Regularization for a class of ill-posed evolution problems in Banach space
In this attractive paper, the authors deal with the issue of Banach space regularization in a very specific sense. They consider ill-posed versions of the evolution problem \[ \frac{du(t)}{dt}= A(t,D) u(t), \quad 0 \leq s \leq t <T, \] \[ u(s)=\chi, \] where the solution \(u(t)\) takes values in a Banach space \(X\), \(i D\) denotes a bounded strongly continuous group on \(X\), and the operator \(A\) attains the form \[ A(t,D)= \sum_{j=1}^k a_j(t) D^j, \] for \(t \in [0,T]\) with \(a_j \in C([0,T]: \mathbb{C})\) and \(1 \leq j \leq k\). To overcome the ill-posedness, a regularization approach with a one-parametric family \(R_\beta(t)\) of regularization operators is exploited. Precisely, for regularization parameters \(0<\beta<1\), the regularized solutions \(v_\beta(t)=R_{\beta(\delta)}(t)\chi\) are defined by solving the evolution problem \[ \frac{dv_\beta(t)}{dt}= f_\beta(t,D)v_\beta(t), \quad 0 \leq s \leq t <T, \] \[ v_\beta(s)=\chi. \] There are given sufficient conditions such that this auxiliary problem is well-posed for all \(\beta\) under consideration and, moreover, as common in all facets of regularization theory, convergence \(\|u(t)-R_{\beta(\delta)}(t)\chi_\delta\| \to 0\) with \(\beta(\delta) \to 0\) as \(\delta \to 0\) takes place whenever \(\|\chi-\chi_\delta\| \leq \delta\), here for all \(0 \leq s \leq t \leq T\). For \(X=L^p(\mathbb{R}),\;p \not=2,\) the situation is made explicit when \(f_\beta(t,D)=A(t,D)-\beta D^{2k}\). Examples of this kind include the linearized Korteweg-de Vries equation as well as the Schrödinger equation. Then, even a Hölder type conditional stability estimate \[ \|u(t)-v_\beta(t)\| \leq C \beta^\frac{T-t}{T-s} M^\frac{t-s}{T-s} \] with two positive constants \(C\) and \(M\) independent of \(\beta\) occurs under additional requirements, which leads to convergence rates \(\|u(t)-R_{\beta(\delta)}(t)\chi_\delta\|=O\left(\delta^{\frac{T-1}{2(T-s)}}\right)\) as \(\delta \to 0\) for an a priori parameter choice \(\beta = \sqrt{\delta}\) and all \(0\leq s \leq t<T\). It is well-known that rates in all varieties of regularization can only hold when the solution is subject to an additional smoothness condition. It seems to be of interest to verify in detail such conditions for the evolution problems under consideration in this paper.
scientific article; zbMATH DE number 6136735
scientific article