Grant PN-II-RU-Te-2014-4-0657, Contract 191 din 01/10/2015
finantat de CNCS-UEFISCDI, ANCS.
Institutia gazda: Institutul de Matematica "Simion Stoilow" al Academiei Romane
Titlul proiectului: Simetrie, fizicalitate si aspecte de regularitate
in teoria tensorilor Q a cristalelor lichide
Rezumat:
Cristalele lichide sunt omniprezente în zilele noastre în tehnologia ce ne
înconjoară şi reprezintă cel mai simplu punct de intrare în fascinanta lume
a materialelor complexe.
Cu toate acestea, în ciuda impresionantelor lor aplicatii tehnologice, ele nu
sunt înţelese la un nivel fundamental şi există mai multe teorii concurente.
Această propunere urmareşte să studieze anumite probleme cheie ale teoriei
tensorilor Q a cristalelor lichide. Deşi a fost propusă în anii 70 de P.G. de Gennes,
medaliat Nobel, şi foarte populară în rândul fizicienilor, teoria tensorilor Q
a generat relativ puţin interes din partea matematicienilor. Aceasta şi din cauza
complexităţii sale, întrucât implică conexiuni ale EDP cu domenii precum topologia,
geometria şi algebra.
Studiul ei din punct de vedere riguros matematic a fost iniţiat în 2005 şi s-a dezvoltat
rapid de atunci, mai ales în ultimii ani. Acum teoria a atins nivelul de maturitate
necesar pentru a adresa anumite probleme cheie, unele dintre acestea
fiind prezentate în această propunere şi anume chestiuni legate de:
-simetrii ascunse ale defectelor punctuale,
-inerţie, pastrarea fizicalităţii şi limita numărului Ericksen larg, pentru sistemul
cuplat Navier-Stokes şi tensori Q
-dinamica statistică a tranziţiei de fază
Realizarea cu succes a propunerii va conduce la progrese nu doar în
teoria tensorilor Q ci şi în aria generală a EDP neliniare, în particular în ceea
ce priveşte aplicaţii fizice din ştiinţa materialelor.
Obiectivele proiectului:
Obtinerea unor rezultate cheie privind teoria tensorilor Q, si anume:
1) simetrii ale texturilor de tip defect
2) inertie, preservarea fizicalitatii si limita numarului Ericksen
larga pentru sisteme cuplate Navier-Stokes si Q-tensori
3) dinamici generice pentru tranzitii de faza relevante
Rezultate estimate:
Obtinerea unor rezultate cu obiectivele mai sus mentionate, ce vor fi publicate
intr-o serie de articole in jurnale internationale de top si de asemenea prezentate
in cadrul unor conferinte internationale relevante.
Echipa: 1) Arghir Zarnescu-Director de Proiect
2) Maria-Magdalena Bourean-Cercetator Postdoctoral (pana in august 2016)
3) Adrian Murza-Cercetator Postdoctoral (din septembrie 2016)
3) Iulian Cimpean-student doctorand
4) Tatiana Ignat- student doctorand
Articole:
1. Articole publicate:
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. Articole acceptate:
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. Articole trimise:
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,
Prezentari in cadrul conferintelor internationale:
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
Prezentari in cadrul seminariilor:
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
Vizite de cercetare:
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
Participari la conferinte:
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)