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arXiv · 2509.01196

Regularity and dynamics of weak solutions for one-dimensional compressible Navier-Stokes equations with vacuum

Jin Tan·Yan-Lin Wang·Lan Zhang
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Abstract

In the spirit of D. Hoff's weak solution theory for the compressible Navier-Stokes equations (CNS) with bounded density, in this paper we establish the global existence and regularity properties of finite-energy weak solutions to an initial boundary value problem of one-dimensional CNS with general initial data and vacuum. The core of our proof is a global in time a priori estimate for one-dimensional CNS that holds for any $H^1$ initial velocity and bounded initial density not necessarily strictly positive: it could be a density patch or a vacuum bubble. We also establish that the velocity and density decay exponentially to equilibrium. As a by-product, we obtain the quantitative dynamics of aforementioned two vacuum states.

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Jin Tan, Yan-Lin Wang, Lan Zhang. 2025-09-01. Regularity and dynamics of weak solutions for one-dimensional compressible Navier-Stokes equations with vacuum. https://arxiv.org/abs/2509.01196

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