scholarly journals Jean Leray. 7 November 1906 — 10 November 1998

2006 ◽  
Vol 52 ◽  
pp. 137-148
Author(s):  
Martin Andler

Jean Leray was one of the major mathematicians of the twentieth century. His primary focus in mathematics came from applications; indeed, a majority of his contributions were in the theory of partial differential equations arising in physics, notably his 1934 paper on the Navier–Stokes equation. World War II, during which he was a prisoner–of–war in Austria for five years, induced him to turn to pure mathematics to avoid helping the German war effort. There he worked in topology, developing two radically new ideas: sheaf theory and spectral sequences. After 1950 he came back to partial differential equations and became interested in complex analysis, writing a remarkable series of papers on the Cauchy problem. Leray remained mathematically active until the end of his life; in the course of his career he wrote 132 papers. His influence on present mathematics is tremendous. On the one hand, sheaf theory and spectral sequences became essential tools in contemporary pure mathematics, reaching far beyond their initial scope in topology. On the other hand, Leray can rightly be considered the intellectual guide of the distinguished French school of applied mathematics.

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tamaz Vashakmadze

Abstract The basic problem of satisfaction of boundary conditions is considered when the generalized stress vector is given on the surfaces of elastic plates and shells. This problem has so far remained open for both refined theories in a wide sense and hierarchic type models. In the linear case, it was formulated by I. N. Vekua for hierarchic models. In the nonlinear case, bending and compression-expansion processes do not split and in this context the exact structure is presented for the system of differential equations of von Kármán–Mindlin–Reisner (KMR) type, constructed without using a variety of ad hoc assumptions since one of the two relations of this system in the classical form is the compatibility condition, but not the equilibrium equation. In this paper, a unity mathematical theory is elaborated in both linear and nonlinear cases for anisotropic inhomogeneous elastic thin-walled structures. The theory approximately satisfies the corresponding system of partial differential equations and the boundary conditions on the surfaces of such structures. The problem is investigated and solved for hierarchic models too. The obtained results broaden the sphere of applications of complex analysis methods. The classical theory of finding a general solution of partial differential equations of complex analysis, which in the linear case was thoroughly developed in the works of Goursat, Weyl, Walsh, Bergman, Kolosov, Muskhelishvili, Bers, Vekua and others, is extended to the solution of basic nonlinear differential equations containing the nonlinear summand, which is a composition of Laplace and Monge–Ampére operators.


Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1547
Author(s):  
Stephen C. Anco ◽  
Bao Wang

A geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. Additionally, for systems of evolution equations, adjoint-symmetries are shown to have another geometrical formulation given by one-forms that are invariant under the flow generated by the system on the solution space. This result is generalized to systems of evolution equations with spatial constraints, where adjoint-symmetry one-forms are shown to be invariant up to a functional multiplier of a normal one-form associated with the constraint equations. All of the results are applicable to the PDE systems of interest in applied mathematics and mathematical physics.


2004 ◽  
Vol 01 (03) ◽  
pp. 265-284 ◽  
Author(s):  
XAVIER GRÀCIA ◽  
MIGUEL C. MUÑOZ-LECANDA ◽  
NARCISO ROMÁN-ROY

In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular differential equations, and partial differential equations in field theories. The geometric structures underlying these systems are presented and commented on. The main results concerning these structures are stated and discussed, as well as their influence on the study of the differential equations with which they are related. In addition, research to be developed in these areas is also commented on.


2018 ◽  
Author(s):  
Steven G. Krantz ◽  
Estela A. Gavosto ◽  
Marco M. Peloso

2005 ◽  
Vol 07 (05) ◽  
pp. 553-582 ◽  
Author(s):  
YURI BAKHTIN ◽  
JONATHAN C. MATTINGLY

We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier–Stokes equation and stochastic Ginsburg–Landau equation.


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