Linear transport equations with \(\mu\)-monotone coefficients (Q5945094)

The authors introduce the new concept of a \(\mu\)-monotone function, which replaces Filippov's condition. This family of conditions is larger than Filippov's class. It allows to study the existence, uniqueness, and stability for a wide class of ODEs as well as linear transport equations. An important case is \(\mu (z)=(|z|_p)^p\), \(p\in [1,\infty)\), which is called \(p\)-monotonicity. If \(p=2\), one obtains the Filippov's condition. It is shown that an ODE with a \(\mu\)-monotone right-hand side has a unique on the left Filippov solution. In the particular case of \(p\)-monotone velocities, the Cauchy problem for the nonconservative linear transport equation has a Lipschitz solution. Under additional hypotheses, this solution is stable. Finally, the uniqueness for weak solutions to conservation laws is established.