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<p>In this paper, we first construct a class of global strong solutions for the 2-D inhomogeneous Navier-Stokes equations under very general assumptions that the initial density is only bounded and the initial velocity is in <i>H</i><sup>1</sup>(ℝ<sup>2</sup>). With suitable assumptions on the initial density, which include the case of a density patch and vacuum bubbles, we prove that Lions’s weak solution is the same as the strong solution with the same initial data. In particular, this gives a complete resolution of the density patch problem proposed by Lions (1996): <i>for the density patch data ρ</i><sub>0</sub> = 1<sub><i>D</i></sub> <i>with a smooth bounded domain D</i> ⊂ ℝ<sup>2</sup>, <i>the regularity of D is preserved by the time evolution of Lions’s weak solution</i>.</p>

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On the density patch problem for the 2-D inhomogeneous Navier-Stokes equations

  • 期刊论文

  • 发表日期：

    2026年2月24日

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  • Tiantian Hao,
  • Feng Shao,
  • Dongyi Wei,
  • Zhifei Zhang

摘要

In this paper, we first construct a class of global strong solutions for the 2-D inhomogeneous Navier-Stokes equations under very general assumptions that the initial density is only bounded and the initial velocity is in H1(ℝ2). With suitable assumptions on the initial density, which include the case of a density patch and vacuum bubbles, we prove that Lions’s weak solution is the same as the strong solution with the same initial data. In particular, this gives a complete resolution of the density patch problem proposed by Lions (1996): for the density patch data ρ0 = 1D with a smooth bounded domain D ⊂ ℝ2, the regularity of D is preserved by the time evolution of Lions’s weak solution.