Global regularity of solution for general degenerate parabolic equations in 1-D
The parabolic Cauchy problem in 1-D \[ u_t+ F(u)_x+ H(u)= G(u)_{xx} \quad\text{in }\mathbb{R}\times (0,T],\quad u(0,x)= u_0(x)\tag{1} \] is considered with a special eye on regularity of weak solutions, where \(F\in C^2\), \(H\in C^1\), \(g(u)= G'(u)\geq 0\) and \(g\in C^2\), \(g(u)= 0\) on a set of measure \(0\). Using approximation by smooth solutions \(u^\varepsilon\) of \((1^\varepsilon)\), where \(G(u)_{xx}\) is replaced by \(((g(u)+ \varepsilon)u_x)_x\) and \(u_0\approx u^\varepsilon_0(x)\in C^\infty\) it is shown that in case \(H(u)= 0\) and \(| u_0|_1= | u_0|_0+| u_{0,x}|_0\leq M\) a unique weak solution of (1) satisfying \(| G(u)|_1\leq M\) exists. In the case \(F(u)= c_1u^n\), \(H(u)= c_2u^h\), \(G(u)u^m\) with \(c_2\), \(n\geq 1\), and \(h, m> 1\), which allows application to infiltration theory, problem (1) has a weak solution satisfying \(| u^{m-1}(x, t)|_1\leq M\) if \(u_0(x)\geq 0\) obeys this estimate \((t= 0)\). Finally, the Navier-Stokes equations for the Broadwell model \[ \rho_t+(\rho u)_x= 0,\quad u_t+ (f'(u)- u)u_x+ (f(u)- u^2){u_x\over\rho_x}= {1\over\rho}(g(u)u_x)_x,\tag{2} \] where \(g(u)> 0\) in \((-1,1)\) and \(g(1)= g(-1)= 0\), is considered. The degenerate parabolic Cauchy problem obtained by putting \(\rho= 1\) in the second equation of (2) has a unique weak solution satisfying \(u\in C^1\), \(| u|_1< \infty\), \(| u|\leq 1\) if \(| u_0|\leq 1\) and \(-{1\over 2}\leq u_{0,x}\leq {1\over 4}\). This interesting paper closes with a-priori estimates for a solution \((u,\rho)\) of (2) with initial values \((u_0,\rho_0)\) \[ | u|\leq 1,\quad u_x\leq {\rho(f(u)- u^2)\over g(u)},\quad \rho\geq {\inf_x\rho_0(x)\over 1+ 4t(\inf_x\rho_0(x))}, \] if the first two inequalities hold for \((u_0,\rho_0)\) in place of \((u,\rho)\) and \(\inf\rho_0> 0\).
Navier-Stokes equations for the Broadwell model
scientific article; zbMATH DE number 1109523
scientific article