Quasilinear Evolution Equations in LPμ-Spaces with Lower Regular Initial Data

We study the Cauchy problem of the quasilinear evolution equations in Lpμ-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.

In general, a sectorial operator  0 does not have the property of   -maximal regularity, even though its negative − generates an exponentially decaying analytic semigroup.However, if  is a UMD space, and  0 admits bounded imaginary powers   0 (|| ≤  for some  > 0), or  0 ∈ BIP for short (see [3,Ch. 3], [5,6]), then  0 has the   -maximal regularity.Further investigation, involving the vector-valued multiplier theorem and  ∞ -functional calculus of sectorial operators, showed that, in a UMD space,   -maximal regularity equals R-sectoriality for a sectorial operator (see [7], [8,Ch. 3]).Recall that, given Ω ⊆ R  a bounded or exterior domain with compact  1 type boundary, Ω = R  or a half space R  + , for each 1 <  < ∞, the Lebesgue space   (Ω) is of UMD type, on which the elliptic differential operators with some boundary conditions are R-sectorial (cf.[7,9]).And the Stokes operator   is of BIP type with the power angle    < /2 (see [10,11]).Hence, we can make  1, (  ) and its closed subspace 0 E  1, (  ).In Section 3, we establish the local and global existence of E  1, (  ) and  , solutions in subcritical and critical cases, respectively, using estimate (4), together with the embedding properties of 0 E  1, (  ) and the smoothing action of the operator R. To illustrate the obtained results, in Section 4, we make    −   estimates for the parabolic evolution equations and the Navier-Stokes equations, using the fractional order Sobolev space  2(−1/), (Ω) and Besov space  2(−1/),  (Ω)  as the state spaces.
Framework of our study can be incorporated into the theory of sectorial operators and analytic semigroups, together with time-weighted   -spaces.Results obtained here have their meaning in the study of quasilinear evolution spaces driven by the sectorial operators.For the relative investigations on the quasilinear evolutions in   or    spaces with different focuses, please refer to [5,16,17,20], with the references therein.
Analogously, we can derive the following. where Remark 4. From [5], we know that, for every sectorial operator  0 lying in MR  (R + , ,  1 ), the corresponding semigroup  − 0 decays exponentially, and consequently the constant   appearing in Propositions 1 and 2 is independent of , whence the subscript  can be erased.
The following lemma tells us that the property of  maximal regularity can be preserved under small perturbations (refer to [5]).

Main Results and Proofs
Firstly, we give some hypotheses on  and  used in this paper.
We shall now discuss the local well posedness of Pr. (1) in two cases.
Introduce the map G : Then solution  of the auxiliary problem ( 22) can be represented by and consequently solution of the original problem ( 1) is exactly the fixed point of F.
Remark 12. Since the regularity of the initial value is allowed to be lower than that of the space where the nonlinear map (, ⋅) acts, here we need not require  1−2/, (Ω) → (Ω), and, consequently, condition  > +2 which was used in [17] is replaced by the weaker one:  > ( + 2)/2 in this paper.