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Abridged Continuous Data Assimilation for the 2D Navier–Stokes Equations Utilizing Measurements of Only One Component of the Velocity Field

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Abstract

We introduce a continuous data assimilation (downscaling) algorithm for the two-dimensional Navier–Stokes equations employing coarse mesh measurements of only one component of the velocity field. This algorithm can be implemented with a variety of finitely many observables: low Fourier modes, nodal values, finite volume averages, or finite elements. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate asymptotically in time, to the unique exact unknown reference solution, of the 2D Navier–Stokes equations, associated with the observed (finite dimensional projection of) velocity.

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Authors and Affiliations

  1. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904-4137, USA

    Aseel Farhat

  2. Department of Mathematics, Indiana University, Bloomington, Bloomington, IN, 47405, USA

    Aseel Farhat

  3. Department of Mathematics, United States Naval Academy, Annapolis, MD, 21401, USA

    Evelyn Lunasin

  4. Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX, 77843-3368, USA

    Edriss S. Titi

  5. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel

    Edriss S. Titi

Authors
  1. Aseel Farhat
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  2. Evelyn Lunasin
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  3. Edriss S. Titi
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Correspondence to Aseel Farhat.

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Communicated by A. V. Fursikov

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Farhat, A., Lunasin, E. & Titi, E.S. Abridged Continuous Data Assimilation for the 2D Navier–Stokes Equations Utilizing Measurements of Only One Component of the Velocity Field. J. Math. Fluid Mech. 18, 1–23 (2016). https://doi.org/10.1007/s00021-015-0225-6

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  • DOI: https://doi.org/10.1007/s00021-015-0225-6

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