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Continuous Data Assimilation for Semilinear Parabolic Equations: A General Approach by Evolution Equations

Gianmarco Del Sarto, Matthias Hieber, Filippo Palma, Tarek Zöchling

Abstract

This article develops a general framework for continuous deterministic data assimilation for semilinear parabolic equations by means of evolution equations. Introducing a nudged model driven by partial observations, the global well-posedness of the reference and the approximating systems is established under natural assumptions. In addition, it is shown that the approximating solution converges exponentially to the solution of the reference system, provided the observational resolution and the nudging parameter are suitably chosen. The approach allows us to consider many systems, such as the Allen-Cahn, Cahn-Hilliard, Sellers-type energy balance, and bidomain systems, for the first time.

Continuous Data Assimilation for Semilinear Parabolic Equations: A General Approach by Evolution Equations

Abstract

This article develops a general framework for continuous deterministic data assimilation for semilinear parabolic equations by means of evolution equations. Introducing a nudged model driven by partial observations, the global well-posedness of the reference and the approximating systems is established under natural assumptions. In addition, it is shown that the approximating solution converges exponentially to the solution of the reference system, provided the observational resolution and the nudging parameter are suitably chosen. The approach allows us to consider many systems, such as the Allen-Cahn, Cahn-Hilliard, Sellers-type energy balance, and bidomain systems, for the first time.
Paper Structure (11 sections, 12 theorems, 100 equations)

This paper contains 11 sections, 12 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Main results
  3. Illustrations of the General Framework - strong solutions
  4. 2D-Navier-Stokes equations
  5. 3D-Primitive Equations
  6. Energy Balance Model coupled to an active fluid
  7. 2D-Bidomain problem with FitzHugh-Nagumo transport
  8. Illustrations of the General Framework - weak solutions
  9. 2D-Navier-Stokes equations - revisited
  10. 1D-Allen-Cahn equation
  11. 1D and 2D Cahn-Hilliard equation

Key Result

Lemma 1

Let $T>0$ and assume that $\bf(A1)-\bf(A3)$ are satisfied. Then for any $u_0 \in \mathcal{H}$, there exists $a = a(u_0)\leq T$ such that problem eq:model admits a unique solution The solution exists on a maximal time interval $[0,a_{\mathrm{max}}(u_0))$ and depends continuously on the data. If the solution does not exist globally in time, i. e., if $a_{\mathrm{max}} <T$, the maximal existence tim

Theorems & Definitions (19)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Corollary 1: Global well-posedness of \ref{['eq:model shift']}
  • proof
  • Proposition 1: Global well-posedness of the Data Assimilation problem \ref{['eq:model data']}
  • proof
  • Remark 3
  • Theorem 2.1: Convergence in the $\mathcal{H}$-norm
  • Corollary 2: Convergence in the $\mathcal{V}^\ast$-norm
  • ...and 9 more