The paper is concerned with a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals. In dimension two, we establish both interior and boundary regularity theorem for such a flow under smallness conditions. As a consequence, we establish the existence of global weak solutions that are smooth away from at most finitely many singular times in any bounded smooth domain of R 2 .

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BackgroundStripe rust (Puccinia striiformis f. sp. tritici; Pst) and powdery mildew (Blumeria graminis f. sp. tritici; Bgt) are important diseases of wheat (Triticum aestivum) worldwide. Similar mechanisms and gene transcripts are assumed to be involved in the host defense response because both pathogens are biotrophic fungi. The main objective of our study was to identify co-regulated mRNAs that show a change in expression pattern after inoculation with Pst or Bgt, and to identify mRNAs specific to the fungal stress response.ResultsThe transcriptome of the hexaploid wheat line N9134 inoculated with the Chinese Pst race CYR 31 was compared with that of the same line inoculated with Bgt race E09 at 1, 2, and 3 days post-inoculation. Infection by Pst and Bgt affected transcription of 23.8% of all T. aestivum genes. Infection by Bgt triggered a more robust alteration in gene expression in N9134 compared with the response to Pst infection. An array of overlapping gene clusters with distinctive expression patterns provided insight into the regulatory differences in the responses to Bgt and Pst infection. The differentially expressed genes were grouped into seven enriched Kyoto Encyclopedia of Genes and Genomes pathways in Bgt-infected leaves and four pathways in Pst-infected leaves, while only two pathways overlapped. In the plant–pathogen interaction pathway, N9134 activated a higher number of genes and pathways in response to Bgt infection than in response to Pst invasion. Genomic analysis revealed that the wheat genome shared some microbial genetic fragments, which were specifically induced in response to Bgt and Pst infection.ConclusionsTaken together, our findings indicate that the responses of wheat N9134 to infection by Bgt and Pst shows differences in the pathways and genes activated. The mass sequence data for wheat–fungus interaction generated in this study provides a powerful platform for future functional and molecular research on wheat–fungus interactions.Electronic supplementary materialThe online version of this article (doi:10.1186/1471-2164-15-898) contains supplementary material, which is available to authorized users.

We investigate the well-posedness of (i) the heat flow of harmonic maps from R n to a compact Riemannian manifold N without boundary for initial data in BMO; and (ii) the hydrodynamic flow (u, d) of nematic liquid crystals on R n for initial data in BMO

For any n-dimensional compact Riemannian manifold (M, g) without boundary and another compact Riemannian manifold (N, h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0, T ), W 1,n ). For the hydrodynamic flow (u, d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈. This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.

For any bounded smooth domain Ω⊂ℝ3 (or Ω=ℝ3), we establish the global existence of a weak solution (u,d):Ω×[0,+∞)→ℝ3×double-struckS2 of the initial boundary value (or the Cauchy) problem of the simplified Ericksen‐Leslie system LLF modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary (or Cauchy) data (u0,d0)∈H×H1(Ω,double-struckS2), with d0(Ω)⊂double-struckS+2 (the upper hemisphere). Furthermore, (u,d) satisfies the global energy inequality.© 2016 Wiley Periodicals, Inc.

For a domain ⊂ R 4 and a compact Riemannian manifold N ⊂ R k without boundary, if u ∈ W 2,2 ( , N ) is an extrinsic (or intrinsic, respectively) biharmonic map, then u ∈ C ∞ ( , N ).

In this paper, we first establish the regularity theorem for suitable weak solutions to the Ericksen-Leslie system in R 2 . Building on such a regularity, we then establish the existence of a global weak solution to the Ericksen-Leslie system in R 2 for any initial data in the energy space, under the physical constraint conditions on the Leslie coefficients ensuring the dissipation of energy of the system, which is smooth away from at most finitely many times. This extends earlier works by Lin-Lin-Wang [23] on a simplified nematic liquid crystal flow.