Enhanced and restored signals as a generalized solution for shock filter models. I: Existence and uniqueness result of the Cauchy problem
In this paper the authors propose generalized one- and two-dimensional shock models for signal enhancement and restoration. In the one-dimensional case the following quasilinear partial differential equation is proposed as shock filter model \[ u_{t}+a(x)F(u_{x^{2}},u_{x})\partial _{x}f(u)=0\text{ \;in \;}\mathbb{R}\times \mathbb{R}^{+}, \tag{1} \] and in the two-dimensional case \[ u_{t}+a_{1}(x,y)F_{1}(\Delta u,u_{x})\partial _{x}f_{1}(u)+a_{2}(x,y)F_{2}(\Delta u,u_{y})\partial _{y}f_{2}(u)=0\text{ \;in \;}\mathbb{R}^{2}\times \mathbb{R}^{+}, \tag{2} \] where the functions \(a,a_{1}\) and \(a_{2}\) are bounded and measurable and the functions \(f,f_{1},f_{2}\) and \(F,F_{1},F_{2}\) are smooth. The Cauchy problems for equations (1) and (2) are studied within the framework of Colombeau generalized functions. These Cauchy problems are rewritten as regularized partial differential equations. This interpretation consists of a regularization, with an adequate choice of mollifiers, of space derivatives. Then the authors prove theorems of existence and uniqueness of Colombeau generalized global solutions for these regularized partial differential equations.
scientific article; zbMATH DE number 1915742