王玉,黃蘭
帶有反應項的可壓縮微極實際氣體模型解的指數(shù)穩(wěn)定性
王玉,黃蘭
(華北水利水電大學 數(shù)學與統(tǒng)計學院,河南 鄭州 450046)
研究了齊次邊界條件下一維粘性帶有反應項的可壓縮微極實際氣體模型解的大時間行為.假定任意初始值(密度不含真空),利用能量估計和各種精細的插值不等式證明密度函數(shù)和溫度函數(shù)的一致上下界,進而證明了解的整體存在性和指數(shù)穩(wěn)定性.
帶有反應項流體;微極實際氣體;存在性;先驗估計;指數(shù)穩(wěn)定性
流體力學主要研究流體(液體和氣體)的力學運動規(guī)律及其應用,經(jīng)典的流體力學模型包括Navier-Stokes方程、磁流體方程、微極流體模型以及磁微極流體模型等.1964年,文獻[1]首次描述了微極流體模型,該模型可用于對具有微觀結構的材料的研究,一開始就受到許多研究者的關注.微極流體的數(shù)學理論研究分為兩個方向:可壓縮微極流體和不可壓縮微極流體.不可壓縮微極流體是一種相對理想的狀態(tài),目前已有大量研究.關于可壓縮微極流體,在三維情況下,文獻[2-6]研究了具有球對稱的粘性可壓縮微極流體模型解的整體存在性和具有圓柱對稱的粘性可壓縮微極流體解的整體存在性以及指數(shù)穩(wěn)定性.而在一維情況下,關于可壓縮微極流體模型解的存在性與正則性也有大量研究[7-13],其中文獻[11-12]研究了理想氣體模型經(jīng)典解的存在性,文獻[13]研究了可壓縮微極實際氣體模型解的整體存在性.在微極實際氣體模型的基礎上增加反應項后,文獻[14]研究了齊次邊界條件下帶有反應項的粘性可壓縮微極實際氣體模型的局部解存在性.但對于該一維模型整體解的存在性以及大時間性態(tài),還沒有相關研究.本文在文獻[14]基礎上進一步研究齊次邊界條件下粘性反應微極實際氣體一維模型解的大時間行為.
在拉格朗日坐標系下,帶有反應項的粘性可壓縮微極實際氣體模型方程為
假設系統(tǒng)(1)~(5)滿足初值條件
和邊界條件
定理1假設初值滿足
令
結合邊界條件(7),并利用插值不等式,由式(12)可得
利用引理2并結合式(22),由式(23)可得
由式(27)和引理1中Young不等式可知
由式(24)(29)可知
由式(28)(30)可知
由式(28)(37),易得
結合式(11)(38),有
將式(42)(43)相加,得
再結合式(8)(38),利用引理3中Poincaré不等式,有
結合式(11)(38)(44)(45),利用引理3中Poincaré不等式,有
由式(11)(38)(47),再利用引理3中Poincaré不等式,得到
結合式(25)(38)(49)(50),利用引理3中Poincaré不等式,得到
再結合式(26)(51)(52),有
結合式(14)(57)(58)(60),得到
證畢.
由式(62)可得定理2解的指數(shù)穩(wěn)定性.
通過嚴密的推理,本文證明了在齊次邊界條件下帶有反應項的可壓縮微極實際氣體模型整體解的存在性與指數(shù)收斂性.但在粘性系數(shù)依賴于溫度和密度的情況下,帶有反應項的可壓縮微極實際氣體模型解的整體存在性和大時間性等一系列性質還有待證明.
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Exponential stability of solutions to the compressible viscous and reactive micropolar real gas model
WANG Yu,HUANG Lan
( School of Mathematics and Statistics,North China University of Water Resources and Electric Power,Zhengzhou 450046,China)
The large-time behavior of solutions to the one-dimensional compressible viscous and reactive micropolar real gas model with homogeneous boundary conditions is studied.Based on the assumption that any initial value (mass density without vacuum),theuniform upper and lower bounds of the density and the temperature function are proved by using energy estimate method and delicate interpolation inequality,and then the global existence and exponential stability of solutions are obtained.
reactive fluid;micropolar real gas;existence;aprior estimate;exponential stability
1007-9831(2023)12-0001-08
O175
A
10.3969/j.issn.1007-9831.2023.12.001
2023-05-02
國家自然科學基金項目(11501199)
王玉(1999-),女,河南開封人,在讀碩士研究生,從事偏微分方程研究.E-mail:wangyu1115598@163.com
黃蘭(1982-),女,河南信陽人,教授,博士,從事偏微分方程研究.E-mail:huanglan82@hotmail.com