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Zbl 0542.35062
Valli, Alberto
Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 10, 607-647 (1983). ISSN 0391-173X

This is a substantial piece of work on a very difficult problem: the boundary-initial value problem for compressible flow with zero boundary conditions. The paper essentially begins with a proof of local existence under weak hypotheses than those of previous writers and follows with a uniqueness theorem and global a-priori estimates. The next section is very interesting and uses energy stability arguments in conjunction with comparisons to the solution to the stationary Stokes problem to derive sufficient conditions to ensure asymptotic stability. Finally, an existence result is established for periodic solutions and this is extended for a time-independent body force to obtain a stationary solution as the period tends to zero.
[B.Straughan]

MSC 2000:: *35Q30 Stokes and Navier-Stokes equations
35A05 General existence and uniqueness theorems (PDE)
35B10 Periodic solutions of PDE
35B45 A priori estimates
35B05 General behavior of solutions of PDE

Keywords: boundary-initial value problem; compressible flow; zero boundary conditions; local existence; uniqueness; global a-priori estimates; energy stability; comparisons; stationary Stokes problem

Cited in: Zbl 0627.76080 Zbl 0611.76082

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