Well-Posedness of the Cauchy Problem

Part of the Applied Mathematical Sciences book series (AMS, volume 152)


The goal of this chapter is to show that the limit found by front tracking, that is, the weak solution of the initial value problem
$$\displaystyle u_{t}+f(u)_{x}=0,\quad u(x,0)=u_{0}(x),$$
is stable in L 1 with respect to perturbations in the initial data. In other words, if \(v=v(x,t)\) is another solution found by front tracking, then
$$\displaystyle{\left\|u(\,\cdot\,,t)-v(\,\cdot\,,t)\right\|}_{1}\leq C{\left\|u_{0}-v_{0}\right\|}_{1}$$
for some constant C. Furthermore, we shall show that under some mild extra entropy conditions, every weak solution coincides with the solution constructed by front tracking.


Weak Solution Shock Front Borel Measure Riemann Problem Radon Measure 
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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.Department of MathematicsUniversity of OsloOsloNorway

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