Solutions to some open problems in Fluid dynamics

Abstract

Let u = u(x, t, u0) represent the global solution of the initial value problem for the one-dimensional fluid dynamics equationwhere α > 0, β ≥ 0, γ ≥ 0, δ ≥ 0 and ε ≥ 0 are constants. This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations.The nonlinear function satisfies the conditions f (0) = 0, |f (u)| → ∞ as |u| → ∞, and f ∈ C 1 (R), and there exist the following limitsSuppose that the initial function u0 ∈ L 1 (R) ∩ H 2 (R). By using energy estimates, Fourier transform, Plancherel's identity, upper limit estimate, lower limit estimate and the results of the linear problemthe author justifies the following limits (with sharp rates of decay)where 0!! = 1, 1!! = 1 and m!! = 1 · 3 · · · · (2m − 3) · (2m − 1). Moreoverif the initial function u0(x) = ρ0 (x), for some function ρ0 ∈ C 1 (R) ∩ L 1 (R) and R ρ0(x)dx = 0.

New results of general n-dimensional incompressible Navier–Stokes equations

“…Very recently, the author obtained some exact limits for the global solutions of some dissipative partial differential equations. See [45,46].…”