Periodic Entropy Solution To A Conservation Law With Nonlocal Source

This thesis is concerned with the Cauchy problem on a scalar conservation law with nonlocal source arising in radiative gas,taking the form of(?)(0.1)with periodic initial data(say of period L)(?)(0.2)The global well-posedness and large-time asymptotic behavior of spatially periodic entropy solution to the Cauchy problem(0.1),(0.2)are established.In particular,it turns out that if the initial data is periodic,the source term induces the solution to decay in L2-norm at an exponential rate to the mean value of initial data over one space period.The thesis involves there parts:In the first part,main theorem in the thesis is given.That is,there exists a unique global L-periodic entropy solution to the Cauchy problem(0.1),(0.2)if initial data u0(x)? L?[0,1]? L1[0,1].In addition,the large-time asymptotic behavior is obtained.It turns out that if the initial data is periodic,the source term induces the solution to decay in L2-norm at an exponential rate to the mean value of initial data over one space period.Finally,in this part we also present the research background and summarize related literature.In fact,the scalar balance law(0.1)can be derived from a hyperbolic-elliptic coupled system arising in the radiative gas,which is also called the Hamer model of radiating gas(?)(0.3)In specific physical situation,the hyperbolic-elliptic system(0.3)gives a good approx-imation to the non-isentropic compressible Euler equations for an ideal gas subject to heat radiation phenomena,i.e.(?)In the second part,two propositions which describe useful properties on some functions(especially periodic functions)are presented.Some of these properties will play an important role in this thesis.In the third part,we give the proof of main theorem.Firstly,we prove global existence of the periodic entropy solution by the vanishing viscosity method.The compactness of the sequence issued by this method is proved with the aid of the L1-contraction and the comparison principle.In addition,asymptotic behavior and time-decay rates of periodic entropy solution are obtained by subtle analysis on nonlocal source and Gronwall's inequality:(?)(0.4)where(?).