The effect of a nonlinear viscous regluarization on the solution of a Cauchy-Riemann equation (Q1353512)

A variant of the Cauchy-Riemann equation is considered, in which the Cauchy-Riemann equation has been regularized with a nonlinear second-order viscous term \([(\varepsilon +|u_x|^2) u_x]_x\). The equation is degenerate of parabolic type when \(\varepsilon=0\) and has a weak solution for all time. An embedding process is used to analyse the properties of the solution. It is shown that the smallest scale of the solution is \(\varepsilon\), the coefficient of the regularizing second-order term. From the sequence of regularized solutions a converging subsequence is extracted. A limit of the subsequence \(u\) belongs to \(H^1([0,1] \times [0,T])\). For each fixed \(t\), the first-order spatial derivative of \(u\) also belongs to \(L^\infty ([0,1])\), \(u\in C^0([0,1] \times [0,T])\) and, for fixed \(t\), \(u\) is Hölder continuous with exponent \(\alpha<1\). Numerically, it is verified that the solution has some of the features of the solution of the porous medium equation even though the equation is complex. The influence of the regularizing second-order diffusion term on the solution is numerically investigated.