We will discuss recent developments of the theory of a-contraction with shifts to study the stability of discontinuous solutions of systems of equations modeling inviscid compressible flows, like the compressible Euler equation.
In the one dimensional configuration, the Bressan theory shows that small BV solutions are stable under small BV perturbations (together with a technical condition known as bounded variations on space-like curve).
The theory of a-contraction allows to extend the Bressan theory to a weak/BV stability result allowing wild perturbations fulfilling only the so-called strong trace property. Especially, it shows that the technical condition of BV on space-like curve is not needed. (joint work with Sam Krupa and Geng Chen).
We will show several applications of the theory of a-contraction with shifts on the barotropic Navier-Stokes equation. Together with Moon-Jin Kang and Yi wang, we proved the conjecture of Matsumura (first mentioned in 1986). It consists in proving the time asymptotic stability of composite waves of viscous shocks and rarefactions.
Together with Moon-Jin Kang, we proved also that inviscid shocks of the Euler equation, are stable among the family of inviscid limits of Navier-Stokes equation (Inventiones 2021). This stability result holds in the class of wild perturbations of inviscid limits, without any regularity restriction (not even strong trace property). This shows that the class of inviscid limits of Navier-Stokes equations is better behaved that the class of weak solutions to the inviscid limit problem.
This is obtained thanks to a stability result at the level of Navier-Stokes, which is uniform with respect to the viscosity, allowing asymptotically infinitely large perturbations (JEMS 2021).
A first multi D result of stability of contact discontinuities without shear, in the class of inviscid limit of Fourier-Navier-Stokes, shows that the same property is true for some situations even in multi D (joint work with Moon-jin Kang and Yi Wang).