List of Publications
Books.
A. Topological approximation methods for evolutionary problems of nonlinear hydrodynamics. Walter de Gruyter, 2008 (with V. Zvyagin).
Chapters in books.
B. The second boundary value problem for equations of viscoelastic diffusion in polymers, In: Advances in mathematics research (ed. Albert R. Baswell) Vol. 10, Nova Sci. Publ., New York, 2010, pp. 249-271 (preprint available at arXiv).
C. Anomalous diffusion in polymers: long-time behaviour, In: Fields Institute Communications, Vol. 64, Infinite Dimensional Dynamical Systems, J. Mallet-Paret, J. Wu, Y. Yi and H. Zhu (Eds.), Springer, 2013, pp. 481-496 (preprint available at arXiv).
Papers in peer-reviewed journals of international level.
2004-2007
1. On the solvability of the initial-boundary value problem for the motion equations of nonlinear viscoelastic medium in the whole space, Nonlinear Anal. TMA, 2004, V. 58, pp.631-656. (with V.G. Zvyagin).
2. On the existence of weak solutions for the initial-boundary value problem in the Jeffreys model of motion of a viscoelastic medium, Abstr. Appl. Anal., 2004, V. 2004, no. 10, 815-829. (with V.G. Zvyagin).
3. On the existence of weak stationary solutions of the boundary value problem in the Jeffreys model of motion of a viscoelastic medium, Izvestiya Vuzov, Ser. Math., 2004, no. 9, pp. 13-17.
4. On convergence of solutions of the regularized problem for motion equations of Jeffreys viscoelastic medium to solutions of the original problem, Journal of Mathematical Sciences, 2007, V.144, No.5, 4398-4408. Translated from Fundamentalnaya i prikladnaya matematika (FPM), 2005, V. 11, no. 4, 49-63. (with V.G. Zvyagin).
5. Topological approach to investigation of acoustic axes in crystals, Crystallography Reports (Kristallografiya), 2006, V.51, no. 1, 104-109 (with B.M. Darinskii and V.G. Zvyagin).
6. On the trajectory and global attractors for the equations of motion of a viscoelastic medium, Russian Mathematical Surveys, Volume 61(2006), Number 2, 368-370. (with V.G. Zvyagin).
7. Uniform attractors for non-autonomous motion equations of viscoelastic medium, J. Math. Anal. Appl., 2007, Volume 325, Issue 1, 438-458 (with V.G. Zvyagin).
8. Existence of solutions for motion equations for an objective model of viscoelastic medium, PAMM - Proc. Appl. Math. Mech. 7, 1060107-1060108 (2007), with V.G. Zvyagin.
9. Weak solutions and attractors for motion equations for an objective model of viscoelastic medium, PAMM - Proc. Appl. Math. Mech. 7, 1060105-1060106 (2007), with V.G. Zvyagin.
2008-2011
10. Trajectory and global attractors of the boundary value problem for autonomous motion equations of viscoelastic medium, J. Math. Fluid Mech., V. 10 (2008), 19-44 (with V.G. Zvyagin).
11. Dissipative solutions for equations of viscoelastic diffusion in polymers, J. Math. Anal. Appl., 2008, V. 339, 876-888.
12. Approximating - topological methods in some problems of hydrodynamics, J. Fixed Point Theory Appl., 2008, Volume 3, Number 1, 23-49 (with V.G. Zvyagin).
13. Longitudinal normals and existence of acoustic axes in crystals, Nonlinear Analysis B: Real World Applications, V. 10 (2009), 798-809 (with B.M. Darinskii and V.G. Zvyagin).
14. Weak solvability for equations of viscoelastic diffusion in polymers with variable coefficients, J. Differential Equations, 2009, V. 246, no. 3, 1038-1056.
15. On repeated concentration and periodic regimes with anomalous diffusion in polymers. Sbornik: Mathematics 201:1 (2010), 57-77. (preprint available at arXiv).
16. On the well-posedness of a two-phase minimization problem. J. Math. Anal. Appl. 378:1 (2011), 159-168 (with J.M. Urbano, preprint available here)
17. The flashing ratchet and unidirectional transport of matter. Discr. Cont. Dyn. Syst. Ser. B, V. 16, no. 3, 2011, 963 – 971 (preprints available at arXiv and here).
18. Asymptotic behaviour of the non-autonomous 3D Navier-Stokes problem with coercive force. J. Differential Equations, 2011, V. 251, 2209–2225 (preprints available at arXiv and here). Corrigendum: J. Differential Equations, 2013, V. 255, 3747.
2012-2015
19. Global generalized solutions for Maxwell-alpha and Euler-alpha equations, Nonlinearity 25 (2012) 309-327 (preprints available at arXiv and here).
20. On a system of adaptive coupled PDEs for image restoration, Journal of Mathematical Imaging and Vision 48 (2014) 35-52. (with V. B. Surya Prasath, preprint available at arXiv).
21. Weak solutions for a bioconvection model related to Bacillus subtilis, Commun. Math. Sci. 12 (2014) 545–563 (preprints available at arXiv and here).
