Local well-posedness of solutions to 2D magnetic Prandtl model in the Prandtl-Hartmann regime

Yuming Qin; Xiuqing Wang; Junchen Liu

Applications of Mathematics (2025)

  • Issue: 2, page 169-202
  • ISSN: 0862-7940

Abstract

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We consider the 2D magnetic Prandtl equation in the Prandtl-Hartmann regime in a periodic domain and prove the local existence and uniqueness of solutions by energy methods in a polynomial weighted Sobolev space. On the one hand, we have noted that the x -derivative of the pressure P plays a key role in all known results on the existence and uniqueness of solutions to the Prandtl-Hartmann regime equations, in which the case of favorable P ( ∂ x P < 0 ) or the case of ∂ x P = 0 (led by constant outer flow U = constant ) was only considered. While in this paper, we have no restriction on the sign of ∂ x P , which has generalized all previous results and definitely gives rise to a difficulty in mathematical treatments. To overcome this difficulty, we shall use the skill of cancellation mechanism which is valid under the monotonicity assumption. One the other hand, we consider the general outer flow U ≠ constant , leading to the boundary data at y = 0 being much more complicated. To deal with these boundary data, some more delicate estimates and mathematical induction method will be used. Therefore, our result also provides an extension of earlier studies by addressing the challenges arising from general outer flow.

How to cite

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Qin, Yuming, Wang, Xiuqing, and Liu, Junchen. "Local well-posedness of solutions to 2D magnetic Prandtl model in the Prandtl-Hartmann regime." Applications of Mathematics (2025): 169-202. <http://eudml.org/doc/299982>.

@article{Qin2025,
abstract = {We consider the 2D magnetic Prandtl equation in the Prandtl-Hartmann regime in a periodic domain and prove the local existence and uniqueness of solutions by energy methods in a polynomial weighted Sobolev space. On the one hand, we have noted that the $x$-derivative of the pressure $P$ plays a key role in all known results on the existence and uniqueness of solutions to the Prandtl-Hartmann regime equations, in which the case of favorable $P$$(\partial _x P<0)$ or the case of $\partial _x P=0$ (led by constant outer flow $U=\{\rm constant\}$) was only considered. While in this paper, we have no restriction on the sign of $\partial _x P$, which has generalized all previous results and definitely gives rise to a difficulty in mathematical treatments. To overcome this difficulty, we shall use the skill of cancellation mechanism which is valid under the monotonicity assumption. One the other hand, we consider the general outer flow $U\ne \{\rm constant\}$, leading to the boundary data at $y=0$ being much more complicated. To deal with these boundary data, some more delicate estimates and mathematical induction method will be used. Therefore, our result also provides an extension of earlier studies by addressing the challenges arising from general outer flow.},
author = {Qin, Yuming, Wang, Xiuqing, Liu, Junchen},
journal = {Applications of Mathematics},
keywords = {Prandtl-Hartmann; boundary layer; local well-posedness},
language = {eng},
number = {2},
pages = {169-202},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Local well-posedness of solutions to 2D magnetic Prandtl model in the Prandtl-Hartmann regime},
url = {http://eudml.org/doc/299982},
year = {2025},
}

TY - JOUR
AU - Qin, Yuming
AU - Wang, Xiuqing
AU - Liu, Junchen
TI - Local well-posedness of solutions to 2D magnetic Prandtl model in the Prandtl-Hartmann regime
JO - Applications of Mathematics
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 2
SP - 169
EP - 202
AB - We consider the 2D magnetic Prandtl equation in the Prandtl-Hartmann regime in a periodic domain and prove the local existence and uniqueness of solutions by energy methods in a polynomial weighted Sobolev space. On the one hand, we have noted that the $x$-derivative of the pressure $P$ plays a key role in all known results on the existence and uniqueness of solutions to the Prandtl-Hartmann regime equations, in which the case of favorable $P$$(\partial _x P<0)$ or the case of $\partial _x P=0$ (led by constant outer flow $U={\rm constant}$) was only considered. While in this paper, we have no restriction on the sign of $\partial _x P$, which has generalized all previous results and definitely gives rise to a difficulty in mathematical treatments. To overcome this difficulty, we shall use the skill of cancellation mechanism which is valid under the monotonicity assumption. One the other hand, we consider the general outer flow $U\ne {\rm constant}$, leading to the boundary data at $y=0$ being much more complicated. To deal with these boundary data, some more delicate estimates and mathematical induction method will be used. Therefore, our result also provides an extension of earlier studies by addressing the challenges arising from general outer flow.
LA - eng
KW - Prandtl-Hartmann; boundary layer; local well-posedness
UR - http://eudml.org/doc/299982
ER -

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