广义BBM_Burgers方程稀疏波解的稳定性_英文_

广义BBM_Burgers方程稀疏波解的稳定性_英文_ Stability of the rarefaction wave for th e generalized BBM2Burgers equa tion J IANG Mi2na , XU Yan2ling (Laboratory of Nonlinear Analysis , Department of Mathematics , )Central China Normal University , Wuhan 430079 Abstract : This paper is concerned with the stability of the rarefaction wave for the generalized BBM2Burgers equa2 tion ( ) u+ f u= u+ u,t x xx xxt ()1 ( ) u | = ux? u , x ? ?. t = 0 o ? R ( ) ( ) () ( ) under the assumption of u < u , the solution u x , tto Cauchy problem 1satisfies sup | u x , t- ux/ t|- + x ?R R ( ) ( ) ?0 as t ??, where ux/ tis the rarefaction wave of the non2viscous Burgers equation u+ f u= 0 with Rie2t x mann initial data u , x < 0 , - ( ) u x , 0= u , x > 0. + 2Key words : BBM2Burgers equation ; rarefaction wave ; a priori estimate ; L2energy method CL C number : O175. 25 Document code : A consider the case of u < u . As in 9 , we find the - + 1 Introduction () () asymptotic behavior of the solution to 1 . 1, 1 . 2as t corresponding Riemann In this paper , we consider the Cauchy problem for ? ? is closely related to the +the generalized BBM2Burgers equation on R ×R problem without viscosity : ( ) u+ u,()u+ f u= 1 . 1 ( ) = 0 , xx xxt u+ f ut x t x with initial data u ,x < 0 , () 1. 3 - R = u ( ) u | x= 0t = 0( )u | = ux ? u , u < u as x ?? ?. o t = 0?- + x > 0. u , + ()1 . 2 (As is well known , when f is convex and u < u , 1 . - + R We assume that f is a smooth convex function. The ) 3has nonlinear wave which is the rarefaction wave u () equation of type 1 . 1is related to the well2known BBM ( ) x/ t given explicitly by equation advocated by Benjamin2Bona2Mahony 1 as a ( ) Φ f′u t , x u , - - refinement of the KdV equation. All kinds of generalized - 1 ( ) ( ) f′x/ t , R ( )ux/ t BBM2Burgers equations have been studied by many peo2 = ( ) ( ) f′u t< f′u t ,x < - + 2 ? ple , cf . 2 , 4 , 10 . TheL and L norms decay ( ) x Ε f′u t . u , + + rates have been obtained. To our knowledge , all the ( )1. 4 study concentrated on the cases of u = u and u > - + - 2 In this paper , we use the L 2energy method which u . But to the case of u < u , i . e . , the Riemann + - + is different from the method in 5 , and get our main problem of the hyperbolic conservation which only con2 () ()result , i . e . , the solution of problem 1 . 1and 1 . 2 ( tains rarefaction wave corresponding to the problem 1 . R ( ) ) (satisfies sup | u x , t - u x/ t | ?0 as t ? ?, ) 1has not been studied up to now. In this paper , we x ?R Received date : 2002204229. R ( ) () ()where ux/ tis defined by 1 . 