### Abstract

We prove the finite-time existence of a solution to the Euler-Lagrange equations corresponding to the necessary conditions for minimization of a functional defining varational a assimilation of observational data into the two-dimensional, incompressible Euler equations. The data are given by linear functionals acting on the space of functions representing vorticity. The d data are sparse and avilable on a fixed space-time domain. The objective of the data assimilation is to obtain an estimate of the vorticity which minimizes a cost functional and is anlogous to a distributed parameter control problem. The cost functional i is the sum of a weighted squared error in the dynamics, the initial condition, and in the misfit to the observed data. Vorticity estimates which minimize the cost functional are obtained by solving the corresponding system of Euler-Lagrange Equations. The Euler-Lagrange system is a coupled two-point boundary value problem in time. An application of the Schauder fixed-point theorem establishes the existence of a least one solution to the system. Iterative methods for generating solutions have proven useful in applications in meterology and oceanography.

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

Pages (from-to) | 437-454 |

Number of pages | 18 |

Journal | Inverse Problems |

Volume | 12 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1996 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Inverse Problems*,

*12*(4), 437-454. https://doi.org/10.1088/0266-5611/12/4/007

**Local existence results for the generalized inverse of the vorticity equation in the plane.** / Hagelberg, Carl R.; Bennett, Andrew F.; Jones, Donald.

Research output: Contribution to journal › Article

*Inverse Problems*, vol. 12, no. 4, pp. 437-454. https://doi.org/10.1088/0266-5611/12/4/007

}

TY - JOUR

T1 - Local existence results for the generalized inverse of the vorticity equation in the plane

AU - Hagelberg, Carl R.

AU - Bennett, Andrew F.

AU - Jones, Donald

PY - 1996/8

Y1 - 1996/8

N2 - We prove the finite-time existence of a solution to the Euler-Lagrange equations corresponding to the necessary conditions for minimization of a functional defining varational a assimilation of observational data into the two-dimensional, incompressible Euler equations. The data are given by linear functionals acting on the space of functions representing vorticity. The d data are sparse and avilable on a fixed space-time domain. The objective of the data assimilation is to obtain an estimate of the vorticity which minimizes a cost functional and is anlogous to a distributed parameter control problem. The cost functional i is the sum of a weighted squared error in the dynamics, the initial condition, and in the misfit to the observed data. Vorticity estimates which minimize the cost functional are obtained by solving the corresponding system of Euler-Lagrange Equations. The Euler-Lagrange system is a coupled two-point boundary value problem in time. An application of the Schauder fixed-point theorem establishes the existence of a least one solution to the system. Iterative methods for generating solutions have proven useful in applications in meterology and oceanography.

AB - We prove the finite-time existence of a solution to the Euler-Lagrange equations corresponding to the necessary conditions for minimization of a functional defining varational a assimilation of observational data into the two-dimensional, incompressible Euler equations. The data are given by linear functionals acting on the space of functions representing vorticity. The d data are sparse and avilable on a fixed space-time domain. The objective of the data assimilation is to obtain an estimate of the vorticity which minimizes a cost functional and is anlogous to a distributed parameter control problem. The cost functional i is the sum of a weighted squared error in the dynamics, the initial condition, and in the misfit to the observed data. Vorticity estimates which minimize the cost functional are obtained by solving the corresponding system of Euler-Lagrange Equations. The Euler-Lagrange system is a coupled two-point boundary value problem in time. An application of the Schauder fixed-point theorem establishes the existence of a least one solution to the system. Iterative methods for generating solutions have proven useful in applications in meterology and oceanography.

UR - http://www.scopus.com/inward/record.url?scp=0000151936&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000151936&partnerID=8YFLogxK

U2 - 10.1088/0266-5611/12/4/007

DO - 10.1088/0266-5611/12/4/007

M3 - Article

AN - SCOPUS:0000151936

VL - 12

SP - 437

EP - 454

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 4

ER -