Global Regularity of Solution for General Degenerate Parabolic Equations in 1-D

Abstract This paper considers the Cauchy problem for the general degenerate parabolic equations (1.1) with initial data (1.2). In the critical condition meas{ u :  g ( u )=0{=0 we obtain the regular estimate G ( u )∈C (1) , where G ( u )=∫ u 0 g ( s )  ds . A new maximum principle is introduced to obtain the estimate and is applied to some special equations such as prous media equation, an infiltration equation to obtain the optimal estimate |( u m −1 ) x |⩽ M . Finally an interesting equation related to the Broadwell model (where g ( u ) has two zero points) is studied and a uniquely regular solution u ∈C (1) is obtained. Moreover the estimates u x ⩽ ρ ( f ( u )− u 2 )/ g ( u ) and ρ ⩾inf x ρ 0 ( x )/(1+4 t (inf x ρ 0 ( x ))) are proved for the solution of the Navier–Stokes equations corresponding to the Broadwell model.