# Low Mach Number Limit and Far Field Convergence Rates of Irrotational Flows in Multidimensional Nozzles with an Obstacle Inside

## Abstract.

In this paper, the low Mach number limit and far field convergence rates are investigated for steady irrotational flows with external forces in three-dimensional infinitely long nozzles with an obstacle inside. First, with the aid of uniform a priori estimates, we proved the well-posedness theory for both incompressible flows and compressible subsonic flows with external forces in a multidimensional nozzle with an obstacle inside. Furthermore, the uniformly subsonic compressible flows tend to the incompressible flows with quadratic order of Mach numbers as the compressibility parameter goes to zero. We finally give the convergence rates of both incompressible and compressible flows at far fields as the boundary of the nozzle goes to flat even when the forces do not admit convergence rates at far fields. These results reveal how the external forces affect the convergence rates of the flows at far fields.

## Acknowledgment.

The authors thank the referees for their helpful comments, which help improve the presentation of the paper.

## References

1.
T. Alazard, Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions, Adv. Differential Equations, 10 (2005), pp. 19–44.
2.
L. Bers, Existence and uniqueness of a subsonic flow past a given profile, Comm. Pure Appl. Math., 7 (1954), pp. 441–504.
3.
L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Surv. Appli. Math. 3, John Wiley & Sons, New York, 1958.
4.
C. Chen, L. Du, C. Xie, and Z. Xin, Two dimensional subsonic Euler flows past a wall or a symmetric body, Arch. Ration. Mech. Anal., 221 (2016), pp. 559–602.
5.
C. Chen and C. Xie, Existence of steady subsonic Euler flows through infinitely long periodic nozzles, J. Differential Equations, 252 (2012), pp. 4315–4331.
6.
G.-Q. Chen, C. Christoforou, and Y. Zhang, Continuous dependence of entropy solutions to the Euler equations on the adiabatic exponent and Mach number, Arch. Ration. Mech. Anal., 189 (2008), pp. 97–130.
7.
G.-Q. G. Chen, F. Huang, T.-Y. Wang, and W. Xiang, Incompressible limit of solutions of multidimensional steady compressible Euler equations, Z. Angew. Math. Phys., 67 (2016), 75.
8.
G.-Q. G. Chen, F.-M. Huang, T.-Y. Wang, and W. Xiang, Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles, Adv. Math., 346 (2019), pp. 946–1008.
9.
G. C. Dong, Nonlinear Partial Differential Equations of Second Order, Transl. Math. Monogr. 95, AMS, Providence, RI, 1991.
10.
G. C. Dong and B. Ou, Subsonic flows around a body in space, Comm. Partial Differential. Equations, 18 (1993), pp. 355–379.
11.
L. Du and B. Duan, Global subsonic Euler flows in an infinitely long axisymmetric nozzle, J. Differential Equations, 250 (2011), pp. 813–847.
12.
L. Du and B. Duan, Subsonic Euler flows with large vorticity through an infinitely long axisymmetric nozzle, J. Math. Fluid Mech., 18 (2016), pp. 511–530.
13.
L. Du, C. Xie, and Z. Xin, Steady subsonic ideal flows through an infinitely long nozzle with large vorticity, Comm. Math. Phys., 328 (2014), pp. 327–354.
14.
L. Du, Z. Xin, and W. Yan, Subsonic flows in a multi-dimensional nozzle, Arch. Ration. Mech. Anal., 201 (2011), pp. 965–1012.
15.
B. Duan and S. Weng, Global smooth axisymmetric subsonic flows with nonzero swirl in an infinitely long axisymmetric nozzle, Z. Angew. Math. Phys., 69 (2018), 135.
16.
L. C. Evans, Partial Differential Equations, 2nd ed., AMS, Providence, RI, 2010.
17.
R. Finn and D. Gilbarg, Asymptotic behavior and uniqueness of plane subsonic flows, Comm. Pure Appl. Math., 10 (1957), pp. 23–63.
18.
R. Finn and D. Gilbarg, Three-dimensional subsonic flows, and asymptotic estimates for elliptic partial differential equations, Acta Math., 98 (1957), pp. 265–296.
19.
F. I. Frankl and M. V. Keldysh, Dieäussere Neumann’she aufgabe für nichtlineare elliptische differentialgleichungen mit anwendung auf die theorie der flügel im kompressiblen gas, Izv. Akad. Nauk SSSR, 12 (1934), pp. 561–607.