4. iii There exists a constant Cdepending on q such q The organization of this paper is as follows. In the that 2 2 WWsection 2 , we construct a smooth approximate rarefaction xx xx ( )( )d x = x , t t 1?WWR xxwave as in 5 . In the section 3 , we reformulate our L 1 11- 1 - - 1 - problem and give the proofs of the theorems. 2 2 q 2 q2 q ) (εε Φ Cmin w~ , w~ t ,q 2 2 W W xxtxxt 2 Smooth a pproximate sol ution to the ( )d x = ( )x , tt 1? W W R xx L rarefaction waves 34 4 ( εεγ( ε) ) Φ Cmin w~+ w~ , q ,, w~ , q where We consider the Riemann problem for non2viscous 1 1 1 4 2 - - - 1 - Burgers equation q q qγ( εεε)(q ,, w~ + w~ 1 + = w~ t R R R - 6w+ w w= 0 , xε) t w~t 3 33w , x < 0 ,()- 2 . 1 3 - - 1 - - 2 q 2 q2 qε R R + w~ t + ( ) ( ) wx , 0= wx= 0 1 1 1 w , x > 0. + 1 - - - 3 - 2 q 2 q2 q εw~ t .When w < w , it is well2known that the solution of - + R ()iv ( ) ( ) up| W x , t- wx/ t | ?0 as t ??. sR ( ) the Riemann problem 2 . 1is the centered rarefaction - 1 R R ( ) ( ) ( ( ) ) Let w x , t = f′W x , t . Then it is easy to ( ) ( ) wave wx , t = wx/ t ,where ()show from 2 . 2 w , x Φ w t , - - ( ) w+ f w= 0 R t x x/ t , w t < x < w t , ( ) wx/ t= - + - 1 ( )( ) ( ) ( ( ) ) 2. 3 w | = wx= f′Wx? u t = 00 0 ?w , x Ε w t . + + R R - 1 x ?? ?.( ) ( ) ( ) rewritten as ux/ t = f′Hence ux/ t is R From Lemma 2 . 1 , we have the following Lemma . ( ) ( ( ) ) w = f′u . The smooth ap2 ×wx/ twith? ? ( ) The problem 2 . 3has a unique Lemma 2 . 2 R ( )proximate solution of the Riemann solution wx/ t is ( ) global smooth solution w x , t satisfying the following constructed as follows : u - u + -W + WW = 0 , t x u~ properties by setting =: 2 1 ()( ) ( ) i u < w x , t< u , wx , t> 0 for each - + x ( ) ( )( )= w + w W x , 0= Wx 0 + - ( )2 2 . 2 ( ) ) x , t ? R ×[ 0 , ?. ε x2 - q (() ) + w~ K1 + y d y . iiFor any p with 1 Φ p Φ ?,there exists a constantq?0 Cdepending on p , q such that p , q 1 ( ) εwhere w~ = w - w > 0 ,> 0 is a constant andp p - 1 p - p +1 + - p( ) (ε ) ‖wt‖Φ Cmin u ~ , u~ t , 2 x L p , q? 2 - q pp ( ) K?1 + yd y = 1 Kis a constant such that Φ C q - ? ‖wt( ) q ‖ xx L p , q 1 p - 1p - 11 ( ) ()- - p - 2 p - 1 p p - 11 - 2 q 2 q 2 q (εεfor each q > .) As in 5 min ~u , ~u t . , 7 , we have the follow 2 2 ()iii There exists a constant Cdepending on q such q ing result . that ( ) The problem 2 . 2has a unique Lemma 2 . 1 2 2 wwxx xx ( ) global smooth solution W x , t satisfying the following ( )( )x , t d x = t 1?ww R xxL properties : 1 1 1 ()2 1 - 1 -- - 2 q2 q()( 2 q( ) )i w < W x , t < w , Wx , t > 0 for each ε) (ε- + x u~ , Φ Cmin ~u t , q 2 2 ( ) ) x , t ?R ×[ 0 , ?. ww xxt xxt ( )x , t d x = ( )t 1?