In this paper we establish a number of implications between various qualitative and quantitative versions of the global regularity problem for the Navier-Stokes equations, in the periodic, smooth finite energy, smooth H, Schwartz, or mild H categories, and with or without a forcing term. In particular, we show that if one has global well-posedness in H for the periodic Navier-Stokes problem wit...

The N-particle free fermion state for quantum particles in the plane subject to a perpendicular magnetic field, and with doubly periodic boundary conditions, is written in a product form. The absolute value of this is used to formulate an exactly solvable one-component plasma model, and further motivates the formulation of an exactly solvable two-species Coulomb gas. The large N expansion of th...

Doubly periodic (periodic both in time and in space) solutions for the Lagrange–Euler equation of the (1 + 1)–dimensional scalar φ theory are considered. The nonlinear term is assumed to be small, and the Poincaré– Lindstedt method is used to find asymptotic solutions in the standing wave form. The principal resonance problem, which arises for zero mass, is solved if the leading-order term is t...

so that we are dealing with an asymptotically linear problem. The forcing term p is T -periodic, and we are interested in T -periodic solutions of (1). We assume that the linear part of the equation is nonresonant, that is a 6= 0 and if c = 0 then a 6= (2πm/T ) for all integer m. Our result shows that for generic forcing term p, when the parameter λ, which measures the strength of the forcing, ...

For 2-dimensional dislocation dynamics (DD) simulations under periodic boundary conditions (PBC) in both directions, the summation of the periodic image stress fields is found to be conditionally convergent. For example, different stress fields are obtained depending on whether the summation in the x-direction is performed before or after the summation in the y-direction. This problem arises be...

The principal term in the asymptotic expansion of the variance of the periodic measure of a ball in R under uniform random shift is proportional to the (d + 1)st power of the grid scaling factor. This result remains valid for a bounded set in R with sufficiently smooth isotropic covariogram under a uniform random shift and an isotropic rotation, and the asymptotic term is proportional also to t...

In a mechanical system x is the displacement and / is the time. We call g(x) a restoring force and ƒ(/) a forcing term. Throughout this paper f{t) will be periodic and when we speak of a periodic solution of (1) we always mean a solution having the same period as ƒ(/). If g(x) is simply a constant multiplied by x, equation (1) represents a linear oscillator. In this case the amplitude of the fo...

In this paper‎, two inverse problems of determining an unknown source term in a parabolic‎ equation are considered‎. ‎First‎, ‎the unknown source term is ‎estimated in the form of a combination of Chebyshev functions‎. ‎Then‎, ‎a numerical algorithm based on Chebyshev polynomials is presented for obtaining the solution of the problem‎. ‎For solving the problem‎, ‎the operational matrices of int...

This work is concerned with the identification problem for what we call the perturbation term or error term in a parabolic partial differential equation, through its approximate periodic solutions. The observation is made over a subregion of the physical domain. The existence and uniqueness problem of the approximate periodic solutions is studied in the first part of the paper. A solution to th...