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 北京化工大学学报(自然科学版)  2018, Vol. 45 Issue (1): 119-123  DOI: 10.13543/j.bhxbzr.2018.01.020 0

### 引用本文

CHEN Xiao, SUN Ying, CHEN YaZhou. Asymptotic stability of the periodic solution of bird-carreau type viscous van der waals fluids[J]. Journal of Beijing University of Chemical Technology (Natural Science), 2018, 45(1): 119-123. DOI: 10.13543/j.bhxbzr.2018.01.020.

### 文章历史

Bird-Carreau型黏性van der Waals流体周期解的渐近稳定性

Asymptotic stability of the periodic solution of Bird-Carreau type viscous van der Waals fluids
CHEN Xiao , SUN Ying , CHEN YaZhou
Faculty of Science, Beijing University of Chemical Technology, Beijing 100029, China
Abstract: In this paper, the asymptotic stability of a one-dimensional compressible viscous van der Waals fluids system is discussed, where the viscosity coefficient is a nonlinear function that satisfies the Bird-Carreau model, and the pressure is a non-convex function. By constructing the energy function and using the energy estimation method and the monotone operator theory, we prove that:under the condition of large viscosity, the solutions of the non-Newtonian fluid are asymptotically stable when the initial value is either located in the stable region, or located in the metastable region under small disturbance conditions.
Key words: Bird-Carreau type viscosity    van der Waals fluids    periodic boundary

1 基本模型及其主要定理

 $\left\{ \begin{gathered} {v_t} - {u_x} = \varepsilon {v_{xx}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;t > 0,x \in \mathbb{R} \hfill \\ {u_t} + p{\left( v \right)_x} = {\left( {\mu \left( {\frac{{{u_x}}}{v}} \right)\frac{{{u_x}}}{v}} \right)_x}\;\;\;\;\;\;\;\;t > 0,x \in \mathbb{R} \hfill \\ \left( {v,u} \right)\left| {_{t = 0}} \right. = \left( {{v_0},{u_0}} \right)\left( x \right)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \mathbb{R} \hfill \\ \left( {v,u} \right)\left( {x,t} \right) = \left( {v,u} \right)\left( {x + 2L,t} \right) \hfill \\ \end{gathered} \right.$ (1)

 $\mu = {\mu _\infty } + \left( {{\mu _0} - {\mu _\infty }} \right){\left( {1 + {{\left( {{u_x}/v} \right)}^2}} \right)^{\frac{\gamma }{2}}}$ (2)

 图 1 p和v之间的关系 Fig.1 Plot of p against v

 $\left\{ \begin{array}{l} p'\left( v \right) < 0,\;\;\;\;\;\;v \in \left( { - \infty ,\alpha } \right) \cup \left( {\beta , + \infty } \right)\\ p'\left( v \right) > 0,\;\;\;\;\;\;v \in \left( {\alpha ,\beta } \right)\\ \left| {p\left( v \right)} \right| \le c,\;\;\;\;\;\;\left| {p'\left( v \right)} \right| \le c \end{array} \right.$ (3)

 $\left\{ \begin{array}{l} \bar v = \frac{1}{{2L}}\int_0^{2L} {{v_0}\left( x \right){\rm{d}}x} \\ \bar u = \frac{1}{{2L}}\int_0^{2L} {{u_0}\left( x \right){\rm{d}}x} \end{array} \right.$ (4)

 $\left\{ \begin{array}{l} \int_0^{2L} {\left( {v - \bar v} \right){\rm{d}}x} = 0\\ \int_0^{2L} {\left( {u - \bar u} \right){\rm{d}}x} = 0 \end{array} \right.$ (5)

 $\begin{array}{l} \mathit{\Phi }\left( v \right) = \int_{\bar v}^v {\left( {p\left( {\bar v} \right) - p\left( \xi \right)} \right){\rm{d}}\xi } = p\left( {\bar v} \right)\left( {v - \bar v} \right) - \\ \int_{\bar v}^v {p\left( \xi \right){\rm{d}}\xi } \end{array}$ (6)

 $\left\{ \begin{array}{l} p\left( {{v_1}} \right)\left( {{v_2} - {v_1}} \right) = \int_{{v_1}}^{{v_2}} {p\left( \xi \right){\rm{d}}\xi } \\ p\left( {{v_1}} \right) = p\left( {{v_2}} \right) \end{array} \right.$ (7)