()ii For any p with 1 Φp Φ ?, there exists a constant R ww xxL 4 34 Cdepending on p , q such that p , q ( εεγ( ε) ) Φ Cmin ~u+ u~ , q ,, ~u , q p - 1 - p +1 pp p ( ) (ε ) ‖Wt‖Φ C min w~ , w~ t . x L p , q where p 1 11 p( ) - ‖Wt ‖ 2 -- 6 - 1 - 4 xx L q q γ ε q ε ε()( ) ε + ~u 1 + ~ut t q , , u~ = ~u 1 1 p - p - 1 ( ) ()- - p - 3 3 3 1 1 1 2 p - 1 p - 11 - p 3 - - - 1 - 1 - - - 3 - 2 q 2 q 2 q (ε ε) Φ Cmin w~ ,w~ t . 2 q 2 q2 q 2 q 2 q2 q εεp , q+ ~u t + w~ t . R t ( ) ( ) ()sup| w x , t- ux/ t| ?0 as t ??.iv 2 2 1 (τ) ( ) ( Rwv ‖‖v t‖+‖ x H ?0 Due to Lemma 2 . 2 , we get the following Lemma . 2 )(τ) τ + ‖v‖ d x εLemma 2 . 3 Let = u ~ , q = 1 . Then the solu2 2 1 ( ( ) ) ()Φ C‖v‖+ h u . 3. 2 ` 0 0 H( ) tion w x , t in Lemma 2 . 2 satisfies : () Multiplying 3 . 1by v and integrating it with Proof ()i ( ( )) u < w x , t < u , wx , t > 0 for each - + x respect to x over R , we have : + ( ) x , t? R ×R . d 1 2 2 ()ii ( ) ( ( )For any p with 1 Φ p Φ ?, there exits a constant v+ vd x + v f v + w x ??2 d tR R Cdepending on p such that p 2 p- p +1 ) ) ( - f wd x +v d xp x x ( ) ( ) ( ) ‖wt‖Φ h ~u C 1 + t , ?x L p R 3 p - 1 p- p 2 ( ( ) () ) ‖wt ‖Φ h u~ C 1 + t , xx ( w+ w) ( ) d x . 3. 3 L p = vxx xxt?R 2 2 3 wwxx xx - 2 () The second term of 3 . 3is estimated as follows : ( ) () ( )( )d x Φ Ch ~u 1 + t, t = x , t 1?ww R xxL 2 2 ( ( ) ( ) ) v f v + w- f wd xx wwxxt xxt ?- 2 R ( ) () ( )Φ Ch ~u 1 + t, ( )d x = x , t t 1 ?ww Rxxv + w L ( ) ( ) f sds - f wv = - ?( ) where h ~u is a function of ~u and satisfies w ?R x ( ) ) ( ( )( )- f′wvw} d x ( ) + f v + w - f w lim h u~ = 0 .x u ~ ?0 R ()( ) ( ) iii sup| w x , t - ux/ t | ?0 as t ??.( ( )( )( ) ) = f v + w - f w - f′wvwd x x R? R α 2 ()Ε w vd x . 3. 4 x 3 Ref ormulation of the problem and ?2R α( ) Where= min f″x, w Φ x Φ v + w . ma in results () The right side of 3 . 3is estimated by the following : By setting u = w + v , we reformulate the initial problem )( wv + w d x xxxxt () () 1 . 1, 1 . 2as follows : ?R ( ( ) ( ) ) v+ f v + w- f w t x= vwd x +vwd xxx xxt ??RR v+ w+ v+ w,= xx xx xxt xxt 2 w 2 αxx 2 ( ) ( ) ( )= vx= ux- wx ?0 , x ?? ?.v | 0 0 0 t = 0 Φ w vd x + d xx ?α?8w R R x()3 . 1 2 w α xxt2 2 + w vd x + d x x () To solve 3 . 1, we define the solution space by ?α?8wR R x 12 2 ( ) ( ) Xt, t= { v | v ? C [ t, t; H,M 1 21 2 w αw 2 2 xxt xx 2 = w vd x + ( )d x +d x . 3. 5 x 2 11 ?α?α?( ) ( ) 4ww R R R v?L t, t; Hand sup ‖v t‖Φ M} . x 1 2 H xx[ t, t] 1 2 ()() ( Combining 3 . 4 and 3 . 5, we can obtain from 3 . In the above solution space , we have the following local )3 existence theorem. 