Hyperbolic systems of conservation laws, well-posedness of free boundary problems, mathematical fluid dynamics, shock waves and characteristic discontinuities
Biography
Junior Researcher (1993–94), Researcher (1994–96), Senior Researcher (1996–2009), Leading Researcher (2009–2018), Principal Researcher (2018–now) of the Sobolev Institute of mathematics.
Visiting positions:
January 1997 – December 1998, Alexander von Humboldt Research Fellow, University of Stuttgart, Germany; August 2002 – April 2004, Visiting Research Associate, University of Hull, UK; January 2005 – September 2005, Visiting Research Associate, University of Hull, UK;
October 2006 – November 2006, Landau Network-Centro Volta-Cariplo Fellow, University of Brescia, Italy; April 2007 – June 2007, Alexander von Humboldt Research Fellow, University of Leipzig, Germany; October 2009 – November 2009, DAAD Visiting Professor, University of Konstanz, Germany; September 2010 – November 2010, Landau Network-Centro Volta-Cariplo Fellow, University of Brescia, Italy; September 2012 – November 2012, Landau Network-Centro Volta-Cariplo Fellow, University of Brescia, Italy, April 2018 – May 2018, Visiting Professor, June 2019, Visiting Professor, University of Konstanz, Germany.
Main publications:
Blokhin A., Trakhinin Y., “Stability of strong discontinuities in fluids and MHD”, Handbook of mathematical fluid dynamics, 1, eds. Friedlander S., Serre D., North-Holland, Amsterdam, 2002, 545–652
Trakhinin Y., “A complete 2D stability analysis of fast MHD shocks in an ideal gas”, Commun. Math. Phys., 236 (2003), 65–92
Trakhinin Y., “Existence of compressible current-vortex sheets: variable coefficients linear analysis”, Arch. Ration. Mech. Anal., 177 (2005), 331–366
Trakhinin Y., “Dissipative symmetrizers of hyperbolic problems and their applications to shock waves and characteristic discontinuities”, SIAM J. Math. Anal., 37 (2006), 1988–2024
Freistühler H., Trakhinin Y., “On the viscous and inviscid stability of magnetohydrodynamic shock waves”, Physica D: Nonlinear Phenomena, 23:1 (2008), 3030–3037
Trakhinin Y., “The existence of current-vortex sheets in ideal compressible magnetohydrodynamics”, Arch. Ration. Mech. Anal., 191 (2009), 245–310
Trakhinin Y., “Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition”, Comm. Pure Appl. Math., 62 (2009), 1551–1594
Trakhinin Y., “On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD”, J. Differential Equations, 249 (2010), 2577–2599
Secchi P., Trakhinin Y., “Well-posedness of the plasma-vacuum interface problem”, Nonlinearity, 27 (2014), 105–169
Alessandro Morando, Yuri Trakhinin, Paola Trebeschi, “Well-posedness of the linearized problem for MHD contact discontinuities”, Journal of Differential Equations, 258 (2015), 2531–2571 , Elsevier BV
Yuri Trakhinin, “Well-posedness of the free boundary problem in compressible elastodynamics”, Journal of Differential Equations, 264 (2018), 1661–1715 , Elsevier BV
Morando A., Trakhinin Y., Trebeschi P., “Local existence of MHD contact discontinuities”, Arch. Ration. Mech. Anal., 228 (2018), 691–742
Morando A., Trakhinin Y., Trebeschi P., “Structural stability of shock waves in 2D compressible elastodynamics”, Mathematische Annalen, 378 (2019), 1471–1504
Lizhi Ruan, Yuri Trakhinin, “Elementary symmetrization of inviscid two-fluid flow equations giving a number of instant results”, Physica D: Nonlinear Phenomena, 391 (2019), 66–71 , Elsevier BV
Trakhinin Y., Wang T., “Well-posedness of Free Boundary Problem in Non-relativistic and Relativistic Ideal Compressible Magnetohydrodynamics”, Arch. Ration. Mech. Anal., 239:2 (2021), 1131–1176
Trakhinin Y., Wang T., “Well-posedness for the free-boundary ideal compressible magnetohydrodynamic equations with surface tension”, Mathematische Annalen, 383 (2022), 761–808
Trakhinin Y., Wang T., “Nonlinear Stability of MHD Contact Discontinuities with Surface Tension”, Arch. Ration. Mech. Anal., 243 (2022), 1091–1149
Trakhinin Y., Wang T., “Well-posedness for moving interfaces with surface tension in ideal compressible MHD”, SIAM J. Math. Anal., 54:6 (2022), 5888–592
Paolo Secchi, Yuri Trakhinin, Tao Wang, “On vacuum free boundary problems in ideal compressible magnetohydrodynamics”, Bulletin of the London Mathematical Society, 55 (2023), 2087–2111 , Wiley
Morando A., Secchi P., Trakhinin Y., Trebeschi P., Yuan D., “Well-Posedness of the Two-Dimensional Compressible Plasma-Vacuum Interface Problem”, Archive for Rational Mechanics and Analysis, 248:4 (2024), 56 , 45 pp.