 $\mathit{\Phi }\left( v \right)\left\{ \begin{array}{l} \ge 0\;\;\;\bar v \in 区域\;C\\ < 0\;\;\;\bar v \in 区域\;A \end{array} \right.$ (8)
 $\mathit{\Phi }\left( v \right)\left\{ \begin{array}{l} \ge 0\;\;\;v \le {v_ * },\bar v \in 区域\;{B_1}\\ < 0\;\;\;v > {v_ * },\bar v \in 区域\;{B_1} \end{array} \right.$ (9)
 $\mathit{\Phi }\left( v \right)\left\{ \begin{array}{l} \ge 0\;\;\;v \ge {v^* },\bar v \in 区域\;{B_2}\\ < 0\;\;\;v < {v^* },\bar v \in 区域\;{B_2} \end{array} \right.$ (10)

 $L_{{\rm{per}}}^2 = \left\{ {f\left( x \right)\left| {f\left( x \right)} \right. = f\left( {x + 2L} \right);f\left( x \right) \in {L^2}\left( {0,2L} \right)} \right\}$ (11)

 $H_{{\rm{per}}}^2 = \left\{ {f\left( x \right)\left| {f\left( x \right)} \right. \in L_{{\rm{per}}}^2,\partial _x^if \in L_{{\rm{per}}}^2\left( \mathbb{R} \right);i = 1, \cdots ,k} \right\}$ (12)

 $H_{{\rm{per,0}}}^k = \left\{ {f\left( x \right)\left| {f\left( x \right) \in H_{{\rm{per}}}^k,\int_0^{2L} {f\left( x \right){\rm{d}}x} = 0} \right.} \right\}$ (13)

 $\left\{ \begin{array}{l} \bar v \in 区域\;C\\ 或\;\bar v \in 区域\;{B_1}\;且\;M \le \frac{{{v_ * } - \bar v}}{{2\sqrt {2L} }}\\ 或\;\bar v \in 区域\;{B_2}\;且\;M \le \frac{{\bar v - {v^ * }}}{{2\sqrt {2L} }} \end{array} \right.$ (14)

 $\varepsilon > \frac{{{2^{\frac{5}{2}}}{L^{\frac{1}{2}}}\left( {{L^2} + {{\rm{ \mathsf{ π} }}^2}} \right)M}}{{{\mu _\infty }{{\rm{ \mathsf{ π} }}^2}}}\mathop {\sup }\limits_{v \in 区域A} \left| {p'\left( v \right)} \right|$ (15)

 $\left( {v,u} \right)\left( {x,t} \right) \in C\left( {0, + \infty ;H_{{\rm{per}}}^1} \right)$

 $\begin{array}{l} \left\| {\left( {v - \bar v,u - \bar u} \right)\left( t \right)} \right\|_1^2 + \int_0^{ + \infty } {\left\| {\left( {v - \bar v,u - } \right.} \right.} \\ \left. {\left. {\bar u} \right)} \right\|_2^2{\rm{d}}t \le C\left\| {\left( {{v_0} - \bar v,{u_0} - \bar u} \right)} \right\|_1^2 \end{array}$ (16)

 $\mathop {\lim }\limits_{t \to + \infty } \left\| {v - \bar v,u - \bar u} \right\| = 0$ (17)
2 问题重构

 $\left\{ \begin{array}{l} {\phi _t} - {\psi _x} = \varepsilon {\phi _{xx}}\\ {\psi _t} = {\left( {p\left( {\phi + \bar v} \right) - p\left( {\bar v} \right)} \right)_x} = {\left( {\mu \frac{{{\psi _x}}}{{\phi + \bar v}}} \right)_x}\\ \left( {\phi ,\psi } \right)\left( {x,t} \right) = \left( {\phi ,\psi } \right)\left( {x + 2L,t} \right)\\ \left( {\phi ,\psi } \right)\left| {_{t = 0}} \right. = \left( {{v_0} - \bar v,{u_0} - \bar u} \right)\left( x \right)\\ \int_0^{2L} {\left( {\phi ,\psi } \right)\left( {x,t} \right){\rm{d}}x} = \left( {0,0} \right) \end{array} \right.$ (18)

