On the stability of the standard Riemann semigroup (Q2782622)

The following hyperbolic system of conservation laws in one space dimension NEWLINE\[NEWLINE u'_t+[f(u)]_x=0 \tag{1}NEWLINE\]NEWLINE is under consideration. Here \( f:\Omega \to \mathbb{R}^n\) is a function that generates an Standard Riemann Semigroup (SRS): \( f\in \text{Hyp}(\Omega)\). Such semigroup \(S^f\) generated by \(f\) is defined by the properties: (i) \(S^f_0=I, \;S^f_t\circ S^f_s=S^f_{t+s};\) (ii)\( S^f\) is Lipschitz continuous: NEWLINE\[NEWLINE\|S^f_t u-S^f_s w\|_{L^1}\leq L_f(|t-s|+\|u-w\|_{L^1})NEWLINE\]NEWLINE for all \(t, s \geq 0\) and all \(u,w\in D^f;\) (iii) if \(u\) is piecewise constant, then for \(t\) small, \(S^f_t u\) coincides with the gluing of standard solutions to Riemann problems. In the present paper the correspondence \(f\to S^f\) is investigated. It is proved that SRS is a Lipschitz function of the flow \(f\), with respect to the \(C^0\) norm of \(Df\). More exactly the main result is Theorem 2.1:NEWLINENEWLINENEWLINELet \( f\in Hyp(\Omega)\). Then, for all \(g\in Hyp(\Omega)\) with \(D^g\subseteq D^f\) and for all \(u\in D^g,\) NEWLINE\[NEWLINE \|S^f_t u-S^g_t u\|_{L^1}\leq L_f\cdot d(f,g)\cdot\int_0^t \text{ Tot.Var.}(S^g_tu)dt, NEWLINE\]NEWLINE where \(d(f,g)= \text{ sup}_{u\in R(D^g)} {{\|S^f_1 u-S^g_1 u\|_{L^1}}\over{|u^+-u^-|}}\) is the 'distance' between \(f\) and \(g\), and \( R(D^f)\) is the set of initial data to the Riemann problem (1)-(2): NEWLINE\[NEWLINE u(0,x)=\{u^- \text{ for } x<0 , \text{ and } x^+ \text{ for } x>0\}. \tag{2}NEWLINE\]NEWLINE{}.