Global Existence Of Bv Solutions To Periodic Cauchy Problem Of Diperna-type Conservation Laws

In this paper,we study the existence of periodic solutions to Diperna-type hyperbolic conservation laws via a modified Glimm scheme.This system of equations is motivated by the isentropic equations of gas dynamics for a polytropic gas.This problem has been studied by Frid.Due to the insufficient consideration of the factor of random choice method in the approximate solutions,it is necessary to add conditions to the initial data.The global existence of the periodic solutions has to be obtained by extension.Using a lemma on the periodic solutions^[44]:at any moment,the average of the solution over a period keeps constant,the above deficiencies can be solved.Based on the above research,we get the result that for the bounded periodic initial data which has the boundedness of total variation in one period,there exists an entropy solution that is bounded and satisfies the boundedness of total variation in one period.In order to get this conclusion,first,we study the geometric properties of shock curves in the Riemann invariant coordinate and get the solution in the large for the Riemann problem.Then we construct the approximate solutions through Glimm scheme and prove the boundedness of both the approximate solution and total variation in one period.Finally,the global existence of periodic solutions is proved according to the framework of Glimm scheme.