Inflow-outflow problems for Euler equations in a rectangular cylinder

Abstract

We prove that some inflow-outflow problems for the Euler equations in a (nonsmooth) bounded cylinder admit a regular solution. The problems considered are symmetric hyperbolic systems with partly characteristic and partly noncharacteristic boundary; for such problems, no general theory is available. Therefore, we introduce particular spaces of functions satisfying suitable additional boundary conditions which allow to determine a regular solution by means of a ''reflection technique''.2000 Mathematics Subject Classification: 35L45, 76N10, 76J20.

Persistence of the Steady Normal Shock Structure for the Unsteady Potential Flow

“…It is worth pointing out that, since the solid boundary is perturbed and no longer flat, the symmetry assumptions proposed in [39,63] fail to be valid in this problem. Therefore, new ideas and methods must be developed to deal with the dihedral singularity, which is also completely different from the one caused by the corner singularity in [37].…”

“…In fact, Osher has given examples in [58,59] showing that hyperbolic equations in cornered space domain may be ill-posed. On the other hand, for the well-posedness problem of hyperbolic equations in space-domains with non-smooth boundaries, there are also positive results, for instance, see [39][40][41]63]. In particular, under certain symmetry assumptions, Gazzola-Secchi [39] studied the inflowoutflow problem in a bounded cylinder.…”

“…On the other hand, for the well-posedness problem of hyperbolic equations in space-domains with non-smooth boundaries, there are also positive results, for instance, see [39][40][41]63]. In particular, under certain symmetry assumptions, Gazzola-Secchi [39] studied the inflowoutflow problem in a bounded cylinder. Then Yuan [63] studied the dynamic stability of normal shock in a duct with flat boundaries in two space dimensions.…”

“…Such symmetry assumptions fail to be valid in the problem (1.3), (1.7), and (1.13)- (1.15) studied in this paper, since the solid wall Γ 0 is a curved surface. Hence the methods developed in [39,63] are not applicable. Nevertheless, the assumption that Γ 0 is a slightly perturbed surface from a flat one implies that there may hold some symmetry properties under certain transformation.…”

Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains

“…The main purpose of the present paper (Theorem 1) is to prove regularity and uniqueness results for (3) in a suitable class of merely Lipschitz domains, the sectors, see Definition 3 below; this class includes all the domains in Figure 1. For the proofs we take advantage of the reflection method introduced in [14] for the Euler equations and subsequently applied in [3,15] to the Navier-Stokes equations. The reflection is possible because we have Navier boundary conditions; under Dirichlet boundary conditions the same argument does not allow smooth extensions of the involved functions and vector fields.…”