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Pure Mathematics 2026
不可压缩多孔介质方程在Lei-Lin空间中解的全局存在性和解析性
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Abstract:
本文考虑二维不可压缩多孔介质方程的Cauchy问题。首先通过Fourier变换将原方程转化为等价积分方程，随后在Lei-Lin型函数空间框架下，系统分析了线性项与非线性项的估计。基于Banach动点定理，我们证明
充分小时，方程存在唯一的全局解。此外，进一步给出了方程解的解析性。
This paper investigates the Cauchy problem for the two-dimensional incompressible porous medium equation. First, we transform the original equation into an equivalent integral equation via the Fourier transform. Subsequently, within the framework of Lei-Lin-type function spaces, we systematically analyze the estimates of both the linear and nonlinear terms. Based on the Banach fixed-point theorem, we prove that when the initial data
is sufficiently small, there exists a unique global solution to the equation. Furthermore, we establish the analyticity of the solution.
| [1] | Castro, Á., Córdoba, D. and Lear, D. (2018) Global Existence of Quasi-Stratified Solutions for the Confined IPM Equation. Archive for Rational Mechanics and Analysis, 232, 437-471. https://doi.org/10.1007/s00205-018-1324-3 |
| [2] | Isett, P. and Vicol, V. (2015) Hölder Continuous Solutions of Active Scalar Equations. Annals of PDE, 1, 1-77. https://doi.org/10.1007/s40818-015-0002-0 |
| [3] | Melo, W.G., Perusato, C. and Rocha, N.F. (2019) On Local Existence, Uniqueness and Blow-Up of Solutions for the Generalized MHD Equations in Lei-Lin Spaces. Zeitschrift für Angewandte Mathematik und Physik, 70, 1-24. https://doi.org/10.1007/s00033-019-1119-x |
| [4] | Bianchini, R., Córdoba, D. and Martínez-Zoroa, L. (2025) Non Existence and Strong Ill-Posedness in H2 for the Stable IPM Equation. Journal of Functional Analysis, 289, Article 111097. https://doi.org/10.1016/j.jfa.2025.111097 |
| [5] | Yuan, B. and Yuan, J. (2009) Global Well-Posedness of Incompressible Flow in Porous Media with Critical Diffusion in Besov Spaces. Journal of Differential Equations, 246, 4405-4422. https://doi.org/10.1016/j.jde.2009.01.022 |
| [6] | Córdoba, D., Gancedo, F. and Orive, R. (2007) Analytical Behavior of Two-Dimensional Incompressible Flow in Porous Media. Journal of Mathematical Physics, 48, Article 065206. https://doi.org/10.1063/1.2404593 |
| [7] | Córdoba, D. and Gancedo, F. (2007) Contour Dynamics of Incompressible 3-D Fluids in a Porous Medium with Different Densities. Communications in Mathematical Physics, 273, 445-471. https://doi.org/10.1007/s00220-007-0246-y |
| [8] | Friedlander, S., Gancedo, F., Sun, W. and Vicol, V. (2012) On a Singular Incompressible Porous Media Equation. Journal of Mathematical Physics, 53, Article 115602. https://doi.org/10.1063/1.4725532 |
| [9] | Castro, Á., Córdoba, D., Fefferman, C. and Gancedo, F. (2013) Breakdown of Smoothness for the Muskat Problem. Archive for Rational Mechanics and Analysis, 208, 805-909. https://doi.org/10.1007/s00205-013-0616-x |
| [10] | Cordoba, D., Faraco, D. and Gancedo, F. (2010) Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation. Archive for Rational Mechanics and Analysis, 200, 725-746. https://doi.org/10.1007/s00205-010-0365-z |
| [11] | Elgindi, T.M. (2017) On the Asymptotic Stability of Stationary Solutions of the Inviscid Incompressible Porous Medium Equation. Archive for Rational Mechanics and Analysis, 225, 573-599. https://doi.org/10.1007/s00205-017-1090-7 |