 $\begin{gathered} {X_{m,M}}\left( I \right) = \left\{ {\left( {\phi ,\psi } \right)\left| {\left( {\phi ,\psi } \right) \in C\left( {I;H_{{\rm{per,0}}}^1} \right),\left( {\phi ,} \right.} \right.} \right. \hfill \\ \left. \psi \right) \in {L^2}\left( {I;H_{{\rm{per,0}}}^2} \right),\mathop {\sup }\limits_{t \in I} {\left\| {\left( {\phi ,\psi } \right)\left( t \right)} \right\|_2} \leqslant M,\mathop {\inf }\limits_{t \in I,x \in \mathbb{R}} \left( {\bar v + } \right. \hfill \\ \left. {\left. \phi \right)\left( {x,t} \right) \geqslant m} \right\} \hfill \\ \end{gathered}$ (19)

 $\left\{ \begin{array}{l} - {U_x} = \varepsilon {V_{xx}}\\ {\left( {p\left( {V + \bar v} \right) - p\left( {\bar v} \right)} \right)_x} = {\left( {\mu \left( {\frac{{{U_x}}}{{V + \bar v}}} \right)\frac{{{U_x}}}{{V + \bar v}}} \right)_x}\\ \left( {V,U} \right)\left( x \right) = \left( {V,U} \right)\left( {x + 2L} \right)\\ \int_0^{2L} {\left( {V,U} \right)\left( x \right){\rm{d}}x} = \left( {0,0} \right) \end{array} \right.$ (20)

 $\begin{gathered} {Y_{m,M}} = \left\{ {\left( {V,U} \right)\left| {\left( {V,U} \right) \in {H_{{\rm{per,0}}}},{{\left\| {\left( {V,U} \right)} \right\|}_1}} \right.} \right. \hfill \\ \left. { \leqslant M,\mathop {\inf }\limits_{x \in \mathbb{R}} \left( {V + \bar v} \right)\left( x \right) \geqslant m > 0} \right\} \hfill \\ \end{gathered}$ (21)

 $\left\{ \begin{array}{l} \left\| f \right\| \le \frac{L}{{\rm{ \mathsf{ π} }}}\left\| {{f_x}} \right\|\\ {\sup _{\left[ {0,2L} \right]}}\left| f \right| \le 2\;\;\;\sqrt {2L} \left\| {{f_x}} \right\| \end{array} \right.$ (22)

 ${\sup _{\left[ {0,2L} \right]}}\left| f \right| \le 2\int_0^{2L} {\left| {f'} \right|{\rm{d}}x} + \frac{1}{{2L}}\int_0^{2L} {\left| f \right|{\rm{d}}x}$ (23)

 $f\left( 0 \right) = \frac{1}{{2L}}\int_0^{2L} {\int_0^x {f'\left( s \right){\rm{d}}s{\rm{d}}x} }$

 $\begin{array}{l} \sup \left| {f\left( x \right)} \right| \le \left| {\int_0^x {f'\left( s \right){\rm{d}}s} } \right| + \left| {\frac{1}{{2L}}\int_0^{2L} {\int_0^x {f'\left( s \right)} } } \right.\\ \left. {{\rm{d}}s{\rm{d}}x} \right| \le 2\sqrt {2L} \left\| {{f_x}} \right\| \end{array}$

 $\left\{ \begin{array}{l} \varepsilon \int_0^{2L} {V_x^2{\rm{d}}x} = \int_0^{2L} {V{U_x}{\rm{d}}x} \\ \int_0^{2L} {\mu \frac{{U_x^2}}{{V + \bar v}}{\rm{d}}x} = \int_0^{2L} {\left( {p\left( {V + \bar v} \right) - p\left( {\bar v} \right)} \right){U_x}{\rm{d}}x} \end{array} \right.$ (24)

 $\int_0^{2L} {\left( {\frac{{\varepsilon {{\rm{ \mathsf{ π} }}^2}}}{{{L^2}}}{V^2} - 2V{U_x} + \frac{{{\mu _\infty }}}{{M\sup \left| {p'} \right|}}U_x^2} \right){\rm{d}}x} \le 0$ (25)

