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Volume 39, Issue 1
Fujita-Kato Theorem for the Inhomogeneous Incompressible Navier-Stokes Equations with Nonnegative Density

Jianzhong Zhang & Hongmei Cao

DOI: 10.4208/cmr.2021-0034

Commun. Math. Res., 39 (2023), pp. 79-106.

Published online: 2022-10

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  • Abstract

In this paper, we prove the global existence and uniqueness of solutions for the inhomogeneous Navier-Stokes equations with the initial data $(\rho_0,u_0)\in L^∞\times H^s_0$, $s>\frac{1}{2}$ and $||u_0||_{H^s_0}\leq \varepsilon_0$ in bounded domain $\Omega \subset \mathbb{R}^3$, in which the density is assumed to be nonnegative. The regularity of initial data is weaker than the previous $(\rho_0,u_0)\in (W^{1,\gamma}∩L^∞)\times H^1_0$ in [13] and $(\rho_0,u_0)\in L^∞\times H^1_0$ in [7], which constitutes a positive answer to the question raised by Danchin and Mucha in [7]. The methods used in this paper are mainly the classical time weighted energy estimate and Lagrangian approach, and the continuity argument and shift of integrability method are applied to complete our proof.

  • Keywords

Inhomogeneous Navier-Stokes equations, nonnegative density, global existence and uniqueness.

  • AMS Subject Headings

35Q30, 76D03

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-39-79, author = {Zhang , Jianzhong and Cao , Hongmei}, title = {Fujita-Kato Theorem for the Inhomogeneous Incompressible Navier-Stokes Equations with Nonnegative Density}, journal = {Communications in Mathematical Research }, year = {2022}, volume = {39}, number = {1}, pages = {79--106}, abstract = {

In this paper, we prove the global existence and uniqueness of solutions for the inhomogeneous Navier-Stokes equations with the initial data $(\rho_0,u_0)\in L^∞\times H^s_0$, $s>\frac{1}{2}$ and $||u_0||_{H^s_0}\leq \varepsilon_0$ in bounded domain $\Omega \subset \mathbb{R}^3$, in which the density is assumed to be nonnegative. The regularity of initial data is weaker than the previous $(\rho_0,u_0)\in (W^{1,\gamma}∩L^∞)\times H^1_0$ in [13] and $(\rho_0,u_0)\in L^∞\times H^1_0$ in [7], which constitutes a positive answer to the question raised by Danchin and Mucha in [7]. The methods used in this paper are mainly the classical time weighted energy estimate and Lagrangian approach, and the continuity argument and shift of integrability method are applied to complete our proof.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2021-0034}, url = {http://global-sci.org/intro/article_detail/cmr/21079.html} }
TY - JOUR T1 - Fujita-Kato Theorem for the Inhomogeneous Incompressible Navier-Stokes Equations with Nonnegative Density AU - Zhang , Jianzhong AU - Cao , Hongmei JO - Communications in Mathematical Research VL - 1 SP - 79 EP - 106 PY - 2022 DA - 2022/10 SN - 39 DO - http://doi.org/10.4208/cmr.2021-0034 UR - https://global-sci.org/intro/article_detail/cmr/21079.html KW - Inhomogeneous Navier-Stokes equations, nonnegative density, global existence and uniqueness. AB -

In this paper, we prove the global existence and uniqueness of solutions for the inhomogeneous Navier-Stokes equations with the initial data $(\rho_0,u_0)\in L^∞\times H^s_0$, $s>\frac{1}{2}$ and $||u_0||_{H^s_0}\leq \varepsilon_0$ in bounded domain $\Omega \subset \mathbb{R}^3$, in which the density is assumed to be nonnegative. The regularity of initial data is weaker than the previous $(\rho_0,u_0)\in (W^{1,\gamma}∩L^∞)\times H^1_0$ in [13] and $(\rho_0,u_0)\in L^∞\times H^1_0$ in [7], which constitutes a positive answer to the question raised by Danchin and Mucha in [7]. The methods used in this paper are mainly the classical time weighted energy estimate and Lagrangian approach, and the continuity argument and shift of integrability method are applied to complete our proof.

Zhang , Jianzhong and Cao , Hongmei. (2022). Fujita-Kato Theorem for the Inhomogeneous Incompressible Navier-Stokes Equations with Nonnegative Density. Communications in Mathematical Research . 39 (1). 79-106. doi:10.4208/cmr.2021-0034
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