Research area C: Models and Approximations
Unit C1: Evolution and asymptotics
Unit C2: Multi-scale phenomena and macroscopic structures
• Invariante Mannigfaltigkeiten für schnelle Diffusionen nahe Auslöschung online
• Transport Equations, mixing and fluid dynamics Advection-diffusion equations are of fundamental importance in many areas of science. They describe systems, in which a quantity is simultaneously diffused and advected by a velocity field. In many applications these velocity fields are highly irregular. In this project, several quantitative aspects shall be investigated. One is related to mixing properties in fluids caused by shear flows. The interplay between the transport by the shear flow and the regularizing diffusion leads after a certain time, to the emergence of a dominant length scales which persist during the subsequent evolution and determine mixing rates. A rigorous understanding of these phenomena is desired. In addition, stability estimates for advection-diffusion equations will be derived. These shall give a deep insight into how solutions depend on coefficients and data. The new results shall subsequently be applied to estimate the error generated by numerical finite volume schemes approximating the model equations. online
$\bullet$ Nonlinear partial differential equations.
$\bullet$ Mathematical fluid dynamics.
$\bullet$ Quantitative analysis of transport equations.
$\bullet$ Degenerate parabolic equations.
Selected Publications of Christian Seis
$\bullet$ C. Seis. A quantitative theory for the continuity equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 34(7):1837–1850, 2017.
$\bullet$ A. Schlichting and C. Seis. Convergence rates for upwind schemes with rough coefficients. SIAM J. Numer. Anal., 55(2):812–840, 2017.
$\bullet$ R. L. Jerrard and C. Seis. On the vortex filament conjecture for Euler flows. Arch. Ration. Mech. Anal., 224(1):135–172, 2017.
$\bullet$ L. Mugnai, C. Seis, and E. Spadaro. Global solutions to the volume-preserving mean-curvature flow. Calc. Var. Partial Differential Equations, 55(1):18, 2016.
$\bullet$ C. Seis. Scaling bounds on dissipation in turbulent flows. J. Fluid Mech., 777:591–603, 2015.
$\bullet$ R. J. McCann and C. Seis. The spectrum of a family of fourth-order nonlinear diffusions near the global attractor. Comm. Partial Differential Equations, 40(2):191–218, 2015.
$\bullet$ C. Seis. Long-time asymptotics for the porous medium equation: the spectrum of the linearized operator. J. Differential Equations, 256(3):1191–1223, 2014.
$\bullet$ C. Seis. Maximal mixing by incompressible fluid flows. Nonlinearity, 26(12):3279–3289, 2013.
$\bullet$ F. Otto and C. Seis. Rayleigh- Bénard convection: improved bounds on the Nusselt number. J. Math. Phys., 52(8):083702, 24, 2011.
$\bullet$ Y. Brenier, F. Otto, and C. Seis. Upper bounds on coarsening rates in demixing binary viscous liquids. SIAM J. Math. Anal., 43(1):114–134, 2011.
$\bullet $ Stefano Ceci and Christian Seis. On the dynamics of point vortices for the two-dimensional Euler equation with $L^p$ vorticity. Philos. Trans. Royal Soc. A, 380(2226):20210046, May 2022. doi:10.1098/rsta.2021.0046.
$\bullet $ David Meyer and Christian Seis. Propagation of regularity for transport equations. a Littlewood-Paley approach. arXiv e-prints, March 2022. arXiv:2203.10860.
$\bullet $ Stefano Ceci and Christian Seis. On the dynamics of vortices in viscous 2D flows. arXiv e-prints, March 2022. arXiv:2203.07185.
$\bullet $ Beomjun Choi, Robert J. McCann, and Christian Seis. Asymptotics near extinction for nonlinear fast diffusion on a bounded domain. arXiv e-prints, February 2022. arXiv:2202.02769.
$\bullet $ Stefano Ceci and Christian Seis. On the dynamics of point vortices for the 2D Euler equation with $L^p$ vorticity. arXiv e-prints, July 2021. arXiv:2107.12820.
$\bullet $ Helena J Nussenzveig Lopes, Christian Seis, and Emil Wiedemann. On the vanishing viscosity limit for 2D incompressible flows with unbounded vorticity. Nonlinearity, 34(5):3112–3121, May 2021. doi:10.1088/1361-6544/abe51f.
$\bullet $ Stefano Ceci and Christian Seis. Vortex dynamics for 2D Euler flows with unbounded vorticity. Rev. Mat. Iberoam., 37(5):1969–1990, May 2021. doi:10.4171/rmi/1255.
$\bullet $ André Schlichting and Christian Seis. The Scharfetter–Gummel scheme for aggregation–diffusion equations. IMA Journal of Numerical Analysis, May 2021. doi:10.1093/imanum/drab039.
$\bullet $ Christian Seis and Dominik Winkler. A well-posedness result for a system of cross-diffusion equations. Journal of Evolution Equations, 21(2):2471–2489, May 2021. doi:10.1007/s00028-021-00690-6.
$\bullet $ Víctor Navarro-Fernández, André Schlichting, and Christian Seis. Optimal stability estimates and a new uniqueness result for advection-diffusion equations. arXiv e-prints, February 2021. arXiv:2102.07759.
$\bullet $ Christian Seis. On the Littlewood–Paley spectrum for passive scalar transport equations. