This paper studied a class of fluid models with non-local effects, which consisted of compressible Euler equations coupled with incompressible Navier-Stokes equations, where the communication kernel of the non-local terms had a fractional Laplacian operator structure. Euler equations were reconstructed into a hyperbolic system with symmetric structure. After overcoming the nonlinear properties of non-local terms, the local well-posedness theory and regularity criterion were established for the initial value problems of such models by using the method of energy estimation.
HUANG Baoyun
,
TONG Lining
. Local classical solutions of coupled fluid models with non-local effects[J]. Journal of Shanghai University, 2026
, 32(4)
: 728
-740
.
DOI: 10.12066/j.issn.1007-2861.2586
[1] Carriillo J A, Feireisl E, Gwiazda P, et al. Weak solutions for Euler systems with non-local interactions [J]. Journal of the London Mathematical Society, 2017, 95(3): 705-724.
[2] Choi Y P. The global Cauchy problem for compressible Euler equations with a nonlocal dissipation [J]. Mathematical Models and Methods in Applied Sciences, 2019, 29(1): 185-207.
[3] Carrillo J A, Wráoblewska-Kamiánska A, Zatorska Z. On long-time asymptotics for viscous hydrodynamic models of collective behavior with damping and nonlocal interactions [J]. Mathematical Models and Methods in Applied Sciences, 2017, 29(1): 31-63.
[4] Shvydkoy R, Tadmor E. Eulerian dynamics with a commutator forcing III: fractional diffusion of order 0< fi < 1[J]. Physica D: Nonlinear Phenomena, 2018, 376: 131-137.
[5] Luk J, Speck J. The stability of simple plane-symmetric shock formation for three-dimensional compressible Euler flow with vorticity and entropy [J]. Analysis & PDE, 2024, 17(3): 831-941.
[6] Tong L N, Chen L. Decay estimate to a compressible Euler system with nonlocal velocity alignment [J]. Communications in Mathematical Sciences, 2020, 18(5): 1469-1475.
[7] Chen L, Tan C H, Tong L N. On the global classical solution to compressible Euler system with singular velocity alignment [J]. Methods and Applications Analysis, 2021, 28(2): 153-172.
[8] Tong L N, Chen L, G?ottlich S, et al. The global classical solution to compressible Euler system with velocity alignment [J]. AIMS Mathematics, 2020, 5(6): 6673-6693.
[9] Gagliardo E. Propriettà di alcune classi di funzioni in più variabili [J]. Matematika, 1961, 5(4): 87-116.
[10] Kusunoki Y, Mori S I. Some remarks on boundary values of harmonic functions with finite Dirichlet integrals [J]. Journal of Mathematics of Kyoto University, 1967, 7(3): 315-324.
[11] Di-Nezza E, Palatucci G, Valdinoci E. Hitchhikerfis guide to the fractional Sobolev spaces [J]. Bulletin des Sciences Matháematiques, 2012, 136(5): 521-573.
[12] Li D. On Kato-Ponce and fractional Leibniz [J]. Revista Matemáatica Iberoamericana, 2019, 35(1): 23-100.
[13] Miao Q Y, Tan C H, Xue L T. Global regularity for a 1d Euler alignment system with misalignment [J]. Mathematical Models and Methods in Applied Sciences, 2021, 31(3): 473-524.
[14] Lee Y, Tan C H. A sharp critical threshold for a traffic flow model with look-ahead dynamics [J]. Communications in Mathematical Sciences, 2022, 20(4): 1151-1172.
[15] Majda A. Compressible fluid flow and systems of conservation laws in several space variables [M]. New York: Springer-Verlag, 1984.
[16] Benzoni-Gavage S, Serre D. Multi-dimensional hyperbolic partial differential equations: firstorder systems and applications [M]. New York: Oxford University Press, 2007.
[17] Ha S Y, Kang M J, Kwon B. A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid [J]. Mathematical Models and Methods in Applied Sciences, 2014, 24(11): 2311-2359.