1 d 2 2 2 v d xx ?t R R 1 ( ) ( ) M > 0 such that ‖v t‖Φ M .Rand there exists H α 2 + wvd x x Then there exists a positive constant tdepending on M ?40 R 2 2 () such that the problem 3. 1admits a unique solution v w w xx xxt 2 Φ d x +d x(). 3. 6 α ??( ) () wwx , t in X0 , t. R R 2 M 0xx The proof of lemma 3. 1 is standard and the details () Integrating 3 . 6with respect to t over [ 0 , t ] and us2 are omitted. To obtain global existence , we need to show ing Lemma 2 . 3 , we get 1 2t ( ) a priori estimates for v x , tin Hby applying L 2ener2 2 21 ( ) ( (τ) ‖v t ‖+‖ w v ‖H x ?0 gy method. 2(τ) ) τ+ ‖v‖d x ( ) Lemma 3 . 2 a priori estimate. Suppose that v 2 1 ( )+ h u `‖v‖)(( ) () 3. 7 () Φ C. x , tis a solution of 3 . 1in X0 , Tfor positive 0 H02 M constant T and M . Then it holds that The proof of Lemma 3 . 2 is completed. ( ) Combining Lemma 3 . 1 and Lemma 3 . 2 , we get | v x , t|sup x ?R 1 1 our main results. 2 2 ( ) ( ) Φ ‖v t ‖‖vt‖?0 as t ? ?. x ( ) Theorem 3 . 3 stability of the rarefaction wave. which completes the proof of Theorem 3 . 3 . 1 υ( ) Suppose that x ? Hand there exists a positive 0 Remark 3 . 5 . Theorem 3 . 3 shows that , under the 2 1 η( ) ηconstant > 0 such that ‖v‖+ h u Φ. Then `0 H () smallness assumption on initial data , the solution of 1() the problem 3 . 1has a unique global solution v ?C R ( ) tends to the rarefaction wave ux/ t: 1( ) ) [ 0 , ?; HsatisfyingR ( ) ( ) sup | u x , t- ux/ t|R ( ) sup | v x , t| ?0 as t ? ?. x ?R ( ) Φ sup | u x , t-( ) w x , t| R Proof Due to Lemma 3 . 1 and Lemma 3 . 2 , in order R ) ( () - u x/ t| ?0 as t ? ?.+ sup | w x , t ( ) () to prove that 3 . 1has a unique solution v x , t ?C R 1 ( ) ) 2 as[0 , ?; H, we need only to show the priori sumption Ackno wledgment : The authors would express their ( ( ) ) v x , t ? X0 , T,2 M sincere gratitude to Professor Changjiang Zhu who pro2 i . e . pose useful advice on this paper . 1 ( ) ‖v t‖( )Φ 2 M , 3 . 8 H is reasonable . References : under the priori assumption , we have In fact , 1 Benjamin T B , Bona J L . Mahony J J . Model equation for long waves η () proved 3 . 2. Let small enough such that in nonlinear dispersive system J . Phil Trans R Soc London , Ser A 2 η CΦ M, () 1972 272: 47,78 . 0q 2 Mei M. L decay rates of solutions for Benjamin2Bona2Mahony2Burg2 from Lemma 3 . 2 we have () ers equationsJ . J Differential Equations , 1999 158: 314,340 . 2 2 2 11 η ( ) ( ( ) )Φ C Φ M, ‖v t‖ Φ C ‖v ‖+ h u `0 0 H 0 H Mei M. Large2time behavior of solution for generalized Benjamin2 3 Then Bona2Mahony2Burgers equations J . Nonlinear Analysis , 1988 () 33: 699,714 . 