一維雙曲平衡律系統(tǒng)的弱解和零松弛極限的研究.pdf

1、南京航空航天大學博士學位論文一維雙曲平衡律系統(tǒng)的弱解和零松弛極限的研究姓名：宋國強申請學位級別：博士專業(yè)：流體力學指導教師：陸云光2009-10一維雙曲平衡律系統(tǒng)的弱解和零松弛極限的研究IIAbstract Abstract Abstract AbstractHyperbolic systems of balance laws have already become one of the hot research fields thes

2、e years.These are among the “hot topics” in this field and can be of great interest, not only to professionalmathematicians, but also for physicists and engineers.The concept of hyperbolic systems of balance laws was int

3、roduced by the works of naturalphilosophers of the eighteenth century, predominantly L. Euler (1755), and has over the past onehundred and fifty years become the natural framework for the study of gas dynamics and, more

4、broadly,of continuum physics. During this period of time great personalities like Stokes, Challis, Riemann,Rankine, Hugoniot, Lord Rayleigh and later Prandtl, Hadamard, H. Lewy, G.I. Taylor and many otherswrote several f

5、undamental papers, thus laying the groundwork for the further development of themathematical theory. Many important scientists like J. Von Neumann, R. Courant, K.O. Friedrichs, H.Bethe and Ya. Zeldowich became interested

6、 in this field and proposed many new key concepts, theinfluence of which remains very great to the present day.Immediately after the Second World War there was a considerable development in mathematicaltheory, with key r

7、esults being obtained by a new generation of great mathematicians like S.K. Godunov,P. Lax, F. John, C. Morawetz and O. Oleinik, who led the field until the mid 1960s, when J. Glimmpublished an outstanding paper which ma

8、rked the most important breakthrough in the history of thisfield. Glimm was able to prove the global existence of general systems in one space dimension, withsmall BV data.In this thesis, we are concerned with the Cauchy

9、 problem of 2 × 2 hyperbolic systems of balancelaws and the singular limits of stiff relaxation and dominant diffusion for general 2 × 2 nonlinearsystems of balance laws in one space dimension by using the meth

10、od of the vanishing viscositytogether with compensated compactness and applied the invariant region or the maximum principle.We establish some existence theorems for the global weak solutions to those systems. This thesi

11、sincludes two kinds of problems，the main research topics include as follows：The first kind problem：1 、 The framework theorem of the weak solutions to the Cauchy problem of the hyperbolicsystems of balance laws in one spa

12、ce dimension.We first obtained the existence of the viscosity solution for the Cauchy problem for the relatedparabolic system under suitable conditions. And then we get the uniform boundedness of the viscositysolution by

13、 using the invariant region or the maximum principle. Therefore there exists a weak (weakstar) convergent subsequence of the viscosity solutions. But in general, weak convergence doesn’tmean the weak continuity of the no

14、nlinear flux function. In order to get the strong compactness of thesequence, we apply the theory of compensated compactness and construct suitable entropy-entropyflux pairs. By using the compactness theorem, we just nee

15、d to prove the Young measure derived byviscosity solution sequence is a Dirac measure.2 、 Existence of global weak solutions to two the Cauchy problems for 2 × 2 the hyperbolicsystems of balance laws in one space di