Cauchy problem of semi-linear parabolic equations with weak data in homogeneous space and application to the Navier-Stokes equations

2003 ◽  
Vol 46 (5) ◽  
pp. 641
Author(s):  
Changxing MIAO
Keyword(s):  
Cauchy Problem ◽  
Homogeneous Space ◽  
Parabolic Equations ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Linear Parabolic Equations

scholarly journals Quasilinear Evolution Equations in LμP-Spaces with Lower Regular Initial Data

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Qinghua Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Parabolic Equations ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Interpolation Spaces ◽  
Navier Stokes Equations ◽  
Sectorial Operators ◽  
The Cauchy Problem

We study the Cauchy problem of the quasilinear evolution equations in Lμp-spaces. Based on the theories of maximal Lp-regularity of sectorial operators, interpolation spaces, and time-weighted Lp-spaces, we establish the local posedness for a class of abstract quasilinear evolution equations with lower regular initial data. To illustrate our results, we also deal with the second-order parabolic equations and the Navier-Stokes equations in Lp,q-spaces with temporal weights.

scholarly journals On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations

2020 ◽  
Author(s):  
Thomas Eiter ◽  
Mads Kyed ◽  
Yoshihiro Shibata
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Periodic Solutions ◽  
Parabolic Equations ◽  
Stokes Equations ◽  
Transmission Conditions ◽  
Navier Stokes ◽  
Two Phase ◽  
Navier Stokes Equations ◽  
Linearized Equations

Abstract This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier–Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal $$L_p$$ L p –$$L_q$$ L q regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of $${\mathcal R}$$ R -solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the $${{\mathcal {R}}}$$ R -bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. 10.3934/cpaa.2018081, 2018; $${{\mathcal {R}}}$$ R boundedness, maximal regularity and free boundary problems for the Navier Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. ($${{\mathcal {R}}}$$ R -solvers and their application to periodic $$L_p$$ L p estimates, Preprint in 2019) for the $$L_p$$ L p boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper.

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