On the role of pressure in the theory of MHD equations

Jiří Neustupa; Minsuk Yang

doi:10.1016/j.nonrwa.2020.103283

On the role of pressure in the theory of MHD equations

Jiří Neustupa, Minsuk Yang

Department of Mathematics

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

3 Citations (Scopus)

Abstract

We consider the system of MHD equations in Ω×(0,T), where Ω is a domain in R³ and T>0, with the no slip boundary condition for the velocity u and the Navier-type boundary condition for the magnetic induction b. We show that an associated pressure p, as a distribution with a certain structure, can be always assigned to a weak solution (u,b). The pressure is a function with some rate of integrability if the domain Ω is “smooth”, see section 3. In section 4, we study the regularity of p in a sub-domain Ω₁×(t₁,t₂) of Ω×(0,T), where u (or, alternatively, both u and b) satisfies Serrin's integrability conditions. Regularity criteria for weak solutions to the MHD equations in terms of [Formula presented] are studied in section 5. Finally, section 6 contains remarks on analogous results in the case of Navier's or Navier-type boundary conditions for the velocity u.

Original language	English
Article number	103283
Journal	Nonlinear Analysis: Real World Applications
Volume	60
DOIs	https://doi.org/10.1016/j.nonrwa.2020.103283
Publication status	Published - 2021 Aug

Bibliographical note

Funding Information:
The authors would like to thank the anonymous reviewer for his/her valuable comments and suggestions. The first author acknowledges the support of the Grant Agency of the Czech Republic , Grant No. GA19-04243S , and the Academy of Sciences of the Czech Republic ( RVO 67985840 ). The second author has been supported by the National Research Foundation of Korea No. 2016R1C1B2015731 , No. 2015R1A5A1009350 and the Yonsei University, Republic of Korea No. 2019-22-0034 .

Publisher Copyright:
© 2021 Elsevier Ltd

All Science Journal Classification (ASJC) codes

Analysis
Engineering(all)
Economics, Econometrics and Finance(all)
Computational Mathematics
Applied Mathematics

Access to Document

10.1016/j.nonrwa.2020.103283

Cite this

@article{0ab0e2ff5e9041188b1c6a5a5074f2a7,

title = "On the role of pressure in the theory of MHD equations",

abstract = "We consider the system of MHD equations in Ω×(0,T), where Ω is a domain in R3 and T>0, with the no slip boundary condition for the velocity u and the Navier-type boundary condition for the magnetic induction b. We show that an associated pressure p, as a distribution with a certain structure, can be always assigned to a weak solution (u,b). The pressure is a function with some rate of integrability if the domain Ω is “smooth”, see section 3. In section 4, we study the regularity of p in a sub-domain Ω1×(t1,t2) of Ω×(0,T), where u (or, alternatively, both u and b) satisfies Serrin's integrability conditions. Regularity criteria for weak solutions to the MHD equations in terms of [Formula presented] are studied in section 5. Finally, section 6 contains remarks on analogous results in the case of Navier's or Navier-type boundary conditions for the velocity u.",

author = "Ji{\v r}{\'i} Neustupa and Minsuk Yang",

note = "Funding Information: The authors would like to thank the anonymous reviewer for his/her valuable comments and suggestions. The first author acknowledges the support of the Grant Agency of the Czech Republic , Grant No. GA19-04243S , and the Academy of Sciences of the Czech Republic ( RVO 67985840 ). The second author has been supported by the National Research Foundation of Korea No. 2016R1C1B2015731 , No. 2015R1A5A1009350 and the Yonsei University, Republic of Korea No. 2019-22-0034 . Publisher Copyright: {\textcopyright} 2021 Elsevier Ltd",

year = "2021",

month = aug,

doi = "10.1016/j.nonrwa.2020.103283",

language = "English",

volume = "60",

journal = "Nonlinear Analysis: Real World Applications",

issn = "1468-1218",

publisher = "Elsevier BV",

}

TY - JOUR

T1 - On the role of pressure in the theory of MHD equations

AU - Neustupa, Jiří

AU - Yang, Minsuk

N1 - Funding Information: The authors would like to thank the anonymous reviewer for his/her valuable comments and suggestions. The first author acknowledges the support of the Grant Agency of the Czech Republic , Grant No. GA19-04243S , and the Academy of Sciences of the Czech Republic ( RVO 67985840 ). The second author has been supported by the National Research Foundation of Korea No. 2016R1C1B2015731 , No. 2015R1A5A1009350 and the Yonsei University, Republic of Korea No. 2019-22-0034 . Publisher Copyright: © 2021 Elsevier Ltd

PY - 2021/8

Y1 - 2021/8

N2 - We consider the system of MHD equations in Ω×(0,T), where Ω is a domain in R3 and T>0, with the no slip boundary condition for the velocity u and the Navier-type boundary condition for the magnetic induction b. We show that an associated pressure p, as a distribution with a certain structure, can be always assigned to a weak solution (u,b). The pressure is a function with some rate of integrability if the domain Ω is “smooth”, see section 3. In section 4, we study the regularity of p in a sub-domain Ω1×(t1,t2) of Ω×(0,T), where u (or, alternatively, both u and b) satisfies Serrin's integrability conditions. Regularity criteria for weak solutions to the MHD equations in terms of [Formula presented] are studied in section 5. Finally, section 6 contains remarks on analogous results in the case of Navier's or Navier-type boundary conditions for the velocity u.

AB - We consider the system of MHD equations in Ω×(0,T), where Ω is a domain in R3 and T>0, with the no slip boundary condition for the velocity u and the Navier-type boundary condition for the magnetic induction b. We show that an associated pressure p, as a distribution with a certain structure, can be always assigned to a weak solution (u,b). The pressure is a function with some rate of integrability if the domain Ω is “smooth”, see section 3. In section 4, we study the regularity of p in a sub-domain Ω1×(t1,t2) of Ω×(0,T), where u (or, alternatively, both u and b) satisfies Serrin's integrability conditions. Regularity criteria for weak solutions to the MHD equations in terms of [Formula presented] are studied in section 5. Finally, section 6 contains remarks on analogous results in the case of Navier's or Navier-type boundary conditions for the velocity u.

UR - http://www.scopus.com/inward/record.url?scp=85100195256&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85100195256&partnerID=8YFLogxK

U2 - 10.1016/j.nonrwa.2020.103283

DO - 10.1016/j.nonrwa.2020.103283

M3 - Article

AN - SCOPUS:85100195256

SN - 1468-1218

VL - 60

JO - Nonlinear Analysis: Real World Applications

JF - Nonlinear Analysis: Real World Applications

M1 - 103283

ER -

On the role of pressure in the theory of MHD equations

Abstract

Bibliographical note

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this