Solutions to some open problems in Fluid dynamics

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) $\mathop {\lim }\limits_{t \to \infty } \left[ {(1 + t)^{m + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \smallint _\mathbb{R} |u_{x^m } (x,t)|^2 dx} \right] = \frac{1}{{2\pi }}\left( {\frac{\pi }{{2\alpha }}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \frac{{m!!}}{{(4\alpha )^m }}\left[ {\smallint _\mathbb{R} u_0 (x)dx} \right]^2 ,$ if $\smallint _\mathbb{R} u_0 (x)dx \ne 0,$ 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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