We consider the Navier-Stokes equations with time-dependent external force, either in the whole time or in positive time with initial data, with domain either the whole space, the half space or an exterior domain of dimension n ≥ 3. We give conditions on the external force sufficient for the unique existence of small solutions in the weak-L n space bounded for all time.In particular, this result gives sufficient conditions for the unique existence and the stability of small time-periodic solutions or almost periodic solutions. This result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak-L n space with time-independent external force.
The fabrication of apatite-organic polymer hybrids is one of several attractive methods for the development of biomaterials as a substitute for bone. Such materials have both bone-bonding ability and mechanical properties analogous to natural bone. The biomimetic process has focused attention on fabricating such hybrids, where bone-like apatite is deposited on an organic polymer surface in solutions that mimic physiological conditions. In this process, a bone-like apatite layer can be coated onto organic substrates either by using a simulated body fluid (SBF) with ion concentrations nearly equal to those of human extracellular fluid, or by using fluids that are supersaturated with respect to apatite at ambient conditions. In this study, we investigated the ability of natural silk and its related materials to facilitate apatite deposition under biomimetic conditions. Cloths made of raw silk or normal silk fibers were soaked in 1.5SBF, which has 1.5 times the ion concentration of SBF. Sericin film, which is made from an extract of degummed raw silk, was soaked in 1.5SBF. The cloth and the film soaked in 1.5SBF then were characterized by scanning electron microscopic (SEM) observation, energy dispersive X-ray microanalysis (EDX), and thin-film X-ray diffraction (TF-XRD). Apatite deposition was observed on the surface of cloth made from raw silk fiber after it was soaked in 1.5SBF, but it was not observed on cloth made from normal silk fibers. The apatite deposition on the raw silk fiber cloth was accelerated when the fibers were subjected to treatment with CaCl(2) solution at a concentration of at least 1 kmol/m(3) before immersion in 1.5SBF. Apatite deposition also was observed on the sericin film after the film was soaked in 1.5SBF for 7 days. These results indicate that apatite deposition on raw silk cloth is attributable to the catalytic effect of sericin because the surface of raw silk consists of sericin whereas that of normal silk contains fibroin. The deposition of the apatite and its crystal growth are accelerated by the presence of calcium ions on the sericin after treatment with CaCl(2) solution. Thus, sericin on natural silk fiber has the potential to facilitate apatite deposition and can be useful as a polymer material in the fabrication of hybrid materials analogous to bone through biomimetic processes.
Acidic proteins play an important role during mineral formation in biological systems, but the mechanism of mineral formation is far from understood. In this paper, we report on the relationship between the structure of a protein and hydroxyapatite deposition under biomimetic conditions. Sericin, a type of silk protein, was adopted as a suitable protein for studying structural effect on hydroxyapatite deposition, since it forms a hydroxyapatite layer on its surface in a metastable calcium phosphate solution, and its structure has been reported. Sericin effectively induced hydroxyapatite nucleation when it has high molecular weight and a b sheet structure. This indicates that the specific structure of a protein can effectively induce heterogeneous nucleation of hydroxyapatite in a biomimetic solution, i.e. a metastable calcium phosphate solution. This finding is useful in understanding biomineralization, as well as for the design of organic polymers that can effectively induce hydroxyapatite nucleation.
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