3 主要定理的证明

 $\left\{ \begin{array}{l} \phi _t^{\left( n \right)} - \varepsilon \phi _{xx}^{\left( n \right)} = \psi _x^{\left( n \right)}\\ \psi _t^{\left( n \right)} - {\left( {\mu \left( {\frac{{\psi _x^{\left( n \right)}}}{{{\phi ^{\left( {n - 1} \right)}} + \bar v}}} \right)\frac{{\psi _x^{\left( n \right)}}}{{{\phi ^{\left( {n - 1} \right)}} + \bar v}}} \right)_x} = - p_x^{\left( n \right)}\\ \left( {{\phi ^{\left( n \right)}},{\psi ^{\left( n \right)}}} \right)\left| {_{t = 0}} \right. = \left( {{v_0} - \bar v,{u_0} - \bar u} \right)\left( x \right)\\ \left( {{\phi ^{\left( n \right)}},{\psi ^{\left( n \right)}}} \right)\left( {x,t} \right) = \left( {{\phi ^{\left( n \right)}},{\psi ^{\left( n \right)}}} \right)\left( {x + 2L,t} \right) \end{array} \right.$ (26)

 $\begin{array}{l} \left\| {\left( {\phi ,\psi } \right)\left( t \right)} \right\|_1^2 + \int_0^t {\left\| {\left( {\phi ,\psi } \right)\left( \tau \right)} \right\|_2^2{\rm{d}}\tau } \le C\\ \left( {\left\| {{\phi _0}} \right\|_1^2 + \left\| {{\psi _0}} \right\|_1^2} \right) \end{array}$ (27)

 $\frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {{\phi ^2}{\rm{d}}x} + \varepsilon \int_0^{2L} {\phi _x^2{\rm{d}}x} = \int_0^{2L} {\phi {\psi _x}{\rm{d}}x}$ (28)

 $\begin{array}{l} \frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {{\psi ^2}{\rm{d}}x} + \int_0^{2L} {\left( {\left( {p\left( {\bar v} \right) - p\left( {\phi + \bar v} \right)} \right){\psi _x} + } \right.} \\ \left. {\frac{{\mu \psi _x^2}}{{\phi + \bar v}}} \right){\rm{d}}x = 0 \end{array}$ (29)

 $\begin{array}{l} \frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {\left( {\frac{1}{2}{\psi ^2} + \mathit{\Phi }\left( {\phi + \bar v} \right)} \right){\rm{d}}x} + \int_0^{2L} {\left( {\frac{\mu }{\phi } + } \right.} \\ \left. {\frac{{\psi _x^2}}{{\bar v}}} \right){\rm{d}}x - \varepsilon \int_0^{2L} {p'\left( {\phi + \bar v} \right)\phi _x^2{\rm{d}}x = 0} \end{array}$ (30)

 $\begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {\left( {\frac{1}{2}{\psi ^2} + \mathit{\Phi }\left( {\phi + \bar v} \right)} \right){\rm{d}}x} + \frac{{{\mu _\infty }}}{{{2^{\frac{3}{2}}}{L^{\frac{1}{2}}}M}}\int_0^{2L} {\psi _x^2{\rm{d}}x} - \\ \varepsilon \int_0^{2L} {p'\left( {\phi + \bar v} \right)\phi _x^2{\rm{d}}x} \le 0 \end{array}$ (31)

 $\begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {\left[ {\frac{{{2^{\frac{3}{2}}}{L^{\frac{5}{2}}}M}}{{{\mu _\infty }{\mathit{\pi }^2}\varepsilon }}\left( {\frac{{{\psi ^2}}}{2} + \mathit{\Phi }\left( {\phi + \bar v} \right)} \right) + \frac{1}{2}{\phi ^2}} \right]{\rm{d}}x} + \\ \int_0^{2L} {\left( {\frac{{{L^2}}}{{2{{\rm{ \mathsf{ π} }}^2}\varepsilon }}\psi _x^2 + \frac{\varepsilon }{2}\phi _x^2} \right){\rm{d}}x} + \int_0^{2L} {\left( {\frac{{{L^2}}}{{2{{\rm{ \mathsf{ π} }}^2}\varepsilon }}\psi _x^2 - {\psi _x}\phi + } \right.} \\ \left. {\frac{{\varepsilon {{\rm{ \mathsf{ π} }}^2}}}{{2{L^2}}}{\phi ^2}} \right){\rm{d}}x \le \frac{{{2^{\frac{3}{2}}}{L^{\frac{5}{2}}}M}}{{{\mu _\infty }{\mathit{\pi }^2}}}\int_0^{2L} {p'\left( {\phi + \bar v} \right)\phi _x^2{\rm{d}}x} \end{array}$ (32)

