Symmetry, physicality and regularity
aspects of the
Q-tensor theory of liquid crystals
Symmetry, physicality and regularity
aspects of the
Q-tensor theory of liquid crystals
Grant PN-II-RU-Te-2014-4-0657, Contract 191 from 01/10/2015
financed by CNCS-UEFISCDI, ANCS.
Host Institution: "Simion Stoilow" Institute of Mathematics of the Romanian Academy
Project title: Symmetry, physicality and regularity aspects
of the Q-tensor theory of liquid crystals
Project abstract:
Liquid crystals are ubiquitous nowadays in the technology around us, and are the simplest entry point into the fascinating world of complex materials. However, despite their impressive technological applications they are not understood at a fundamental level and there exist several competing theories.
This proposal aims to study some key problems of the Q-tensor theory of liquid crystals. Despite being proposed by the Nobel medalist P.G. de Gennes in the 70s and very popular among physicists, the Q-tensor theory has attracted relatively little attention from mathematicians. This is perhaps because of its complexity, as it involves connections of PDE with areas such as topology, geometry, and algebra.
Its rigorous mathematical study was initiated in 2005 and it has seen a rapid development ever since, particularly in the last few years. Now it has reached the maturity level necessary for addressing certain key questions, some of which are presented in this proposal, namely concerning issues related to:
- hidden symmetries of point defects
- inertia, physicality preservation and large Ericksen number limit of the coupled Navier-Stokes and Q-tensor system
- the statistical dynamics of the phase transitions
The successful accomplishment of the programme will lead to progresses not just in the Q-tensor theory but also in the general area of nonlinear PDE, particularly in what concerns physical applications to material sciences.
Project objectives: Obtaining some key results concerning the Q-tensor theory, more specifically:
1) symmetries of defect patterns
2) inertia, physicality preservation and the large Ericksen number
limit for coupled systems Navier-Stokes and Q-tensors
3) generic dynamics for relevant phase transitions
Estimated results:
Obtaining substantial results with the above mentioned objectives,
which will be published in a series of articles in top international journals
and also presented at relevant international conferences.
Team: 1) Arghir Zarnescu-PI
2) Maria-Magdalena Boureanu-Postdoctoral Researcher (until August 2016)
3) Adrian Murza-Postdoctoral Researcher (since September 2016)
4) Iulian Cimpean-Ph.D. student
5) Tatiana Ignat-Ph.D. student
Papers:
1. Published papers:
1. R. Ignat, L. Nguyen, V. Slastikov and A. Zarnescu,
Stability of point defects of degree .+/- 1/2 in a two-dimensional nematic liquid crystal model,
Calc. Var. Partial Differential Equations, 55 (2016), no. 5, 55-119
2. M.-M. Boureanu, V. Radulescu and D. Repovs,
On a p( )-biharmonic problem with no-flux boundary conditions
Comput. Math. Appl. 72:9 (2016), 2505-2515
3. G.Kitavtsev, J. M. Robbins, V. Slastikov and A. Zarnescu
Liquid crystal defects in the Landau-de Gennes theory in two
dimensions-beyond the one-constant approximation ,
Math. Models Methods Appl. Sci., 26, 2769-2808 (2016)
4. Liviu Ignat, Tatiana Ignat
Long-time behaviour for a nonlocal convection-diffusion equation,
Journal of Mat. Anal. and App., Vol. 455, Issue 1,
816-831, 2017
2. Papers under revision/accepted:
1. S. Day, A. Zarnescu,
Sphere-valued harmonic maps with surface energy and the
K13 problem,
under revision in Advances in Calculus of Variations
2. F. de Anna, A. Zarnescu,
Global well-posedness and twist-wave solutions for the
inertial Qian-Sheng model of liquid crystals,
to appear in Journal of Differential Equations
3. L. Beznea, I.Cimpean
Invariant, super and quasi-martingale functions of a Markov process,
to appear in Stochastic Partial Differential Equations
and Related Fields, Springer Proceedings in Mathematics
& Statistics, 2017
4. A. Murza
Heteroclinic Cycles in ODEs with the Symmetry of the Quaternionic
Q8 Group,
to appear in Mathematical Reports
5. E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu,
On a hyperbolic system arising in liquid crystals modeling,
to appear in Journal of Hyperbolic Diff. Eqs.
