Proof By a contradiction, we may assume u ≥ 0. By using a standard regularity argument and [[17], Lemmas 3.2 and 3.3], we have u ∈ C 1, α for some α ∈ ( 0, 1 ).
By using an iteration procedure, regularity estimates for the linear semigroups and a classical existence theorem of a global attractor, they proved that problem (2 -(4) possesses a global attractor in the Sobolev space H k for all k ≥ 0, which attracts any bounded subset of H k in the H k -norm.
For the elliptic systems (1.1), Zheng [15] and Zheng and Feng [16] derived an optimal partial regularity in Morrey spaces by using a reverse Hölder inequality and perturbation argument when (A x,u)) is VMO in (xinOmega) and continuous in (uinmathbb {R}^{N}) with controllable growth and natural growth, respectively.
Regularities of lattice constant in ideal perovskites are investigated by using a total of 132 ABX3-type compounds, including oxides and halides.
Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation.
In addition, the continuity, such as ((eta, u,q in C^{0}([0,T],H^{7}times H^{6}timathcal{})), (mathcal{G}(t)) and so on, can be verified by using the regularity of ((eta,u,q)), the transformed MHD equations and a standard regularized method.
These results show that by using magnetogram regularity to reflect the magnetosphere activity, a theoretical analysis of the external geomagnetic field based on local power-law exponents is possible.
Finally, our conclusion will follow by using the regularity result for the elliptic transmission problem (see [11]).
Accordingly, retinal adaptation should improve coding efficiency by using the regularity and scale invariance of contrasts and other visual features in the natural scenes [4], [8], [10], [14].
Currently, some methods have been developed that reveal TP by using regularity in symbol preferences over different triplet positions in the DNA sequence.
