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| Solutions to Some Open Problems in Fluid Dynamics |
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Citation： |
Linghai ZHANG.Solutions to Some Open Problems in Fluid Dynamics[J].Chinese Annals of Mathematics B,2008,29(2):179~198 |
| Page view： 1032
Net amount： 1152 |
Authors： |
Linghai ZHANG; |
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| Abstract： |
Let $u=u(x,t,u_0)$ represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation $$u_t-\varepsilon u_{xxt}+\delta u_x+\gamma Hu_{xx}+\beta u_{xxx}+f(u)_x =\alpha u_{xx},\quad u(x,0)=u_0(x),$$ where $\alpha>0$, $\beta\geq0$, $\gamma\geq0$, $\delta\geq0$ and $\varepsilon\geq0$ 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 $f(0)=0$, $|f(u)|\to\infty$ as $|u|\to\infty$, and $f\in C^1({\mathbb R})$, and there exist the following limits $$L_0=\limsup_{u\to0}\frac{f(u)}{u^3}\quad\mbox{and}\quad L_{\infty}=\limsup_{u\to\infty}\frac{f(u)}{u^5}.$$ Suppose that the initial function $u_0\in L^1({\mathbb R})\cap H^2({\mathbb R})$. By using energy estimates, Fourier transform, Plancherel's identity, upper limit estimate, lower limit estimate and the results of the linear problem $$ v_t-\varepsilon v_{xxt}+\delta v_x+\gamma Hv_{xx}+\beta v_{xxx}
=\alpha v_{xx},\quad v(x,0)=v_0(x),$$ the author justifies the following limits (with sharp rates of decay) $$\lim_{t\to\infty}\Big[(1+t)^{m+1/2}\int_{\mathbb R}|u_{x^m}(x,t)|^2\d x\Big] =\frac{1}{2\pi}\Big(\frac{\pi}{2\alpha}\Big)^{1/2}\frac{m!!}{(4\alpha)^m} \Big[\int_{\mathbb R}u_0(x)\d x\Big]^2,$$ if
$$ \int_{\mathbb R}u_0(x)\d x\neq0,$$ where $0!!=1$, $1!!=1$ and $m!!=1\cdot 3\, \cdots\, \cdot (2m-3)\cdot(2m-1)$. Moreover $$ \lim_{t\to\infty}\Big[(1+t)^{m+3/2}\int_{\mathbb R}|u_{x^m}(x,t)|^2\d x\Big] =\frac{1}{2\pi}\Big(\frac{\pi}{2\alpha}\Big)^{1/2}\frac{(m+1)!!} {(4\alpha)^{m+1}} \Big[\int_{\mathbb R}\rho_0(x)\d x\Big]^2,$$ if the initial function $u_0(x)={\rho_0}'(x)$, for some function $\rho_0\in C^1({\mathbb R})\cap L^1({\mathbb R})$ and $$ \int_{\mathbb R}\rho_0(x)\d x\neq0.$$ |
Keywords： |
Exact limits, Sharp rates of decay, Fluid dynamics equation, Global smooth solutions |
Classification： |
35Q20 |
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