Electronic Research Archive

2021, Issue 4: 2719-2739. doi: 10.3934/era.2021010

Local well-posedness of perturbed Navier-Stokes system around Landau solutions

• Received: 01 December 2020 Published: 22 February 2021
• Primary: 35Q30; Secondary: 49K40

• For the incompressible Navier-Stokes system, when initial data are uniformly locally square integral, the local existence of solutions has been obtained. In this paper, we consider perturbed system and show that perturbed solutions of Landau solutions to the Navier-Stokes system exist locally under $L^q_{\text{uloc}}$-perturbations, $q\geq 2$. Furthermore, when $q\geq 3,$ the solution is well-posed. Precisely, we give the explicit formula of the pressure term.

Citation: Jingjing Zhang, Ting Zhang. Local well-posedness of perturbed Navier-Stokes system around Landau solutions[J]. Electronic Research Archive, 2021, 29(4): 2719-2739. doi: 10.3934/era.2021010

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• For the incompressible Navier-Stokes system, when initial data are uniformly locally square integral, the local existence of solutions has been obtained. In this paper, we consider perturbed system and show that perturbed solutions of Landau solutions to the Navier-Stokes system exist locally under $L^q_{\text{uloc}}$-perturbations, $q\geq 2$. Furthermore, when $q\geq 3,$ the solution is well-posed. Precisely, we give the explicit formula of the pressure term.

 [1] A. Basson, Solutions Spatialement Homog$\grave{e}$nes Adapt$\acute{e}$es des $\acute{e}$quations de Navier-Stokes, Thesis. University of Evry., 2006. [2] Self-similar solutions to the Navier-Stokes equations: A survey of recent results. Nonlinear Analysis in Geometry and Applied Mathematics (2018) 2: 159-181. [3] Navier-Stokes flow in ${\bf R}^3$ with measures as initial vorticity and Morrey spaces. Comm. Partial Differential Equations (1989) 14: 577-618. [4] Note on a theorem of Hilbert. Math. Z. (1920) 6: 314-317. [5] An inequality between integrals. Messenger of Mathematics (1925) 54: 150-156. [6] Well-posedness of nematic liquid crystal flow in \begin{document}$L^3_{\rm uloc } (\Bbb{R}^3)$\end{document}. Arch. Ration. Mech. Anal. (2013) 210: 177-218. [7] Asymptotic stability of Landau solutions to Navier-Stokes system. Arch. Ration. Mech. Anal. (2011) 202: 115-131. [8] $L^2$-asymptotic stability of singular solutions to the Navier-Stokes system of equations in $\mathbb{R}^3$. J. Math. Pures Appl. (2017) 108: 14-40. [9] Strong $L^{p}$-solutions of the Navier-Stokes equation in R$^{m}$, with applications to weak solutions. Math. Z. (1984) 187: 471-480. [10] N. Kikuchi and G. Seregin, Weak solutions to the Cauchy problem for the Navier-Stokes equations satisfying the local energy inequality, Nonlinear Equations and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2,220, Amer. Math. Soc., Providence, RI, 220 (2007), 141–164. doi: 10.1090/trans2/220/07 [11] Well posednesss for the Navier-Stokes equations. Adv. Math. (2001) 157: 22-35. [12] Global Navier-Stokes flows for non-decaying initial data with slowly decaying oscillation. Comm. Math. Phys. (2020) 375: 1665-1715. [13] A new exact solution of Navier-Stokes equations. C. R. (Doklady) Acad. Sci. URSS (N.S.) (1944) 43: 286-288. [14] P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, 2002. doi: 10.1201/9781420035674 [15] (2016) The Navier-Stokes Problem in the 21st Century.CRC Press. [16] Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. (1934) 63: 193-248. [17] Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅰ. One singularity. Arch. Ration. Mech. Anal. (2018) 227: 1091-1163. [18] Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅱ. Classification of axisymmetric no-swirl solutions. J. Differential Equations (2018) 264: 6082-6108. [19] Vanishing viscosity limit for homogeneous axisymmetric no-swirl solutions of stationary Navier-Stokes equations. J. Funct. Anal. (2019) 277: 3599-3652. [20] Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅲ. Two singularities. Discrete Contin. Dyn. Syst. (2019) 39: 7163-7211. [21] On Landau's solutions of the Navier-Stokes equations, Problems in mathematical analysis, No. 61. J. Math. Sci. (N.Y.) (2011) 179: 208-228. [22] Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations. Comm. Partial Differential Equations (1992) 17: 1407-1456. [23] One-point singular solutions to the Navier-Stokes equations. Topol. Methods Nonlinear Anal. (1998) 11: 135-145. [24] Forward discretely self-similar solutions of the Navier-Stokes equations. Comm. Math. Phys. (2014) 328: 29-44. [25] T.-P. Tsai, Lectures on Navier-Stokes Equations, American Mathematical Society, Providence, RI, vol. 192, 2018. doi: 10.1090/gsm/192 [26] J. Zhang and T. Zhang, Global existence of discretely self-similar solutions to the generalized MHD system in Besov space, J. Math. Phys., 60 (2019), 081515, 18 pp. doi: 10.1063/1.5092787 [27] A Liouville theorem for axially symmetric $D$-solutions to steady Navier-Stokes equations. Nonlinear Anal. (2019) 187: 247-258.
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