Trakhinin Y., “Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition”, Comm. Pure Appl. Math., 62 (2009), 1551–1594
Blokhin A., Trakhinin Y., “Stability of strong discontinuities in fluids and MHD”, Handbook of mathematical fluid dynamics, 1, eds. Friedlander S., Serre D., North-Holland, Amsterdam, 2002, 545–652
Trakhinin Y., “On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD”, J. Differential Equations, 249 (2010), 2577–2599
Paolo Secchi, Yuri Trakhinin, “Well-posedness of the linearized plasma-vacuum interface problem”, Interfaces and Free Boundaries, 15 (2013), 323–357 , European Mathematical Society - EMS - Publishing House GmbH
Trakhinin Y., Wang T., “Well-posedness of Free Boundary Problem in Non-relativistic and Relativistic Ideal Compressible Magnetohydrodynamics”, Arch. Ration. Mech. Anal., 239:2 (2021), 1131–1176
Yuri Trakhinin, “Well-posedness of the free boundary problem in compressible elastodynamics”, Journal of Differential Equations, 264 (2018), 1661–1715 , Elsevier BV
Trakhinin Y., Wang T., “Well-posedness for the free-boundary ideal compressible magnetohydrodynamic equations with surface tension”, Mathematische Annalen, 383 (2022), 761–808
Alessandro Morando, Yuri Trakhinin, Paola Trebeschi, “Well-posedness of the linearized problem for MHD contact discontinuities”, Journal of Differential Equations, 258 (2015), 2531–2571 , Elsevier BV
Trakhinin Y., “Dissipative symmetrizers of hyperbolic problems and their applications to shock waves and characteristic discontinuities”, SIAM J. Math. Anal., 37 (2006), 1988–2024
Alessandro Morando, Paolo Secchi, Yuri Trakhinin, Paola Trebeschi, “Stability of an incompressible plasma–vacuum interface with displacement current in vacuum”, Mathematical Methods in the Applied Sciences, 43 (2020), 7465–7483 , Wiley
Trakhinin Y., Wang T., “Well-posedness for moving interfaces with surface tension in ideal compressible MHD”, SIAM J. Math. Anal., 54:6 (2022), 5888–592
Yuri Trakhinin, “On violent instability of a plasma-vacuum interface for an incompressible plasma flow and a nonzero displacement current in vacuum”, Communications in Mathematical Sciences, 18 (2020), 321–337 , International Press of Boston
Yuri Trakhinin, “Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics”, Zeitschrift für angewandte Mathematik und Physik, 71 (2020) , Springer Science and Business Media LLC
Paolo Secchi, Yuri Trakhinin, Tao Wang, “On vacuum free boundary problems in ideal compressible magnetohydrodynamics”, Bulletin of the London Mathematical Society, 55 (2023), 2087–2111 , Wiley
A. M. Blokhin, Yu. L. Trakhinin, I. Z. Merazhov, “On the stability of shock waves in a continuous medium with a space charge”, J. Appl. Mech. Tech. Phys., 39:2 (1998), 184–193
30.
A. M. Blokhin, Yu. L. Trakhinin, “Symmetrization of a system of equations of radiative hydrodynamics, and global solvability of the Cauchy problem”, Siberian Math. J., 37:6 (1996), 1101–1109
31.
