### Abstract

Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto‐fluid dynamics including new constructive local existence theorems for the time‐singular limit equations. In particular, the authors give an entirely self‐contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier‐Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.

Original language | English (US) |
---|---|

Pages (from-to) | 481-524 |

Number of pages | 44 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 34 |

Issue number | 4 |

DOIs | |

State | Published - Jul 1981 |

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### All Science Journal Classification (ASJC) codes

- Applied Mathematics
- Mathematics(all)

### Cite this

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**Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids.** / Klainerman, Sergiu; Majda, Andrew.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids

AU - Klainerman, Sergiu

AU - Majda, Andrew

PY - 1981/7

Y1 - 1981/7

N2 - Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto‐fluid dynamics including new constructive local existence theorems for the time‐singular limit equations. In particular, the authors give an entirely self‐contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier‐Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.

AB - Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto‐fluid dynamics including new constructive local existence theorems for the time‐singular limit equations. In particular, the authors give an entirely self‐contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier‐Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.

UR - http://www.scopus.com/inward/record.url?scp=84980156074&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84980156074&partnerID=8YFLogxK

U2 - https://doi.org/10.1002/cpa.3160340405

DO - https://doi.org/10.1002/cpa.3160340405

M3 - Article

VL - 34

SP - 481

EP - 524

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 4

ER -