This feature reflects the physical phenomenon of breaking of waves and development of shock waves . in the fields of fulid dynamics , ( 0 . 2 . 1 ) is an approximation of small visvosity phenomenon . if viscosity ( or the diffusion term , two derivatives ) are added to ( 0 . 2 . 1 ) , it can be researched in the classical way which say that the solutions become very smooth immediately even for coarse inital data because of the diffusion of viscosity . a natural idea ( method of regularity ) is obtained as follows : solutions of the viscous convection - diffusion pr oblem approachs to the solutions of ( 0 . 2 . 1 ) when the viscosity goes to zeros . another method is numerical method such as difference methods , finite element method , spectrum method or finite volume method etc . numerical solutions which is constructed from the numerical scheme approximate to the solutions of the hyperbolic con - ervation laws ( 0 . 2 . 1 ) as the discretation parameter goes to zero . the aim of these two methods is to construct approximate solutions and then to conside the stability of approximate so - lutions ( i , e . the upper bound of approximate solutions in the suitable norms , especally for that independent of the approximate parameters ) . using the compactness framework ( such as bv compactness , l1 compactness and compensated compactness etc ) and the fact that the truncation is small , the approximate function consquence approch to a function which is exactly the solutions of ( 0 . 2 . 1 ) in some sense of definiton 當(dāng)考慮粘性后,即在數(shù)學(xué)上反映為( 0 . 1 . 1 )中多了擴(kuò)散項(xiàng)(二階導(dǎo)數(shù)項(xiàng)) ,即使很粗糙的初始數(shù)據(jù),解在瞬間內(nèi)變的很光滑,這由于流體的粘性擴(kuò)散引起,這種對(duì)流-擴(kuò)散問(wèn)題可用古典的微分方程來(lái)研究。自然的想法就是當(dāng)粘性趨于零時(shí),帶粘性的對(duì)流-擴(kuò)散問(wèn)題的解在某意義下趨于無(wú)粘性問(wèn)題( 0 . 1 . 1 )的解,這就是正則化方法。另一辦法從離散(數(shù)值)角度上研究?jī)H有對(duì)流項(xiàng)的守恒律( 0 . 1 . 1 ) ,如構(gòu)造它的差分格式,甚至更一般的有限體積格式,有限元及譜方法等,從這些格式構(gòu)造近似解(常表現(xiàn)為分片多項(xiàng)式)來(lái)逼近原守恒律的解。