3. Submitted papers:
1. R. Ignat, L. Nguyen, V. Slastikov, A. Zarnescu,
On the uniqueness of minimisers of Ginzburg-Landau functionals,
2. A. E. Teruel, A. C. Murza, A. Zarnescu,
Shear flow dynamics in the Beris-Edwards model of nematic liquid crystals,
3. H. Wu, X. Xu, A. Zarnescu,
Dynamics and flow effects in the Beris-Edwards system modelling nematic liquid crystals,
4. A.C. Murza,
Heteroclinic Cycles in Systems with Z2xZ2x Z2 Symmetry, Revisited,
5. J.Llibre, A.C. Murza,
Polynomial vector fields on the Clifford torus,
6. J.Llibre, A.C. Murza,
Darboux integrability for real polynomial vector fields on S^n
7. J.Llibre, A.C. Murza, A. E. Teruel,
First integrals of a class of Lotka-Volterra differential systems,
International Conferences Presentations:
1) Arghir Zarnescu, Mathematical Congress of the Americas, Montreal, July 2017
2) Arghir Zarnescu, Conference on Elliptic and Parabolic Problems, Gaeta, Italy,
May 2017
3) Arghir Zarnescu, Phase Transitions Models Workshop, Banff, Canada, May 2017
4) Iulian Cimpeanu, Stochastic Partial Differential Equations and Related Fields,
Bielefeld University, Germany, October 2016
5) Arghir Zarnescu, Symposium on Trends in Application of Mathematics to
Mechanics (STAMM), INdAM-ISIMM workshop, Rome, Italy, September 2016
6) Arghir Zarnescu, Colloque Franco-Roumain en Mathematiques Appliques, Iasi,
Romania, August 2016
7) Arghir Zarnescu, \Modeling materials and fluids using variational methods",
Berlin, Germany, February 2016
8) Arghir Zarnescu, One-day meeting in PDEs, Pavia, Italy, February 2016
9) Arghir Zarnescu, North-East Ohio Applied Mathematics Workshop, Kent, Ohio,
USA, January 2016
10) Arghir Zarnescu, Partial Order: Mathematics, Simulations and Applications,
IPAM, Los Angeles, USA, January 2016
11) Arghir Zarnescu, NYU-Oxford Workshop on Mathematical Models of Defects
and Patterns, Courant Institute, New-York, USA, January 2016
12) Arghir Zarnescu, Mathematical Analysis of Liquid Crystals Minisymposium at
the SIAM PDE Conference 2015 in Scottsdale, Arizona, USA
13) Arghir Zarnescu, Mathematics and Mechanics in the 22nd Century: Seven
Decades and Counting... , Eugene, OR, USA, October 2015
Seminar talks:
1) Arghir Zarnescu, PDE Seminar, Oxford University, Oxford, United Kingdom, October 2016
2) Arghir Zarnescu, PDE Seminar, Chinese Academy of Sciences, Beijing, China, June 2016
3) Arghir Zarnescu, Materials Science Seminar, Peking University, Beijing, China, June 2016
4) Arghir Zarnescu, PDE Seminar, Peking University, Beijing, China, June 2016
5) Arghir Zarnescu, BCAM Scienti.c Seminar, Bilbao, Spain, February 2016
6) Arghir Zarnescu, Joint PDE and Dynamical System Seminar, University of Surrey,
United Kingdom, October 2015
Research visits
1) Arghir Zarnescu, Collaboration visit to University of Bristol, UK, September 2017
2) Tatiana Ignat, Collaboration visit to University of Alicante, Spain, July 2017
3) Arghir Zarnescu, Collaboration visit to University of Athens, Greece, January 2017
4) Arghir Zarnescu, Collaboration visit to University of Bristol, UK-March/April 2016
5) Arghir Zarnescu, Collaboration visit to University of Bristol, UK -May/June 2016
6) Arghir Zarnescu, Collaboration visit (combined with conference attendance) at
Sapienza University, Rome, Italy, September 2016
7) Iulian Cimpean, Collaboration visit at Bielefeld University, Germany,
December 2016
Participations to conferences
1) Iulian Cimpean, Japanese-German Open Conference on Stochastic Analysis,
TU Kaiserslautern, Germany, September 2017 Benasque, Spain, August 2017
3) Tatiana Ignat, Summer School of Romanian Mathematical Sciences Society,
Busteni, August 2017
4) Iulian Cimpean, 3rd BCN Summer School on Stochastic Analysis: a 2016 EMS
Summer School, CRM, Barcelona, June/July 2016
5) Arghir Zarnescu, XIII-.eme Colloque Franco-Roumain, Iasi, Romania,
August 2016
6) Arghir Zarnescu-Indam Workshop on Trends on Applications of Mathematics
to Mechanics, Rome, September 2016
(just a one-way ticket was covered by grant)