Singular Perturbations Of Flood Wave Equations With Relaxation

In this paper, we establish the singular perturbation expansions to flood wave equations which describe the fluid of the river. The flood wave equations depict the relationship between the depth of the river, the cross section area of the riverbed and the corresponding quantity in one minute. This is a classical model in hydrodynamics.The singular perturbation method is a useful tool to work with the mathematical and physical problems. This theory has been applied extensively in hydrodynamics, celestial mechanics and quantum mechanics, etc. So it's one of the important subjects studied in the world recently.We develop the singular perturbation theory for initial-value problems of flood wave equations with relaxation term in the present paper. First, we construct the formal asymptotic approximations of the initial-layer solution to the flood wave equations by the weaker stability conditions. Then, we discuss the regularity of the expansions and do some necessary estimates. In the end, we prove the validity of the singular perturbation expansions and get some useful deductions about the existence of the classical solution to the flood wave equations.