22. Analytical aspects of the Brownian motor effect in randomly flashing ratchets, J. Math. Biol., 68:1677–1705 (2014), an old preprint available here, a new one in arXiv.
23. Weighted and well-balanced anisotropic diffusion scheme for image denoising and restoration. Nonlinear Analysis B: Real World Applications 17 (2014) 33–46 (with V. B. Surya Prasath).
24. Multiscale Tikhonov-total variation image restoration using spatially varying edge coherence exponent scheme, with multiple co-authors, IEEE Trans. Image Proc. 24 (2015) 5220-5235.
25. Analysis of adaptive forward-backward diffusion flows with applications in image processing, with V. B. Surya Prasath and J.M. Urbano, Inverse Problems 31 (2015) 105008 (preprint available here).
2016-
26. A new optimal transport distance on the space of finite Radon measures, with S. Kondratyev and L. Monsaingeon, Adv. Differential Equations 21 (2016) 1117-1164 (preprints available at arXiv (an amended version) and here).
27. On the bulk velocity of Brownian ratchets, with S. Kondratyev and J.M. Urbano, SIAM J. Math. Anal. 48 (2016) 950-980 (preprints available at arXiv and here).
28. Generalized solutions for inextensible string equations, with Y. Şengül, J. Diff. Equ., 262(6):3610–3641, 2017.
29. Adaptive diffusion constrained total variation scheme with application to cartoon + texture + edge image decomposition, submitted (with Juan C. Moreno, V. B. Surya Prasath, K. Palaniappan and H. Proença, preprint available at ArXiv and here).
30. A fitness-driven cross-diffusion system from polulation dynamics as a gradient flow (with S. Kondratyev and L. Monsaingeon), J. Differential Equ. 261 (2016) 2784–2808, preprint available at ArXiv and here).
31. A new multicomponent Poincaré-Beckner inequality (with S. Kondratyev and L. Monsaingeon), to appear in J. Funct. Anal.
32. On time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing, submitted (with V. B. Surya Prasath, preprint available at ArXiv).
33. The gradient flow of the potential energy on the space of arcs, with W. Shi, submitted.
Papers in peer-reviewed journals of national level.
i. On a variant of the implicit function theorem for equations with skew-symmetries. (Russian) Tr. Mat. Fak., Voronezh. Gos. Univ. (N.S.) 3(19), 25-29 (1998).
ii. On the behavior of solutions of one system of Navier- Stokes type in a bounded domain on the plane with large viscosities, Vestnik VSU, Ser. phys.-math., 2000, no.1, pp. 105-106.(in Russian).
iii. On the Navier-Stokes problem in subdomains of $R^n$, Vestnik VSU, Ser. phys.-math., 2001, no.1, pp. 65-74. (in Russian).
iv. On the motion of a nonlinear-viscous fluid in $R^n$, Vestnik VSU, Ser. phys.-math., 2002, no.1, p. 102-120 . (in Russian).
v. Hopf bifurcations of the three-dimensional Navier- Stokes flow under the wave-like external perturbations, Trudy Molodykh Uchenykh VSU, no.2(2002), p. 11-13. (in Russian).
vi. Energy inequality and uniqueness of a weak solution of the initial-boundary value problem for the motion equations of viscoelastic medium, Vestnik VSU, Ser. phys.-math., 2004, no. 1, pp.96-102.(in Russian).
vii. Continuous dependence of solutions on data of initial-value problem for motion equations of nonlinear visoelastic medium, Vestnik VSU, Ser. phys.-math., 2005, no. 1, pp. 148-151.(in Russian).
viii. On the initial-boundary value problem for equations of anomalous diffusion in polymers, Vestnik VSU, Ser. phys.-math., 2008, no. 1, 157-161.
ix. A review of results and open problems on mathematicals models of motion of viscoelastic media of Jeffreys' type, Vestnik VSU, Ser. phys.-math., 2009, no. 2, 30-50, with V. Zvyagin (in Russian). (preprint available at arXiv).
Manuals (small monographs for students).
x. Degree theory of finite-dimensional maps. Manual, Voronezh, VSU, 2002. (in Russian). (with V.G. Zvyagin).
xi. Mathematical models of non-Newtonian fluids. Manual, Voronezh, VSU, 2004.(in Russian). (with V.G. Zvyagin).
xii. Bifurcations of equations with variational operators in Hilbert spaces, Manual, Voronezh, VSU, 2005.(in Russian).
Other papers in scientific editions.
xiii. Individual approach to the gifted ones during the teaching of constructing a mathematical theory of a problem, Trudy Molodykh Uchenykh Mat. Fak. VSU (2001), pp. 43-48.(in Russian). (with Dontsov V.N., Makarenkov O.Y.)
xiv. On a criterion of properness of multimaps, Trudy Molodykh Uchenykh Mat. Fak. VSU, 2003, pp. 23-27.(in Russian) (preprint available at arXiv).
xv. On an example of birth of a cycle near stationary flow, In: Materials of Seminars REC VSU''Wave processes in inhomogeneous and nonlinear media'', Voronezh, 2003, pp. 48-51.(in Russian).