Linear stability analysis of the homogeneous Couette flow in a 2D isentropic compressible fluid

In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain $$\mathbb {T}\times \mathbb {R}$$ T × R . In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we prove that their $$L^2$$ L 2 norm grows as $$t^{1/2}$$ t 1 / 2 and this confirms previous observations in the physics literature. On the contrary, the solenoidal component of the velocity field experiences inviscid damping, namely it decays to zero even in the absence of viscosity. For a viscous compressible fluid, we show that the perturbations may have a transient growth of order $$\nu ^{-1/6}$$ ν - 1 / 6 (with $$\nu ^{-1}$$ ν - 1 being proportional to the Reynolds number) on a time-scale $$\nu ^{-1/3}$$ ν - 1 / 3 , after which it decays exponentially fast. This phenomenon is also called enhanced dissipation and our result appears to be the first to detect this mechanism for a compressible flow, where an exponential decay for the density is not a priori trivial given the absence of dissipation in the continuity equation.

Maximal $$L^q$$-Regularity for Parabolic Hamilton–Jacobi Equations and Applications to Mean Field Games

In this paper we investigate maximal $$L^q$$ L q -regularity for time-dependent viscous Hamilton–Jacobi equations with unbounded right-hand side and superlinear growth in the gradient. Our approach is based on the interplay between new integral and Hölder estimates, interpolation inequalities, and parabolic regularity for linear equations. These estimates are obtained via a duality method à la Evans. This sheds new light on the parabolic counterpart of a conjecture by P.-L. Lions on maximal regularity for Hamilton–Jacobi equations, recently addressed in the stationary framework by the authors. Finally, applications to the existence problem of classical solutions to Mean Field Games systems with unbounded local couplings are provided.