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Discovery Projects - Grant ID: DP150101275

Title
Discovery Projects - Grant ID: DP150101275
Funding
ARC | Discovery Projects
Contract (GA) number
DP150101275
Start Date
2015/01/01
End Date
2017/12/31
Open Access mandate
no
Organizations
-
More information
http://purl.org/au-research/grants/arc/DP150101275

 

  • Well-posedness of the Ericksen-Leslie System for the Oseen-Frank Model in L^3_{uloc}(\mathbb{R}^3)

    We investigate the Ericksen-Leslie system for the Oseen-Frank model with unequal Frank elastic constants in $\mathbb{R}^3$. To generalize the result of Hineman-Wang \cite{HW}, we prove existence of solutions to the Ericksen-Leslie system with initial data having small $L^3_{uloc}$-norm. In particular, we use a new idea to obtain a local $L^3$-estimate through interpolation inequalities and a covering argument, which is different from the one in \cite{HW}. Moreover, for uniqueness of solutions...

    Finite time blowup of the $n$-harmonic flow on $n$-manifolds

    Cheung, Leslie Hon-Nam; Hong, Min-Chun (2017)
    Projects: ARC | Discovery Projects - Grant ID: DP150101275 (DP150101275)
    We generalize the no-neck result of Qing-Tian \cite{QT} to show that there is no neck during blowing up for the $n$-harmonic flow as $t\to\infty$. As an application of the no-neck result, we settle a conjecture of Hungerb\"uhler \cite {Hung} by constructing an example to show that the $n$-harmonic map flow on an $n$-dimensional Riemannian manifold blows up in finite time for $n\geq 3$.

    Biharmonic hypersurfaces with constant scalar curvature in space forms

    Let $M^n$ be a biharmonic hypersurface with constant scalar curvature in a space form $\mathbb M^{n+1}(c)$. We show that $M^n$ has constant mean curvature if $c>0$ and $M^n$ is minimal if $c\leq0$, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen's conjecture and Generalized Chen's conjecture. As a consequence, we prove that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space $\mathbb E^{n+1}...

    The gauge fixing theorem with applications to the Yang-Mills flow over Riemannian manifolds

    In 1982, Uhlenbeck \cite {U2} established the well-known gauge fixing theorem, which has played a fundamental role for Yang-Mills theory. In this paper, we apply the idea of Uhlenbeck to establish a parabolic type of gauge fixing theorems for the Yang-Mills flow and prove existence of a weak solution of the Yang-Mills flow on a compact $n$-dimensional manifold with initial value $A_0$ in $W^{1,n/2}(M)$. When $n=4$, we improve a key lemma of Uhlenbeck (Lemma 2.7 of \cite {U2}) to prove uniquen...
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