Maximal $$L^1$$-regularity of the heat equation and application to a free boundary problem of the Navier-Stokes equations near the half-space

Ogawa, Takayoshi; Shimizu, Senjo

doi:10.1007/s41808-021-00133-w

Cited by 6 publications

(5 citation statements)

References 58 publications

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“…In [9], Danchin, Hieber, Mucha and Tolksdorf address this aspect in the context of free boundary value problems in the half space by using Da Prato-Grisvard theory, allowing for maximal L 1 -regularity in the critical (homogeneous) Besov space. For a different approach to such problems, we also refer to the work of Ogawa and Shimizu [29]. One can also ask whether the results established in [6], [2], [3] remain valid in the present situation of the parabolic-hyperbolic problem.…”

Section: Discussionmentioning

confidence: 99%

Rigorous Analysis and Dynamics of Hibler’s Sea Ice Model

et al. 2022

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This article develops for the first time a rigorous analysis of Hibler’s model of sea ice dynamics. Identifying Hibler’s ice stress as a quasilinear second-order operator and regarding Hibler’s model as a quasilinear evolution equation, it is shown that a regularized version of Hibler’s coupled sea ice model, i.e., the model coupling velocity, thickness and compactness of sea ice, is locally strongly well-posed within the $$L_q$$ L q -setting and also globally strongly well-posed for initial data close to constant equilibria.

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Section: Discussionmentioning

confidence: 99%

Rigorous Analysis and Dynamics of Hibler’s Sea Ice Model

et al. 2022

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“…We then turn into our attention to the initial boundary value problem of the heat equation with the Neumann boundary condition (cf. [38] and [37]).…”

Section: The Solution Formula For the Stokes Equationsmentioning

confidence: 98%

“…Then the estimates for the rest of the velocity components v follow from the estimate for (4.14) and the pressure with (4.15) (cf. [37][38][39]). To this end, we prepare the following estimate.…”

Section: Estimate For the Velocitymentioning

confidence: 99%

Free boundary problems of the incompressible Navier–Stokes equations with non-flat initial surface in the critical Besov space

Ogawa,

Shimizu

2024

Math. Ann.

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Global well-posedness of the Navier–Stokes equations with a free boundary condition is considered in the scaling critical homogeneous Besov spaces $${\dot{B}}_{p,1}^{-1+n/p}({\mathbb {R}}^n_+)$$ B ˙ p , 1 - 1 + n / p ( R + n ) with $$n-1< p< 2n-1$$ n - 1 < p < 2 n - 1 . To show the global well-posedness, we establish end-point maximal $$L^1$$ L 1 -regularity for the initial-boundary value problem of the Stokes equations. Such an estimate is obtained via related estimate for the initial-boundary value problem of the heat equation with the inhomogeneous Neumann data as well as the pressure estimate in the critical Besov space framework. The proof heavily depends on the explicit expression of the fundamental integral kernel of the Lagrange transformed linearized Stokes equations and the almost orthogonal estimates with the space-time Littlewood–Paley dyadic decompositions. Our result here improves the initial space and boundary state than previous results by Danchin–Hieber–Mucha–Tolksdorf (Free boundary problems via Da Prato–Grisvard theory. arXiv:2011.07918v2) and ourselves (Ogawa and Shimizu in J Evol Equ 22(30):67, 2022; Ogawa and Shimizu in J Math Soc Jpn. arXiv:2211.06952v3).

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“…In exterior domains, Shibata [17] proved the global wellposedness without surface tension in L p -L q setting. In the half-space, the global well-posedbess was obtained by Ogawa and Shimizu [10] without surface tension in Ẇ1 (0, ∞; Ḃ−1+ n p p,1…”

Section: Introductionmentioning

confidence: 95%

On the global well-posedness and decay of a free boundary problem of the Navier-Stokes equation in unbounded domains

Oishi¹,

Shibata²

2022

Preprint

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In this paper, we establish the unique existence and some decay properties of a global solution of a free boundary problem of the incompressible Navier-Stokes equations in Lp in time and Lq in space framework in a uniformly H 2 ∞ domain Ω ⊂ R N for N ≥ 4. We assume the unique solvability of the weak Dirichlet problem for the Poisson equation and the Lq-Lr estimates for the Stokes semigroup. The novelty of this paper is that we do not assume the compactness of the boundary, which is essentially used in the case of exterior domains proved by Shibata [17]. The restriction N ≥ 4 is required to deduce an estimate for the nonlinear term G(u) arising from div v = 0. However, we establish the results in the half space R N + for N ≥ 3 by reducing the linearized problem to the problem with G = 0, where G is the right member corresponding to G(u).

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Maximal $$L^1$$-regularity of the heat equation and application to a free boundary problem of the Navier-Stokes equations near the half-space

Cited by 6 publications

References 58 publications

Rigorous Analysis and Dynamics of Hibler’s Sea Ice Model

Rigorous Analysis and Dynamics of Hibler’s Sea Ice Model

Free boundary problems of the incompressible Navier–Stokes equations with non-flat initial surface in the critical Besov space

On the global well-posedness and decay of a free boundary problem of the Navier-Stokes equation in unbounded domains

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