We propose a time discretization of the Navier-Stokes equations inspired by the theory of gradient flows. This discretization produces Leray/Hopf solutions in any dimension and suitable solutions in dimension 3. We also show that in dimension 3 and for initial datum in H, the scheme converges to strong solutions in some interval [0, T ) and, if the datum satisfies the classical smallness condit...

The variational formulations of guiding-center Vlasov-Maxwell theory based on Lagrange, Euler, and Euler-Poincaré variational principles are presented. Each variational principle yields a different approach to deriving guiding-center polarization and magnetization effects into the guiding-center Maxwell equations. The conservation laws of energy, momentum, and angular momentum are also derived ...

Metric regularity is a central concept in variational analysis for the study of solution mappings associated with “generalized equations”, including variational inequalities and parameterized constraint systems. Here it is employed to characterize the distance to irregularity or infeasibility with respect to perturbations of the system structure. Generalizations of the Eckart-Young theorem in n...

This study aims at the theoretical development of a method of “four-dimensional analysis,” namcly the numerical variational analysis. The three basic types of variational formalism in the nurncrical variational analysis method are discussed. The basic formalisms are categorized into three areas: (1) “timewise localized” formalism, ( 2 ) formalism with strong constraint, and (3) formalism with w...

the purpose of this study is to implement a new modification of the variational iteration method (h-s-vim), which is a combination of spectral method and variational iteration method for heat transfer problems with high nonlinearity order. the merit of this method is that it does not require the solution of any linear or nonlinear system of equations unlike spectral method. furthermore the prop...

We consider propagating, spatially localized waves in a class of equations that contain variational and nonvariational terms. The dynamics of the waves is analyzed through a collective coordinate approach. Motivated by the variational approximation, we show that there is a natural choice of projection onto collective variables for reducing the governing (nonlinear) partial differential equation...

In this paper we introduce a new method of spatio-temporal discretization for partial differential equations in variational form. This method generalizes the method of Marsden et al in that it uses a systematic approach to discrete jet spaces based on the finite element method. The resulting method is used to derive integrators for the Nonlinear Schrödinger (NLS) equation which exhibit superior...