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We provide an unconditional L2 upper bound for the boundary layer separation of Leray–Hopf solutions in a smooth bounded domain. By layer separation, we mean the discrepancy between a (turbulent) low-viscosity Leray–Hopf solution u ν and a fixed (laminar) regular Euler solution u ¯ with similar initial conditions and body force. We show an asymptotic upper bound C | | u ¯ | | L ∞ 3 T on the layer separation, anomalous dissipation, and the work done by friction. This extends the previous result when the Euler solution is a regular shear in a finite channel. The key estimate is to control the boundary vorticity in a way that does not degenerate in the vanishing viscosity limit. [ABSTRACT FROM AUTHOR]

Dafermos [Arch. Rational Mech. Anal. 70 (1979), pp. 167–179] proved the weak/strong principle for conservation laws. It states that Lipschitz solutions to conservation laws endowed with convex entropies are unique and stable among weak solutions. The method, based on relative entropy, was extended by Di Perna [Indiana Univ. Math. J. 28 (1979), pp. 137–188] to show the uniqueness of shocks among weak solutions with strong traces. This theory has been recently revisited with the notion of weighted contractions up to shifts. We review in this paper recent applications of this method, including the weak/BV principle and the stability of discontinuous solutions among inviscid double limits of Navier-Stokes systems. [ABSTRACT FROM AUTHOR]

In this paper, we study the a -contraction property of small extremal shocks for 1-d systems of hyperbolic conservation laws endowed with a single convex entropy, when subjected to large perturbations. We show that the weight coefficient a can be chosen with amplitude proportional to the size of the shock. The main result of this paper is a key building block in the companion paper [G. Chen, S. G. Krupa and A. F. Vasseur, Uniqueness and weak-BV stability for 2 × 2 conservation laws, Arch. Ration. Mech. Anal. 246(1) (2022) 299–332] in which uniqueness and BV-weak stability results for 2 × 2 systems of hyperbolic conservation laws are proved. [ABSTRACT FROM AUTHOR]

Consider the steady solution to the incompressible Euler equation ū = Ae1 in the periodic tunnel Ω = Td-1 × (0, 1) in dimension d = 2, 3. Consider now the family of solutions üv to the associated Navier--Stokes equation with the no-slip condition on the flat boundaries, for small viscosities v = A/Re, and initial values in L². We are interested in the weak inviscid limits up to subsequences uv → u∞ when both the viscosity v converges to 0, and the initial value uv0 converges to Ae1 in L². Under a conditional assumption on the energy dissipation close to the boundary, Kato showed in 1984 that uu converges to Ae1 strongly in L² uniformly in time under this double limit. It is still unknown whether this inviscid limit is unconditionally true. The convex integration method produces solutions uE to the Euler equation with the same initial values Ae1 which verify at time 0 < T < T0: || uE(T) Ae1||²L²(Ω) approx A³T. This predicts the possibility of a layer separation with an energy of order A³T. We show in this paper that the energy of layer separation associated with any asymptotic u∞ obtained via double limits cannot be more than ||u∞ (T) Ae1||² L²(Ω) ≲ A³T. This result holds unconditionally for any weak limit of Leray--Hopf solutions of the Navier--Stokes equation. Especially, it shows that, even if the limit is not unique, the shear flow pattern is observable up to time 1/A. This provides a notion of stability despite the possible nonuniqueness of the limit predicted by the convex integration theory. The result relies on a new boundary vorticity estimate for the Navier--Stokes equation. This new estimate, inspired by previous work on higher regularity estimates for Navier--Stokes, provides a nonlinear control scalable through the inviscid limit. [ABSTRACT FROM AUTHOR]