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Solutions to Some Open Problems in Fluid Dynamics

Authors:

Linghai ZHANG

Institution:

Department of Mathematics, Lehigh University, 14 East Packer Avenue, Bethlehem, Pennsylvania 18015, USA

Abstract:

Let u = u(x, t, u ₀) represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation

$u_t - \varepsilon u_{xxt} + \delta u_x + \gamma Hu_{xx} + \beta u_{xxx} + f(u)_x = \alpha u_{xx} ,u(x,0) = u_0 (x),$

where 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 limits

$L_0 = \mathop {lim sup}\limits_{u \to 0} \frac{{f(u)}} {{u^3 }}andL_\infty = \mathop {lim sup}\limits_{u \to \infty } \frac{{f(u)}} {{u^5 }}.$

. Suppose that the initial function u ₀ ∈ 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 problem

$v_t - \varepsilon v_{xxt} + \delta v_x + \gamma Hv_{xx} + \beta v_{xxx} = \alpha v_{xx} ,v(x,0) = v_0 (x),$

the author justifies the following limits (with sharp rates of decay)

$\mathop {\lim }\limits_{t \to \infty } \left {(1 + t)^{m + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \smallint _\mathbb{R} |u_{x^m } (x,t)|^2 dx} \right] = \frac{1} {{2\pi }}\left( {\frac{\pi } {{2\alpha }}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \frac{{m!!}} {{(4\alpha )^m }}\left {\smallint _\mathbb{R} u_0 (x)dx} \right]^2 ,$

$\smallint _\mathbb{R} u_0 (x)dx \ne 0,$

where 0!! = 1, 1!! = 1 and m!! = 1 · 3 ⋯ · (2m–3) · (2m − 1). Moreover

$\mathop {\lim }\limits_{t \to \infty } \left {(1 + t)^{m + {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \smallint _\mathbb{R} |u_{x^m } (x,t)|^2 dx} \right] = \frac{1} {{2\pi }}\left( {\frac{\pi } {{2\alpha }}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \frac{{(m + 1)!!}} {{(4\alpha )^{m + 1} }}\left {\smallint _\mathbb{R} \rho _0 (x)dx} \right]^2 ,$

if the initial function u ₀(x) = ρ ₀′ (x), for some function ρ ₀ ∈ C ¹(ℝ) ∩ L ¹(ℝ) and

$\smallint _\mathbb{R} \rho _0 (x)dx \ne 0.$

Keywords:

Exact limits Sharp rates of decay Fluid dynamics equation Global smooth solutions

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