Data assimilation for the primitive equations in <i>H</i> ²

Abstract In this paper, we prove that the solution to the primitive equations is predicted by the corresponding nudging-type data assimilation (DA) equations in H 2 . Although the DA equation does not include the direct information about the base solution and its initial conditions, the solution to the DA equation exponentially converges to the base (original) solution when the external forces are known even before they are observed. Additionally, when the external forces are not entirely known but their spatially dense observations are available, the DA is stable, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>.</mml:mo> <mml:mi>e</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> , the DA solution lies in a sufficiently small neighbourhood of the base solution. The convergence and stability are proved by energy estimates and the maximal Hölder regularity of the perturbed hydrostatic Stokes operator. The H 2 convergence shown in this paper is stronger than the results by Pei and by the author.