional solution of the two component Camassa-Holm equation is a distributional solution of the Camassa-Holm equation if the set of points where the height of the wave equals its speed, is of measure zero. This includes peakon or cuspon traveling wave solutions.<p>We also develop a suitable modification of the classical Lax pair formalism to deal with singular solutions. We show that the Lax pair formalism can be extended to a distributional weak Lax pair which is appropriate for dealing with the peakon solutions of the Camassa-Holm equation.

This dissertation deals with a class of nonlinear wave equations of the type discovered by R. Camassa and D. D. Holm which includes the Camassa-Holm, the Degasperis-Procesi, and the two component Camassa-Holm equations. All these equations admit certain non-smooth soliton-like solutions, called peakons as well as other non-smooth solutions like cuspons. We apply the techniques of the theory of distributions of L. Schwartz to study these solutions. In particular, every non-smooth traveling wave which is a distributional solution of the two component Camassa-Holm equation is a distributional solution of the Camassa-Holm equation if the set of points where the height of the wave equals its speed, is of measure zero. This includes peakon or cuspon traveling wave solutions.<p>We also develop a suitable modification of the classical Lax pair formalism to deal with singular solutions. We show that the Lax pair formalism can be extended to a distributional weak Lax pair which is appropriate for dealing with the peakon solutions of the Camassa-Holm equation.

Thesis (Ph. D.)--University of Missouri--Rolla, 1993. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed January 25, 2010) Includes bibliographical references (p. 150-155).

Thesis (Ph. D.)--Florida State University, 2002. / Advisor: Dr. Joseph B. Schlenoff, Florida State University, College of Arts and Sciences, Dept. of Chemistry. Title and description from dissertation home page (viewed Sept. 29, 2003). Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis--M. Sc.(Eng.), University of Hong Kong. / Mimeographed.

The first part of this thesis is devoted to a regularity criterion for solutions of the Incompressible Navier-Stokes equations in terms of regularity of the solutions along the streamlines. More precisely, we prove that we can ensure the full regularity of a given suitable weak solution provided we have good control on the second derivative of the velocity along the direction of the streamlines of the fluid. In the second part of this thesis, we will show that the global regularity of a suitable weak solution u for the incompressible Navier-Stokes equations holds under the condition that [mathematical equation] is integrable in space time variables. / text

In this thesis, we establish the asymptotic analysis of the singularly perturbed reaction diffusion equation [cataloger unable to transcribe mathematical equations].... Our results establish the specific dependency on the coefficients of this equation and the size of the parameter [delta] with respect to [epsilon]. The analyses include equation subject to Dirichlet and Neumann boundary conditions. In both cases, the solutions u[superscript epsilon] converge locally uniformally to the equilibria of the reaction term f. We characterize the limiting behavior of the solutions through the viscosity solution of a variational inequality. To construct the coefficients defining the variational inequality, we apply concepts developed for the homogenization of elliptic operators. In chapter two, we derive the convergence results in the Neumann case. The third chapter is dedicated to the analysis of the Dirichlet case. / text

In this dissertation, we study a free boundary problem obtained as a limit as [epsilon omplies 0] to the following regularizing family of semilinear equations [Delta]u = [Beta subscript epsilon](u)F([gradient]u), where [Beta subscript epsilon] approximates the Dirac delta in the origin and F is a Lipschitz function bounded away from 0 and infinity. The least supersolution approach is used to construct solutions satisfying geometric properties of the level surfaces that are uniform. This allows to prove that the free boundary of the limit has the "right" weak geometry, in the measure theoretical sense. By the construction of some barriers with curvature, the classification of global profiles for the blow-up analysis is carried out and the limit function is proven to be a viscosity and pointwise solution (a.e) to a free boundary problem. Finally, the free boundary is proven to be a C[superscript 1, alpha] surface around H[superscript n-1] a.e. point.