Existence In The Large For Pressure-Gradient Equations

This paper discusses the existence of weak solutions and periodic solutions of the large data problem of the pressure-gradient equations of Cauchy problemby using the improved Glimm scheme,including large data problem and periodic two cases.The equations are approximated by the Euler equations considering only the pressure effect For larger initial data problems,we studied the geometrical properties of the shock curve of the pressure-gradient equations in the Riemann invariant coordinates prove that with the change of the left state,left state of shock wave curve satisfy Diperna's conditions.Then,an approximate solution is constructed using the Glimm scheme,taking full account of the random points in the approximate solution consider the study of fundamental wave interactions in a "diamond",the basic wave interaction in establishing the fundamental wave accurate estimates of the number of interactions and through appropriate Glimm functional structure,the strength of a wave is monotonous,and the approximate solution and the total variation of boundedness,resulting in the existence of large data solutions.For the problem of periodicsolution,Frid studied the existence of periodic solutions for this type of equation,but did not fully consider the role of random points in the Glimm scheme in approximate solutions,since there is no full consideration Glimm scheme randomly take on the role of approximate solution,need strong condition,and conditions for the initial value and continuation to get the existence of periodic solutions.This problem has been improved in the[56].Proposed in[56]:At any moment,the average value of the solution over a period remains unchangedand proves the existence of periodic solutions Thus the nature of the shock curves of the pressure-gradient equations,construct approximate solution and the Glimm scheme are obtained and the total variation bounded solution.