Abstract
This paper is concerned with the convergence rates of subsonic flows for airfoil problem and infinite long largely-open nozzle problem, which is an improvement of Finn and Gilbarg (Acta Math 98:265-296, 1957); Payne and Weinberger (Acta Math 98:297-299, 1957); Dong and Ou (Commun Partial Differ Equ 18(1–2):355–379, 1993); and Liu and Yuan (Calc Var Partial Differ Equ 49(1–2):1–36, 2014). The maximum principle is applied to estimate the potential function, by choosing the proper compared functions. Then, by the weighted Schauder estimates, the convergence rates of velocity at the far field are shown as \(|{\textbf{x}}|^{-n+1}\). Furthermore, we construct the examples to show the optimality of our convergence rates and the expansion of the incompressible airfoil flow at infinity, indicating the higher convergence rates \(|{\textbf{x}}|^{-n}\).
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References
Bers, L.: Existence and uniqueness of a subsonic flow past a given profile. Commun. Pure Appl. Math. 7, 441–504 (1954)
Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics. Surveys in Applied Mathematics, vol. 3. John Wiley & Sons, New York (1958)
Chen, G.-Q.G., Huang, F.-M., Wang, T.-Y., Xiang, W.: Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles. Adv. Math. 346, 946–1008 (2019)
Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience, New York (1948)
Dong, G.C.: Nonlinear Partial Differential Equations of Second Order. Translations of Mathematical Monographs, vol. 95. American Mathematical Society, Providence (1991)
Dong, G.C., Ou, B.: Subsonic flows around a body in space. Commun. Partial Differ. Equ. 18(1–2), 355–379 (1993)
Du, L., Xin, Z., Yan, W.: Subsonic flows in a multi-dimensional nozzle. Arch. Ration. Mech. Anal. 201(3), 965–1012 (2011)
Du, L., Xie, C., Xin, Z.: Steady subsonic ideal flows through an infinitely long nozzle with large vorticity. Commun. Math. Phys. 328(1), 327–354 (2014)
Finn, R., Gilbarg, D.: Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations. Acta Math. 98, 265–296 (1957)
Finn, R., Gilbarg, D.: Asymptotic behavior and uniquenes of plane subsonic flows. Commun. Pure Appl. Math. 10, 23–63 (1957)
Frankl, F.I., Keldysh, M.V.: Dieäussere neumann’she aufgabe für nichtlineare elliptische differentialgleichungen mit anwendung auf die theorie der flügel im kompressiblen gas. Izvestiya Akademii Nauk SSSR 12, 561–607 (1934)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics, Springer, Berlin (2001)
Han, Q., Lin, F.: Elliptic Partial Differential Equations. Courant Lecture Notes in Mathematics, vol. 1, 2nd edn. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence (2011)
Liu, L., Yuan, H.: Steady subsonic potential flows through infinite multi-dimensional largely-open nozzles. Calc. Var. Partial Differ. Equ. 49(1–2), 1–36 (2014)
Ma, L.: The optimal convergence rates of non-isentropic subsonic Euler flows through the infinitely long three-dimensional axisymmetric nozzles. Math. Methods Appl. Sci. 43(10), 6553–6565 (2020)
Ma, L., Xie, C.: Existence and optimal convergence rates of multi-dimensional subsonic potential flows through an infinitely long nozzle with an obstacle inside. J. Math. Phys. 61(7), 071514 (2020)
Ma, L., Wang, T.-Y., Xie, C.: Low Mach number limit and far field convergence rates of irrotational flows in multidimensional nozzles with an obstacle inside. SIAM J. Math. Anal. 55(1), 36–67 (2023)
Ou, B.: An irrotational and incompressible flow around a body in space. J. Partial Differ. Equ. 7(2), 160–170 (1994)
Payne, L.E., Weinberger, H.F.: Note on a lemma of Finn and Gilbarg. Acta Math. 98, 297–299 (1957)
Shiffman, M.: On the existence of subsonic flows of a compressible fluid. Proc. Nat. Acad. Sci. USA 38(5), 434–438 (1952)
Wang, T.-Y., Zhang, J.: Low Mach number limit of steady flows through infinite multidimensional largely-open nozzles. J. Differ. Equ. 269(3), 1863–1903 (2020)
Xie, C., Xin, Z.: Global subsonic and subsonic-sonic flows through infinitely long nozzles. Indiana Univ. Math. J. 56(6), 2991–3023 (2007)
Xie, C., Xin, Z.: Existence of global steady subsonic Euler flows through infinitely long nozzles. SIAM J. Math. Anal. 42(2), 751–784 (2010)
Acknowledgements
The research of Lei Ma was partially supported in partial by NSFC Grants 12301288. The research of Tian-Yi Wang was supported in partial by NSFC Grants 11971307 and 12061080. The authors would like to thank Professor Chunjing Xie and Professor Wen Yang for helpful discussions.
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Ma, L., Wang, TY. On the convergence rates of multi-dimensional subsonic irrotational flows in unbounded domains. Z. Angew. Math. Phys. 74, 240 (2023). https://doi.org/10.1007/s00033-023-02128-0
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DOI: https://doi.org/10.1007/s00033-023-02128-0