Local well-posedness for the initial boundary value problem of the scaling critical compressible Navier–Stokes equations in nearly half space

Noboru Chikami; Takayoshi Ogawa; Senjo Shimizu; Noboru Chikami; Takayoshi Ogawa; Senjo Shimizu

doi:10.3934/cam.2025030

Communications in Analysis and Mechanics

2025, Volume 17, Issue 3: 749-778. doi: 10.3934/cam.2025030

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Local well-posedness for the initial boundary value problem of the scaling critical compressible Navier–Stokes equations in nearly half space

1.
Graduate School of Engneering, Nagoya Institute of Technology, Nagoya 466-8555, Japan
2.
Department of Pure and Applied Mathematics, Waseda University, Tokyo 169-8555, Japan
3.
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
4.
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan

Received: 31 December 2024 Revised: 09 July 2025 Accepted: 09 July 2025 Published: 26 August 2025
primary 35K20, 35Q30, 76N06, secondary 35K05, 35K61, 42B25

We consider the initial boundary value problem for the compressible Navier-Stokes equations of barotropic type. It is well-known that the equations maintain its structure nearly invariant under the scaling invariance and by the Fujita–Kato principle, the problem can be solved in the scaling invariant homogeneous Besov space locally in time for the Cauchy problem. We show the initial boundary value problem of the system with $ 0 $-Dirichlet boundary condition has a unique local solution in the nearly half Euclidean space. We employ the Lagrangian transform to transfer the equation in the Lagrangian coordinate, and flattening the boundary function to the half space and show the local well-posedness of the system. The key for the proof is employing end-point $ L^1 $ maximal regularity for the parabolic equations in half space shown in ^[1].
Citation: Noboru Chikami, Takayoshi Ogawa, Senjo Shimizu. Local well-posedness for the initial boundary value problem of the scaling critical compressible Navier–Stokes equations in nearly half space[J]. Communications in Analysis and Mechanics, 2025, 17(3): 749-778. doi: 10.3934/cam.2025030

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Abstract

We consider the initial boundary value problem for the compressible Navier-Stokes equations of barotropic type. It is well-known that the equations maintain its structure nearly invariant under the scaling invariance and by the Fujita–Kato principle, the problem can be solved in the scaling invariant homogeneous Besov space locally in time for the Cauchy problem. We show the initial boundary value problem of the system with $ 0 $-Dirichlet boundary condition has a unique local solution in the nearly half Euclidean space. We employ the Lagrangian transform to transfer the equation in the Lagrangian coordinate, and flattening the boundary function to the half space and show the local well-posedness of the system. The key for the proof is employing end-point $ L^1 $ maximal regularity for the parabolic equations in half space shown in ^[1].

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