Nonlinear parabolic equations with regularized derivatives
Nonlinear systems of differential equations \[ \partial_tu+\partial_xf(u)+g(u)=\Delta u\tag{1} \] are considered on \(\mathbb R_+^{n+1}\) \((t>0)\), where \(u=(u_1,\dots,u_m)^T\), \(f(u)=(f_1(u),\dots,f_n(u))^T\), \(\partial_xf(u)=\sum_{i=1}^n\partial_{x_i}f_i(u)\). This differential equation is regularized as follows: \[ \partial_tu_\varepsilon +\partial_xf(u_\varepsilon)*\varphi_\varepsilon +g(u_\varepsilon)=\Delta u_\varepsilon*\varphi_\varepsilon* \check\varphi_\varepsilon, \tag{2} \] where \((u_\varepsilon)_{\varepsilon>0}\) is a standard net of mollifiers. It should be noted that per derivative one mollifying convolution is taken. The main theorems are as follows: If \(f\) and \(g\) are smooth, globally Lipschitz, and \(f(0)=0\), \(g(0)=0\), then the Cauchy problem for (2) has a unique solution \(u_\varepsilon\) in \(C([0,\infty),L^2(\mathbb R^n))\), \((u_\varepsilon)_{\varepsilon>0}\) has a limit as \(\varepsilon\to 0\), and this limit is the weak solution of the Cauchy problem (1). For the latter convergence it is crucial to have a mollifier of the form \(\varphi_\varepsilon*\check\varphi_\varepsilon\) on the right hand side as its Fourier transform is a non-negative function. Using the results concerning (2) it is shown that the corresponding Cauchy problem in the Colombeau algebra of generalized functions has a unique solution as well. In the last section an unconditionally convergent difference scheme for numerical approximation of the solution to (1) is introduced.
unconditionally convergence difference scheme
scientific article; zbMATH DE number 1295270