The genuinely nonlinear Graetz–Nusselt ultraparabolic equation

We study a second-order quasilinear ultraparabolic equation whose matrix of the coefficients of the second derivatives is nonnegative, depends on the time and spatial variables, and can change rank in the case when it is diagonal and the coefficients of the first derivatives can be discontinuous. We prove that if the equation is a priori known to enjoy the maximum principle and satisfies the additional “genuine nonlinearity” condition then the Cauchy problem with arbitrary bounded initial data has at least one entropy solution and every uniformly bounded set of entropy solutions is relatively compact in $L_{\mathrm{loc}}^1$. The proofs are based on introduction and systematic study of the kinetic formulation of the equation in question and application of the modification of the Tartar $H$-measures proposed by E. Yu. Panov.