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A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three
Applications of Mathematics ( IF 0.6 ) Pub Date : 2020-04-06 , DOI: 10.21136/am.2020.0228-19
Huanyuan Li

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 $$\parallel\triangledown\varrho\parallel_{{L^\infty}(0,T;W^{1,q})}+\parallel{u}\parallel_{{L^s}(0,T;L_\omega^r)}<\infty$$ ∥ ▽ 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 $$L_\omega^r$$ L ω r denotes the weak L r space.

中文翻译:

三维非齐次 Navier-Stokes-Korteweg 方程强解的爆破判据

本文证明了具有真空的 3D 密度相关 Navier-Stokes-Korteweg 方程的 Serrin 型膨胀准则。表明如果密度 ϱ 和速度场 u 满足 $$\parallel\triangledown\varrho\parallel_{{L^\infty}(0,T;W^{1,q})}+\parallel{u} \parallel_{{L^s}(0,T;L_\omega^r)}<\infty$$ ∥ ▽ ϱ ∥ L ∞ ( 0 , T ; W 1 , q ) + ∥ u ∥ L s ( 0 , T ; L ω r ) < ∞ 对于某些 q > 3 和任何 ( r , s ) 满足 2/ s + 3/ r ⩽ 1, 3, < r ⩽ ∞,然后是密度依赖 Navier-Stokes 的强解-Korteweg 方程可以在 [0, T ] 上全局存在。这里$$L_\omega^r$$ L ω r 表示弱L r 空间。
更新日期:2020-04-06
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