 $\frac{1}{2}\frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {\phi _x^2{\rm{d}}x} + \varepsilon \int_0^{2L} {\phi _{xx}^2{\rm{d}}x} = - \int_0^{2L} {{\phi _{xx}}{\psi _x}{\rm{d}}x}$ (33)

 $\begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {\left[ {\frac{{{2^{\frac{3}{2}}}{L^{\frac{1}{2}}}M}}{{{\mu _\infty }\varepsilon }}\left( {\frac{1}{2}{\psi ^2} + \mathit{\Phi }\left( {\phi + \bar v} \right)} \right) + \frac{1}{2}\phi _x^2} \right]{\rm{d}}x} + \\ \frac{1}{{2\varepsilon }}\int_0^{2L} {\psi _x^2{\rm{d}}x} + \frac{\varepsilon }{2}\int_0^{2L} {\phi _{xx}^2{\rm{d}}x} + \int_0^{2L} {\left( {\frac{1}{{2\varepsilon }}\psi _x^2 - {\psi _x}{\phi _{xx}} + } \right.} \\ \left. {\frac{\varepsilon }{2}\phi _{xx}^2} \right){\rm{d}}x \le \frac{{{2^{\frac{3}{2}}}{L^{\frac{1}{2}}}M}}{{{\mu _\infty }}}\int_0^{2L} {p'\left( {\phi + \bar v} \right)\phi _x^2{\rm{d}}x} \end{array}$ (34)

 $\begin{array}{l} \frac{{\rm{d}}}{{{\rm{d}}t}}\int_0^{2L} {\frac{{{2^{\frac{3}{2}}}{L^{\frac{1}{2}}}M\left( {{L^2} + {\pi ^2}} \right)}}{{{\mu _\infty }{\mathit{\pi }^2}\varepsilon }}\left( {\frac{1}{2}{\psi ^2} + \mathit{\Phi }\left( {\phi + \bar v} \right)} \right)} + \\ \left. {\frac{1}{2}\left( {\phi _x^2 + {\phi ^2}} \right)} \right]{\rm{d}}x + \frac{{{L^2} + {\pi ^2}}}{{2{\mathit{\pi }^2}\varepsilon }}\int_0^{2L} {\psi _x^2{\rm{d}}x} + \frac{\varepsilon }{2}\int_0^{2L} {\left( {\phi _x^2 + } \right.} \\ \left. {\phi _{xx}^2} \right){\rm{d}}x \le \frac{{{2^{\frac{3}{2}}}{L^{\frac{1}{2}}}M\left( {{L^2} + {\pi ^2}} \right)}}{{{\mu _\infty }{\mathit{\pi }^2}}}\int_0^{2L} {p'\left( {\phi + \bar v} \right)\phi _x^2{\rm{d}}x} \end{array}$ (35)

 $\begin{array}{l} {\left\| {\psi \left( t \right)} \right\|^2} + \left\| {\phi \left( t \right)} \right\|_1^2 + \int_0^{ + \infty } {\left( {\left\| {\psi} \right\|_1^2 + } \right.} \\ \left. {\left\| {\phi} \right\|_2^2} \right){\rm{d}}x \le C\left( {\left\| {{\phi _0}} \right\|_1^2 + {{\left\| {{\psi _0}} \right\|}^2}} \right) \end{array}$ (36)

 $\begin{array}{l} {\left\| {{\psi _x}\left( t \right)} \right\|^2} + \int_0^{ + \infty } {\left( {{{\left\| {{\psi _{xx}}} \right\|}^2}} \right){\rm{d}}x} \le C\left( {\left\| {\left( {{\phi _0}} \right)} \right\|_1^2 + } \right.\\ \left. {\left\| {\left( {{\psi _0}} \right)} \right\|_1^2} \right) \end{array}$ (37)

4 结束语

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