Incompressible Limit for Solutionsof the Isentropic Navier–Stokes Equationswith Dirichlet Boundary Conditions

The incompressible limit of solutions of the two‐dimensional compressible Euler system with degenerating initial data

“…To get the simplest system possible, we set v (t, x) =γc 0ṽ (γc 0 t, x) and c (t, x) = c 0 + γc 0c (γc 0 t, x). This problem, of course, has already been studied in numerous articles; our reference list, although far from exhaustive, contains more items than we will refer to in the text [1,6,7,8,19,20,21,22,24,26]. Fluids evolving in different domains have been considered (R N , torus, bounded domain), essentially under two kinds of hypotheses: in the well-prepared case [19,20], one assumes that divṽ 0, → 0 andc 0, → 0 as → 0; in the ill-prepared case, one only assumes thatṽ 0, andc 0, are bounded in certain Sobolev spaces (for example) and that the incompressible part ofṽ 0, tends to some field v 0 .…”

The mathematical theory of low Mach number flows

“…Surprisingly, when both fast waves and boundaries are present then the interaction of fast waves with a boundary layer usually makes them decay fast in bounded domains, so that v ε converges to the solution of the slow equations [12].…”

Asymptotic Analysis of Acoustic Waves in a Porous Medium: Microincompressible Flow

“…Secondly, the time behavior that impacts the micro-acoustic part of the flow, namely the boundary layer in time of size ε (3−β)/2 and associated time oscillations are non-trivial and are generated by a boundary layer in space of size ε (β+1)/2 . This time behavior associated to micro-acoustic flow is essentially given in [6] (see also [15] for a more precise construction) for the case β = 2, and we refer the reader to [8], [9] for further details in the general case 1 < β. Lastly, we invite the reader to compare with Figure 1 and note that boundary layer in time of size ε 2−β breaks when β = 2, hence the coupling of the pressure and velocity of micro-incompressible flow.…”