1 ( ) ‖v t ‖Φ M . H Mei M. Convergence to diffusion waves of the solutions for Benjamin2 4 () These imply that the priori assumption 3 . 8is reason2 Bona2Mahony2Burgers equationsJ . J Differential Equations , 1999 () 158:314,340 . ( ) able. So we show that 3 . 1has a unique solution v Matsumura A , Nishihara K. Global stability of the rarefaction waves ( ) ( ) ( ) x , t ?X0 , ?. To show sup| v x , t | ?0 as t 2 M 5 R of a one2dimension model system for compressible viscous gas J . ??, we use the following Lemma : () Comm Math Phys , 1992 144: 325,335 . ( ) g t Ε 0 , ( ) Lemma 3 . 4 see 7 . Suppose that Matsumura A , Nishihara K. Asymptotic toward the rarefaction waves 6 1 1 ( ) ( )() Then g t ?0 asg ?L 0 , ? g′?L 0 , ?. and of the solutions of a gas one2dimension model system for compressible () viscous gasJ . Japan J Appl Math , 1986 3:1,13 . t ??. tNishihara N. A note on the stability of travelling wave solution of 2 ) ) ( ) ( ( Proof Since | g t- g t| = | ?g′t d t 12t 7 1() Burgers equation J . Japan J Indust Appl Math , 1985 2: 27,35 . ( ) | ?0 as t, t??, then g t?gas t ??for some 1 2 0 Zhao H J , Zhu C J , Yu Z. Existance and convergence of solution to 8 1 ( ) nonnegative constant g. By g ?L 0 , ?, g= 0.0 0 a singular perturbed higher order partial differential equation J . () Nonlinear Analysis , TMA , 1995 24: 1 435,1 455 . This proves Lemma 3 . 4 . 2 II’in A M , Oleinik O A. Asymptotic behavior of the solutions of the ( ) ( ) Taking g t = ‖vt ‖in Lemma 3 . 4 , we x 9 Cauchy problem for certain quasilinear equations for large time J . 1 ( ) ( ) ( ) ( )have from 3 . 2 g t ?L 0 , ?and g′t () Math USSR Sb , 1960 51: 191,216 . ( ( ) ( ) ) = 2〈 v, v〉= - 2〈 v, v〉= 2〈 f v + w - f w x xt t xx xZhao H J , Xuan B J . Existence and convergence of solutions for the 10 1 ( ) generalized BBM2Burgers equation with dissipative termsJ . Non 2 w, v〉?L 0 , ?, - v- w-v- xxt xx xx xx xxt 2() linear Anal , 1997 28: 1 835,1 849 . where〈?,?〉is the L 2inner product . Hence , ‖v x ( ) t ‖?0 as t ??. By the Sobolev inequality 广义 BBM2Burgers 方程稀疏波解的稳定性 蒋咪娜 , 徐艳玲 () 华中师范大学 数学系 ,非线性 分析 定性数据统计分析pdf销售业绩分析模板建筑结构震害分析销售进度分析表京东商城竞争战略分析 实验室 , 武汉 430079 摘 要 : 考察了如下广义 BBM2Burgres 方程 ( ) u+ f u= u+ u,t x xx xxt ()1 ( ) u | = ux? u , x ? ?. t = 0 o ? () ( ) u x , t- 稀疏波解的稳定性 , 即在 u < u 的假设条件下 , 当 t ? ?时 , Cauchy 问 题 快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题 1的解满足 sup |- + x ?R R R ( ) ( ) ux/ t| ?0 , 其中 ux/ t是无粘 Burgers 方程黎曼问题 ( ) u+ f u= 0 , t x u , x < 0 , - R ( ) u | = ux= 0 t = 0 u , x > 0 , + 的解 . 2关键词 : BBM2Burgers 方程 ; 稀疏波 ; 先验估计 ; L2能量 方法 快递客服问题件处理详细方法山木方法pdf计算方法pdf华与华方法下载八字理论方法下载 () 作者简介 : 蒋咪娜 1978 - ,女 ,河南焦作人 ,硕士研究生 ,主要从事偏微分方程研究.