A. M. Blokhin, Yu. L. Trakhinin, “On stability of shock waves in magnetohydrodynamics with anisotropic pressure”, Siberian Math. J., 34:6 (1993), 1005–1016
32.
Yu. L. Trakhinin, “On local well-posedness of problems with characteristic free boundary for hyperbolic systems of conservation laws”, Russian Math. Surveys, 79:2 (2024), 325–360
33.
Morando A., Secchi P., Trakhinin Y., Trebeschi P., Yuan D., “Well-Posedness of the Two-Dimensional Compressible Plasma-Vacuum Interface Problem”, Archive for Rational Mechanics and Analysis, 248:4 (2024), 56 , 45 pp.
Yu. L. Trakhinin, “Local solvability of problems with free boundaries in the magnetic hydrodynamics of an ideal compressible fluid with and without account for surface tension”, J. Appl. Mech. Tech. Phys., 62:4 (2021), 684–691
35.
Yu. L. Trakhinin, “Local existence of contact discontinuities in relativistic magnetohydrodynamics”, Siberian Adv. Math., 30:2 (2020), 55–76
36.
Yuri Trakhinin, “Existence and Stability of Compressible and Incompressible Current-Vortex Sheets”, Analysis and Simulation of Fluid Dynamics, Birkhäuser Basel, 2006, 229–246
A. M. Blokhin, Yu. L. Trakhinin, “Shock-wave stability for one model of radiation hydrodynamics”, Journal of Applied Mechanics and Technical Physics, 37 (1996), 775–784 , Springer Science and Business Media LLC
A. M. Blokhin, Yu. L. Trakhinin, “Shock-wave stability for one model of radiation hydrodynamics”, J. Appl. Mech. Tech. Phys., 37:6 (1996), 775–784
39.
A. M. Blokhin, Yu. L. Trakhinin, “Rotational discontinuity in magnetohydrodynamics with anisotropic pressure. I”, Siberian Math. J., 35:1 (1994), 9–20
40.
A. M. Blokhin, Yu. L. Trakhinin, “Rotational discontinuity in magnetohydrodynamics with anisotropic pressure. II”, Siberian Math. J., 35:2 (1994), 250–259
41.
Y. Trakhinin, “Stabilizing effect of surface tension for the linearized MHD–Maxwell free interface problem”, J. Differential Equations, 427 (2025), 143–162 (to appear)
42.
Yuri Trakhinin, “Well-posedness for moving interfaces in anisotropic plasmas”, Zeitschrift für angewandte Mathematik und Physik, 74 (2023), 142 (12 pp.)
43.
Yuri Trakhinin, “Well-posedness of the Free Boundary Problem for Non-relativistic and Relativistic Compressible Euler Equations with a Vacuum Boundary Condition”, Series in Contemporary Applied Mathematics, Co-Published with Higher Education Press, 2012, 692–699
44.
Y. Trakhinin, “On Compressible Current–Vortex Sheets”, Hyperbolic Problems: Theory, Numerics, Applications, Springer Berlin Heidelberg, 2008, 209–220
45.
Yuri L. Trakhinin, “On Stability of Fast Shock Waves in Classical and Relativistic MHD”, Hyperbolic Problems: Theory, Numerics, Applications, Birkhäuser Basel, 2001, 911–919
46.
A. M. Blokhin, Yu. L. Trakhinin, “Stability of a fast magnetohydrodynamic shock wave in plasma with anisotropic pressure”, J. Appl. Mech. Tech. Phys., 36:4 (1995), 496–512
47.
Yu. L. Trakhinin, “On existence of shock waves in compressible neo-Hookean elastic materials”, Theoret. and Math. Phys., 225:1 (2025), 1756–1772
48.
Yu. L. Trakhinin, “On well-posedness of the free boundary problem for ideal compressible MHD equations and Maxwell equations in vacuum”, Nonlocal and nonlinear problems, CMFD, 71, no. 1, PFUR, M., 2025, 176–193
Presentations in Math-Net.Ru
1.
О существовании ударных волн в эластодинамике Yu. L. Trakhinin The Sixth International Conference "Supercomputer Technologies of Mathematical Modelling" (SCTeMM'25) July 15, 2025 11:20
