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

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 estimateG(u) C(1), whereG(u)=∫u0 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 (um−1)x M. Finally an interesting equation related to the Broadwell model (whereg(u) has two zero points) is studied and a uniquely regular solutionu C(1)is obtained. Moreover the estimatesux ρ(f(u)−u2)/g(u) andρ infx ρ0(x)/(1+4t(infx ρ0(x))) are proved for the solution of the Navier–Stokes equations corresponding to the Broadwell model.