黏性不可压流体建模

章节摘录

版权页：   插图：   The uniqueness of weak solutions is completely open in alldimensions even in two dimensions.Of course,the uniqueness ofsolutions is close to the regularity of solutions.It has been wellknown that the solution which is regular enough is unique and anyweak solution is equal to a strong one if the later exists [38[.However,we can't expect full regularity results to be known since they would imply regularityresults for the homogeneous Navier-Stokes equations (1.6). The existence of strong solutions was obtained by Kazhikov andhis collaborators.They assumed that μ is a constant and po isbounded away from 0 and proved the local existence of unique strongsolution for all sufficiently regular data.This result was later extendedby Ladyzhenskaya and Solonnikov,Padula,Salvi.But theyall required that the initial density may not vanish (i.e.non-vacuum).Later,Choe and Kim obtained an local existence result on strongsolutions with nonnegative densities in case that μ is a constant.Recently,they proved the local existence of unique strongsolutions in a bounded domain Ω of Rn(n = 2,3) for all initial datasatisfying a natural compatibility condition in the case when μdepends on p and the initial density p0 may vanish in an open subsetofΩ.

书籍目录

Chapter 1 Introduction 1.1 The main models 1.2 Notations and some preliminary lemmas Chapter 2 The Navier-Stokes Equations with Damping 2.1 Introduction 2.2 Existence of weak solutions 2.3 Existence and uniqueness of strong solutions Chapter 3 Decay of Navier-Stokes Equations with Damping 3.1 Introduction 3.2 A priori estimates on upper bound of decay 3.3 A priori estimates on lower bound of decay 3.4 The decay of the weak solutions Chapter 4 Stokes Approximation of Non-homogeneous Navier Stokes Equations 4.1 Introduction 4.2 Existence of weak solutions 4.3 Existence and uniqueness of strong solutions Chapter 5 Large Solutions to Non-homogeneous Navier-Stokes Equations 5.1 Introduction 5.2 The global existence of solutions 5.3 The global stability of solutions Chapter 6 Some Remarks on Planar Boussinesq Equations 6.1 Introduction and the main results 6.2 The case of smooth initial data Referenees

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