On maximum modulus estimates of the navier–stokes equations with nonzero boundary data

Tongkeun Chang; Hi Jun Choe; Kyungkeun Kang

doi:10.1137/17M1152565

On maximum modulus estimates of the navier–stokes equations with nonzero boundary data

Tongkeun Chang, Hi Jun Choe, Kyungkeun Kang

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We consider discontinuous influx for the Navier–Stokes flow and construct a solution that is unbounded in a neighborhood of a discontinuous point of given bounded boundary data for any dimension larger than or equal to two. This is an extension of the result in [T. Chang and H. Choe, J. Differential Equations, 254 (2013), pp. 2682–2704] that a blow-up solution exists with a bounded and discontinuous boundary data for the Stokes flow. If the normal component of bounded boundary data is Dini-continuous in space or log-Dini-continuous in time, then the constructed solution becomes bounded and a maximum modulus estimate is valid.

Original language	English
Pages (from-to)	3147-3171
Number of pages	25
Journal	SIAM Journal on Mathematical Analysis
Volume	50
Issue number	3
DOIs	https://doi.org/10.1137/17M1152565
Publication status	Published - 2018

Bibliographical note

Funding Information:
∗Received by the editors October 17, 2017; accepted for publication (in revised form) April 6, 2018; published electronically June 19, 2018. http://www.siam.org/journals/sima/50-3/M115256.html Funding: The work of the first author was supported by 2017R1D1A1B03033427. The work of the second author was supported by NRF-2015R1A5A1009350. The work of the third author was supported by NRF-2017R1A2B4006484. †Department of Mathematics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul, 120-749, South Korea (chang7357@yonsei.ac.kr, choe@yonsei.ac.kr, kkang@yonsei.ac.kr).

Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.

All Science Journal Classification (ASJC) codes

Analysis
Computational Mathematics
Applied Mathematics

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title = "On maximum modulus estimates of the navier–stokes equations with nonzero boundary data",

abstract = "We consider discontinuous influx for the Navier–Stokes flow and construct a solution that is unbounded in a neighborhood of a discontinuous point of given bounded boundary data for any dimension larger than or equal to two. This is an extension of the result in [T. Chang and H. Choe, J. Differential Equations, 254 (2013), pp. 2682–2704] that a blow-up solution exists with a bounded and discontinuous boundary data for the Stokes flow. If the normal component of bounded boundary data is Dini-continuous in space or log-Dini-continuous in time, then the constructed solution becomes bounded and a maximum modulus estimate is valid.",

author = "Tongkeun Chang and Choe, {Hi Jun} and Kyungkeun Kang",

note = "Funding Information: ∗Received by the editors October 17, 2017; accepted for publication (in revised form) April 6, 2018; published electronically June 19, 2018. http://www.siam.org/journals/sima/50-3/M115256.html Funding: The work of the first author was supported by 2017R1D1A1B03033427. The work of the second author was supported by NRF-2015R1A5A1009350. The work of the third author was supported by NRF-2017R1A2B4006484. †Department of Mathematics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul, 120-749, South Korea (chang7357@yonsei.ac.kr, choe@yonsei.ac.kr, kkang@yonsei.ac.kr). Publisher Copyright: {\textcopyright} 2018 Society for Industrial and Applied Mathematics.",

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AU - Chang, Tongkeun

AU - Choe, Hi Jun

AU - Kang, Kyungkeun

N1 - Funding Information: ∗Received by the editors October 17, 2017; accepted for publication (in revised form) April 6, 2018; published electronically June 19, 2018. http://www.siam.org/journals/sima/50-3/M115256.html Funding: The work of the first author was supported by 2017R1D1A1B03033427. The work of the second author was supported by NRF-2015R1A5A1009350. The work of the third author was supported by NRF-2017R1A2B4006484. †Department of Mathematics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul, 120-749, South Korea (chang7357@yonsei.ac.kr, choe@yonsei.ac.kr, kkang@yonsei.ac.kr). Publisher Copyright: © 2018 Society for Industrial and Applied Mathematics.

PY - 2018

Y1 - 2018

N2 - We consider discontinuous influx for the Navier–Stokes flow and construct a solution that is unbounded in a neighborhood of a discontinuous point of given bounded boundary data for any dimension larger than or equal to two. This is an extension of the result in [T. Chang and H. Choe, J. Differential Equations, 254 (2013), pp. 2682–2704] that a blow-up solution exists with a bounded and discontinuous boundary data for the Stokes flow. If the normal component of bounded boundary data is Dini-continuous in space or log-Dini-continuous in time, then the constructed solution becomes bounded and a maximum modulus estimate is valid.

AB - We consider discontinuous influx for the Navier–Stokes flow and construct a solution that is unbounded in a neighborhood of a discontinuous point of given bounded boundary data for any dimension larger than or equal to two. This is an extension of the result in [T. Chang and H. Choe, J. Differential Equations, 254 (2013), pp. 2682–2704] that a blow-up solution exists with a bounded and discontinuous boundary data for the Stokes flow. If the normal component of bounded boundary data is Dini-continuous in space or log-Dini-continuous in time, then the constructed solution becomes bounded and a maximum modulus estimate is valid.

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On maximum modulus estimates of the navier–stokes equations with nonzero boundary data

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