A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three

Huanyuan Li

Applications of Mathematics (2021)

  • Volume: 66, Issue: 1, page 43-55
  • ISSN: 0862-7940

Abstract

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This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density ρ and velocity field u satisfy ∥ ∇ ρ ∥ L ∞ ( 0 , T ; W 1 , q ) + ∥ u ∥ L s ( 0 , T ; L ω r ) < ∞ for some q > 3 and any ( r , s ) satisfying 2 / s + 3 / r ≤ 1 , 3 < r ≤ ∞ , then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over [ 0 , T ] . Here L ω r denotes the weak L r space.

How to cite

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Li, Huanyuan. "A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three." Applications of Mathematics 66.1 (2021): 43-55. <http://eudml.org/doc/297374>.

@article{Li2021,
abstract = {This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density $\rho $ and velocity field $u$ satisfy $\Vert \nabla \rho \Vert _\{L^\{\infty \}(0,T; W^\{1,q\})\} + \Vert u\Vert _\{L^s(0,T; L^r_\{\omega \})\}< \infty $ for some $q>3$ and any $(r,s)$ satisfying $2/s+3/r \le 1$, $3 <r \le \infty ,$ then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over $[0,T]$. Here $L^r_\{\omega \}$ denotes the weak $L^r$ space.},
author = {Li, Huanyuan},
journal = {Applications of Mathematics},
keywords = {Navier-Stokes-Korteweg equations; capillary fluid; blow-up criterion; vacuum; strong solutions},
language = {eng},
number = {1},
pages = {43-55},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three},
url = {http://eudml.org/doc/297374},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Li, Huanyuan
TI - A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 43
EP - 55
AB - This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density $\rho $ and velocity field $u$ satisfy $\Vert \nabla \rho \Vert _{L^{\infty }(0,T; W^{1,q})} + \Vert u\Vert _{L^s(0,T; L^r_{\omega })}< \infty $ for some $q>3$ and any $(r,s)$ satisfying $2/s+3/r \le 1$, $3 <r \le \infty ,$ then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over $[0,T]$. Here $L^r_{\omega }$ denotes the weak $L^r$ space.
LA - eng
KW - Navier-Stokes-Korteweg equations; capillary fluid; blow-up criterion; vacuum; strong solutions
UR - http://eudml.org/doc/297374
ER -

References

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