# American Institute of Mathematical Sciences

March  2004, 3(1): 97-113. doi: 10.3934/cpaa.2004.3.97

## On the slightly compressible MHD system in the half-plane

 1 Dipartimento di Matematica, Università di Brescia, Facoltà di Ingegneria, Via Valotti 9, 25133 Brescia, Italy

Received  February 2003 Revised  October 2003 Published  January 2004

In this paper we prove the existence of a smooth compressible solution for the MHD system in the half-plane. It is well-known that, as the Mach number goes to zero, the compressible MHD problem converges to the incompressible one, which has a global solution in time. Hence, it is natural to expect that, for Mach number sufficiently small, the compressible solution exists on any arbitrary time interval, with no restriction on the size of the initial velocity. In order to obtain the existence result, we decompose the solution as the sum of the solution of the irrotational Euler problem, the solution of the incompressible MHD system and the solution of the remainder problem which describes the interaction between the first two components. We show that the solution of the remainder part exists on any arbitrary time interval. Since this holds also for the solution of the irrotational Euler problem, this yields the existence of the smooth compressible solution for the MHD system.
Citation: Paola Trebeschi. On the slightly compressible MHD system in the half-plane. Communications on Pure and Applied Analysis, 2004, 3 (1) : 97-113. doi: 10.3934/cpaa.2004.3.97
 [1] Jishan Fan, Fucai Li, Gen Nakamura. Low Mach number limit of the full compressible Hall-MHD system. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1731-1740. doi: 10.3934/cpaa.2017084 [2] Lan Zeng, Guoxi Ni, Yingying Li. Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5503-5522. doi: 10.3934/dcdsb.2019068 [3] Fucai Li, Yanmin Mu. Low Mach number limit for the compressible magnetohydrodynamic equations in a periodic domain. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1669-1705. doi: 10.3934/dcds.2018069 [4] Jishan Fan, Fucai Li, Gen Nakamura. Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain. Conference Publications, 2015, 2015 (special) : 387-394. doi: 10.3934/proc.2015.0387 [5] Fucai Li, Yanmin Mu, Dehua Wang. Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces. Kinetic and Related Models, 2017, 10 (3) : 741-784. doi: 10.3934/krm.2017030 [6] Eduard Feireisl, Hana Petzeltová. Low Mach number asymptotics for reacting compressible fluid flows. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 455-480. doi: 10.3934/dcds.2010.26.455 [7] Jingrui Su. Global existence and low Mach number limit to a 3D compressible micropolar fluids model in a bounded domain. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3423-3434. doi: 10.3934/dcds.2017145 [8] Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 [9] Thomas Alazard. A minicourse on the low Mach number limit. Discrete and Continuous Dynamical Systems - S, 2008, 1 (3) : 365-404. doi: 10.3934/dcdss.2008.1.365 [10] Baoquan Yuan, Xiaokui Zhao. Blowup of smooth solutions to the full compressible MHD system with compact density. Kinetic and Related Models, 2014, 7 (1) : 195-203. doi: 10.3934/krm.2014.7.195 [11] Young-Pil Choi. Compressible Euler equations interacting with incompressible flow. Kinetic and Related Models, 2015, 8 (2) : 335-358. doi: 10.3934/krm.2015.8.335 [12] Shuxing Chen, Gui-Qiang Chen, Zejun Wang, Dehua Wang. A multidimensional piston problem for the Euler equations for compressible flow. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 361-383. doi: 10.3934/dcds.2005.13.361 [13] Qing Chen, Zhong Tan. Time decay of solutions to the compressible Euler equations with damping. Kinetic and Related Models, 2014, 7 (4) : 605-619. doi: 10.3934/krm.2014.7.605 [14] Jianwei Yang, Dongling Li, Xiao Yang. On the quasineutral limit for the compressible Euler-Poisson equations. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022020 [15] Quentin Chauleur. The isothermal limit for the compressible Euler equations with damping. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022059 [16] Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure and Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503 [17] Ming Lu, Yi Du, Zheng-An Yao. Blow-up phenomena for the 3D compressible MHD equations. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1835-1855. doi: 10.3934/dcds.2012.32.1835 [18] Ming He, Jianwen Zhang. Global cylindrical solution to the compressible MHD equations in an exterior domain. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1841-1865. doi: 10.3934/cpaa.2009.8.1841 [19] Ming Lu, Yi Du, Zheng-An Yao, Zujin Zhang. A blow-up criterion for the 3D compressible MHD equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1167-1183. doi: 10.3934/cpaa.2012.11.1167 [20] Donatella Donatelli, Bernard Ducomet, Šárka Nečasová. Low Mach number limit for a model of accretion disk. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3239-3268. doi: 10.3934/dcds.2018141

2021 Impact Factor: 1.273