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Global wellposedness of the 3D compressible Navier-Stokes equations with free surface in the maximal regularity class
Analysis of PDEs · Submitted 7 Feb 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Combining the maximal $L_p$-$L_q$ estimate and the $L_p$-$L_q$ decay estimate of solutions to the linearized equations, we prove the unique existence of global in time solutions in the time weighted maximal $L_p$-$L_q$ regularity class for some $p>2$ and $q>3.$ Namely, the solution is bounded as $L_p$ in time and $L_q$ in space.
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This paper concerns the global well posedness issue of the Navier-Stokes equations (CNS) describing barotropic compressible fluid flow with free surface occupied in the three dimensional exterior domain. Combining the maximal $L_p$-$L_q$ estimate and the $L_p$-$L_q$ decay estimate of solutions to the linearized equations, we prove the unique existence of global in time solutions in the time weighted maximal $L_p$-$L_q$ regularity class for some $p>2$ and $q>3.$ Namely, the solution is bounded as $L_p$ in time and $L_q$ in space. Compared with the previous results of the free boundary value problem of (CNS) in unbounded domains, we relax the regularity assumption on the initial states, which is the advantage by using the maximal $L_p$-$L_q$ regularity framework. On the other hand, the equilibrium state of the moving boundary of the exterior domain is not necessary the sphere. To our knowledge, this paper is the first result on the long time solvability of the free boundary value problem of (CNS) in the exterior domain.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper concerns the global well posedness issue of the Navier-Stokes equations (CNS) describing barotropic compressible fluid flow with free surface occupied in the three dimensional exterior domain. | background 0.86 | 0.27 | 0.04 | asserted 0.77 |
| s1 | Combining the maximal $L_p$-$L_q$ estimate and the $L_p$-$L_q$ decay estimate of solutions to the linearized equations, we prove the unique existence of global in time solutions in the time weighted maximal $L_p$-$L_q$ regularity class for some $p>2$ and $q>3.$ Namely, the solution is bounded as $L_p$ in time and $L_q$ in space. | result 0.78 | 0.97 | 0.90 | asserted 1.00 |
| s2 | Compared with the previous results of the free boundary value problem of (CNS) in unbounded domains, we relax the regularity assumption on the initial states, which is the advantage by using the maximal $L_p$-$L_q$ regularity framework. | contribution 0.62 | 0.68 | 0.07 | asserted 1.00 |
| s3 | On the other hand, the equilibrium state of the moving boundary of the exterior domain is not necessary the sphere. | result 0.81 | 0.55 | 0.04 | asserted 0.98 |
| s4 | To our knowledge, this paper is the first result on the long time solvability of the free boundary value problem of (CNS) in the exterior domain. | contribution 0.50 | 0.55 | 0.12 | asserted 0.92 |
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Original abstract
This paper concerns the global well posedness issue of the Navier-Stokes equations (CNS) describing barotropic compressible fluid flow with free surface occupied in the three dimensional exterior domain. Combining the maximal $L_p$-$L_q$ estimate and the $L_p$-$L_q$ decay estimate of solutions to the linearized equations, we prove the unique existence of global in time solutions in the time weighted maximal $L_p$-$L_q$ regularity class for some $p>2$ and $q>3.$ Namely, the solution is bounded as $L_p$ in time and $L_q$ in space. Compared with the previous results of the free boundary value problem of (CNS) in unbounded domains, we relax the regularity assumption on the initial states, which is the advantage by using the maximal $L_p$-$L_q$ regularity framework. On the other hand, the equilibrium state of the moving boundary of the exterior domain is not necessary the sphere. To our knowledge, this paper is the first result on the long time solvability of the free boundary value problem of (CNS) in the exterior domain.
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To apply the semigroup theory established in [ref] , the construction of $(ρ^2,^2)$ is necessary, which can be regarded as one difficulty from the quasilinear type conditions [ref] in our problem.
The $L_p$-$L_q$ decay estimate for the multidimensional compressible flow with free surface in the exterior domain · Reformulation of [ref] and main idea · 0.99
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