| [12] | Kiselev, A. and Yao, Y. (2022) Small Scale Formations in the Incompressible Porous Media Equation. Archive for Rational Mechanics and Analysis, 247, Article No. 1. https://doi.org/10.1007/s00205-022-01830-z |
| [13] | Székelyhidi Jr, L. (2012) Relaxation of the Incompressible Porous Media Equation. Annales Scientifiques de l’École normale supérieure, 45, 491-509. https://doi.org/10.24033/asens.2171 |
| [14] | Xue, L. (2009) On the Well-Posedness of Incompressible Flow in Porous Media with Supercritical Diffusion. Applicable Analysis, 88, 547-561. https://doi.org/10.1080/00036810902913821 |
| [15] | Lei, Z. and Lin, F. (2011) Global Mild Solutions of Navier‐Stokes Equations. Communications on Pure and Applied Mathematics, 64, 1297-1304. https://doi.org/10.1002/cpa.20361 |
| [16] | Melo, W.G., de Souza, M. and Santos, T.S.R. (2022) On the Generalized Magnetohydrodynamicsα Equations with Fractional Dissipation in Lei-Lin and Lei-Lin-Gevrey Spaces. Zeitschrift für Angewandte Mathematik und Physik, 73, Article No. 44. https://doi.org/10.1007/s00033-022-01677-0 |
| [17] | Benhamed, M. and Abusalim, S.M. (2020) Long Time Behavior of the Solution of the Two-Dimensional Dissipative QGE in Lei-Lin Spaces. International Journal of Mathematics and Mathematical Sciences, 2020, 1-6. https://doi.org/10.1155/2020/6409609 |
| [18] | Yuan, B. and Xiao, Y. (2023) The Global Well-Posedness of Strong Solutions to 2D MHD Equations in Lei-Lin Space. Acta Mathematicae Applicatae Sinica, English Series, 39, 647-655. https://doi.org/10.1007/s10255-023-1068-1 |
| [19] | Xiao, C., Qiu, H. and Yao, Z. (2024) The Global Existence and Analyticity of a Mild Solution to the 3D Regularized MHD Equations. Acta Mathematica Scientia, 44, 973-983. https://doi.org/10.1007/s10473-024-0311-z |
| [20] | Benameur, J. and Bennaceur, M. (2020) Large Time Behaviour of Solutions to the 3D-NSE in xσ Spaces. Journal of Mathematical Analysis and Applications, 482, Article 123566. https://doi.org/10.1016/j.jmaa.2019.123566 |
| [21] | Benameur, J. (2015) Long Time Decay to the Lei-Lin Solution of 3D Navier-Stokes Equations. Journal of Mathematical Analysis and Applications, 422, 424-434. https://doi.org/10.1016/j.jmaa.2014.08.039 |
| [22] | Benameur, J. and Abdallah, S.B. (2021) Asymptotic Behavior of Critical Dissipative Quasi-Geostrophic Equation in Fourier Space. Journal of Mathematical Analysis and Applications, 497, Article 124873. https://doi.org/10.1016/j.jmaa.2020.124873 |
| [23] | Benameur, J. and Jlali, L. (2020) Long Time Decay of 3D-NSE in Lei-Lin-Gevrey Spaces. Mathematica Slovaca, 70, 877-892. https://doi.org/10.1515/ms-2017-0400 |
| [24] | Bae, H. (2015) Existence and Analyticity of Lei-Lin Solution to the Navier-Stokes Equations. Proceedings of the American Mathematical Society, 143, 2887-2892. https://doi.org/10.1090/s0002-9939-2015-12266-6 |
| [25] | Ye, Z. (2015) Global Well-Posedness and Decay Results to 3D Generalized Viscous Magnetohydrodynamic Equations. Annali di Matematica Pura ed Applicata (1923-), 195, 1111-1121. https://doi.org/10.1007/s10231-015-0507-x |
| [26] | Ferreira, L.C.F. and Villamizar-Roa, E.J. (2011) Exponentially-Stable Steady Flow and Asymptotic Behavior for the Magnetohydrodynamic Equations. Communications in Mathematical Sciences, 9, 499-516. https://doi.org/10.4310/cms.2011.v9.n2.a7 |
