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Observability inequalities for degenerate transport equations

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Abstract

In this paper, we prove an observability inequality for a degenerate transport equation with time-dependent coefficients. First we introduce a local in time Carleman estimate for the degenerate equation, then we apply it to obtain a global in time observability inequality by using also an energy estimate.

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Acknowledgements

This work was supported by Grant-in-Aid for Scientific Research (S) Grant Number JP15H05740, Grant-in-Aid for JSPS Fellows Grant Number JP20J11497, and Istituto Nazionale di Alta Matematica (IN\(\delta \)AM), through the GNAMPA Research Project 2020, titled “Problemi inversi e di controllo per equazioni di evoluzione e loro applicazioni,” coordinated by the first author. Moreover, this research was performed in the framework of the French-German-Italian Laboratoire International Associé (LIA), named COPDESC, on Applied Analysis, issued by CNRS, MPI, and IN\(\delta \)AM, during the IN\(\delta \)AM Intensive Period-2019, “Shape optimization, control and inverse problems for PDEs,” held in Napoli in May–June–July 2019. The authors thank Prof. Piermarco Cannarsa and Prof. Masahiro Yamamoto for the useful and interesting discussions about the topics of this paper.

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Authors and Affiliations

  1. Dipartimento di Patrimonio, Architettura e Urbanistica (PAU), Via dell’Ùniversità 25, Università Mediterranea di Reggio Calabria, 89124, Reggio Calabria, Italy

    Giuseppe Floridia

  2. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Hiroshi Takase

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  1. Giuseppe Floridia
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  2. Hiroshi Takase
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Correspondence to Hiroshi Takase.

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Floridia, G., Takase, H. Observability inequalities for degenerate transport equations. J. Evol. Equ. 21, 5037–5053 (2021). https://doi.org/10.1007/s00028-021-00740-z

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  • DOI: https://doi.org/10.1007/s00028-021-00740-z

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