20.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer, Berlin, 2001.
21.
X. Gu and T.-Y. Wang, On subsonic and subsonic-sonic flows with general conservatives force in exterior domians, Acta Math. Sci. Ser. B Engl. Ed., to appear.
22.
X. Gu and T.-Y. Wang, On subsonic and subsonic-sonic flows in the infinity long nozzle with general conservatives force, Acta Math. Sci. Ser. B Engl. Ed., 37 (2017), pp. 752–767.
23.
F. Huang, T. Wang, and Y. Wang, On multi-dimensional sonic-subsonic flow, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), pp. 2131–2140.
24.
S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), pp. 481–524.
25.
S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), pp. 629–651.
26.
M. Li, T.-Y. Wang, and W. Xiang, Low mach number limit of multidimensional steady flows on the airfoil problem, Calc. Var. Partial Differential Equations, 59 (2020), 68.
27.
M. Li, T.-Y. Wang, and W. Xiang, Low mach number limit of steady Euler flows in multi-dimensional nozzles, Commun. Math. Sci., 18 (2020), pp. 1191–1220.
28.
L. Ma and C. Xie, Existence and optimal convergence rates of multi-dimensional subsonic potential flows through an infinitely long nozzle with an obstacle inside, J. Math. Phys., 61 (2020), 071514.
29.
N. Masmoudi, Examples of singular limits in hydrodynamics, in Handbook of Differential Equations: Evolutionary Equations, Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, pp. 195–275.
30.
G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), pp. 61–90.
31.
O. A. Oleĭnik and G. A. Iosif’yan, The behavior at infinity of the solutions of second-order elliptic equations in domains with a noncompact boundary, Mat. Sb. (N.S.), 112 (1980), pp. 588–610.
32.
M. Schiffer, Analytical theory of subsonic and supersonic flows, in Handbuch der Physik, Vol. 9, Part 3, Springer-Verlag, Berlin, 1960, pp. 1–161.
33.
S. Schochet, The mathematical theory of low mach number flows, Math. Model. Numer. Anal., 39 (2005), pp. 441–458.
34.
M. Shiffman, On the existence of subsonic flows of a compressible fluid, Proc. Natl. Acad. Sci. USA, 38 (1952), pp. 434–438.
35.
S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ., 26 (1986), pp. 323–331.
36.
M. Van Dyke, Perturbation Methods in Fluid Mechanics, Appl. Math. Mech. 8, Academic Press, New York, 1964.
37.
T.-Y. Wang and J. Zhang, Low Mach number limit of steady flows through infinite multidimensional largely-open nozzles, J. Differential Equations, 269 (2020), pp. 1863–1903.
38.
C. Xie and Z. Xin, Global subsonic and subsonic-sonic flows through infinitely long nozzles, Indiana Univ. Math. J., 56 (2007), pp. 2991–3023.
39.
C. Xie and Z. Xin, Existence of global steady subsonic Euler flows through infinitely long nozzles, SIAM J. Math. Anal., 42 (2010), pp. 751–784.
40.
C. Xie and Z. Xin, Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles, J. Differential Equations, 248 (2010), pp. 2657–2683.
41.
V. A. Zorich, Mathematical Analysis, Springer, Berlin, 2004.

## Information & Authors

### Information

#### Published In

SIAM Journal on Mathematical Analysis
Pages: 36 - 67
ISSN (online): 1095-7154

#### History

Submitted: 20 January 2022
Accepted: 22 September 2022
Published online: 24 January 2023

### Authors

#### Affiliations

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China.
Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan, Hubei 430070, China.
School of Mathematical Sciences, Institute of Natural Sciences, Ministry of Education Key Laboratory of Scientific and Engineering Computing, and CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China.

#### Funding Information

Funding: The research of the second author was supported in part by NSFC grants 11601401 and 11971024. The research of the third author was partially supported by NSFC grants 11971307 and 11631008, Natural Science Foundation of Shanghai grant 21ZR1433300, and Shanghai Science and Technology Commission grant 22XD1421400.

## Metrics & Citations

### Citations

#### Cited By

There are no citations for this item