| Hyperbolic systems of balance laws have already become one of the hot research fields these years. These are among the“hot topics”in this field and can be of great interest, not only to professional mathematicians, but also for physicists and engineers.The concept of hyperbolic systems of balance laws was introduced by the works of natural philosophers of the eighteenth century, predominantly L. Euler (1755), and has over the past one hundred and fifty years become the natural framework for the study of gas dynamics and, more 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 others wrote several fundamental papers, thus laying the groundwork for the further development of the mathematical theory. Many important scientists like J. Von Neumann, R. Courant, K.O. Friedrichs, H. Bethe and Ya. Zeldowich became interested in this field and proposed many new key concepts, the influence of which remains very great to the present day.Immediately after the Second World War there was a considerable development in mathematical theory, with key results 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. Glimm published an outstanding paper which marked the most important breakthrough in the history of this field. Glimm was able to prove the global existence of general systems in one space dimension, with small BV data.In this thesis, we are concerned with the Cauchy problem of 2×2 hyperbolic systems of balance laws and the singular limits of stiff relaxation and dominant diffusion for general 2×2 nonlinear systems of balance laws in one space dimension by using the method of the vanishing viscosity together 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 thesis includes 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 hyperbolic systems of balance laws in one space dimension.We first obtained the existence of the viscosity solution for the Cauchy problem for the related parabolic system under suitable conditions. And then we get the uniform boundedness of the viscosity solution by using the invariant region or the maximum principle. Therefore there exists a weak (weak star) convergent subsequence of the viscosity solutions. But in general, weak convergence doesn't mean the weak continuity of the nonlinear flux function. In order to get the strong compactness of the sequence, we apply the theory of compensated compactness and construct suitable entropy-entropy flux pairs. By using the compactness theorem, we just need to prove the Young measure derived by viscosity solution sequence is a Dirac measure.2?Existence of global weak solutions to two the Cauchy problems for 2×2 the hyperbolic systems of balance laws in one space dimension.The first Cauchy problem for 2×2 the hyperbolic systems of balance laws in one space dimension is a symmetrically hyperbolic system with a source. Under the framework theorem we obtained the existence of global bounded weak solutions for this system. Firstly we get the uniform boundedness of the viscosity solution by using the maximum principle. Secondly by applying theories of BV compactness and compensated compactness coupled with the scalar conservation law, we have an existence theorem of global bounded weak solutions to this system when inhomogeneous terms satisfy some conditions, and we give some examples for some inhomogeneous terms.The second Cauchy problem for 2×2 the hyperbolic systems of balance laws in one space dimension is a special system of Euler equation with a source. There are two special pressure functions ,linear and general source in these systems. We apply the maximum principle, the vanishing viscosity approach and the theory of compensated compactness to establish some existence theorem for global weak solutions to the Cauchy problem of the non-strictly hyperbolic system—a special system of Euler equation with a kind of source terms which satisfy some conditions. And then we point out that the general source term contains some special cases which have been studied in other article.The second kind problem:1?We are concerned with singular limits of stiff relaxation and dominant diffusion for general 2×2 nonlinear systems of balance laws, that is,?= o(?),??0 the relaxation time?tends to zero faster than the diffusion parameter?. We apply the method of compensated compactness to establish the following general framework: If there is an a priori L~?bound that is uniformly with respect to?for the solutions of a system, then the solution sequence converges to the corresponding equilibrium solution of this system.2?With the theory of the invariant region,this framework theorem can be applied to some important nonlinear systems with relaxation terms and inhomogeneous terms, such as the system of quadratic flux, the LeRoux system, the system of elasticity and the extended models of traffic flows. |
PDF Full Text Request