Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier-Stokes equations. (Existence et stabilité de solutions non caractéristiques et stationnaires des équations de Navier-Stokes non isentropiques sur un intervalle.) (English. French summary) Zbl 1541.35337
The 1D compressible nonsteady Navier-Stokes equations are considered, with noncharacteristic inflow-outflow boundary conditions. The existence, uniqueness, and stability of steady solutions of the full (nonisentropic) above system on a bounded interval are studied, for large-amplitude data. The spectral stability is studied by using numerical Evans function investigations – see [B. Barker et al., SIAM J. Appl. Dyn. Syst. 17, No. 2, 1766–1785 (2018; Zbl 1395.65131)]. Important tools are the Morse index of the linearized operator about the wave, the Brouwer degree and a “Cauchy-to-boundary value” map \(\Psi\). Roughly speaking, \(\Psi\) realizes the correspondence between the conditions of the problem and the solution – see (1.7), (1.8). The properties of \(\Psi\) and the existence theorem are given in Sections 2–3. The uniqueness is proved in Proposition 4.1: if \(\Psi\) is bijective for some particular data, then we have uniqueness; otherwise “there is at least one choice of data possessing multiple solutions”. I would say that, to some extent, the conclusion seems to be contained in the hypothesis (which is very strong). The spectral stability and the Evans function are explained, by using the stability index. Numerical results for simple gases are given in Section 6. The computation of the Evans function is obtained with the package STABLAB. Very interesting plots illustrate the obtained results. Some very useful numerical and analytical examples concerning the non uniqueness cases are given in the last section, where an abstract bifurcation result is also obtained. Details regarding the MATLAB-based package STABLAB (used here) are given in the appendix – see [B. Barker et al., Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci. 376, No. 2117, Article ID 20170184, 25 p. (2018; Zbl 1402.65052)].
Reviewer: Gelu Paşa (Bucureşti)
MSC:
| 35Q30 | Navier-Stokes equations |
| 76N06 | Compressible Navier-Stokes equations |
| 76N15 | Gas dynamics (general theory) |
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 35B32 | Bifurcations in context of PDEs |
| 35B35 | Stability in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35P05 | General topics in linear spectral theory for PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
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