Relative energy method for weak–strong uniqueness of the inhomogeneous Navier–Stokes equations far from vacuum - Crin-Barat, Timothée; Skondrić, Stefan; Violini, Alessandro (Journal of Evolution Equations, 2025)
We present a weak–strong uniqueness result for the inhomogeneous Navier–Stokes equations in ℝd (d=2,3) for bounded initial densities that are far from vacuum. Given a strong solution, i.e. a solution satisfying the equation as an identity in L2, and a Leray–Hopf weak solution, we establish that they coincide if the initial data agree. Our proof strategy is based on the relative energy method and new W-1,p-type stability estimates for the density. A key point lies in proving that every Leray–Hopf weak solution originating from initial densities far from vacuum remains distant from vacuum at all times.
CRIS-Team Scopus Importer:2025-02-17
Keywords: Inhomogeneous Navier–Stokes equations; Leray–Hopf weak solutions; Relative energy method; Weak–strong uniqueness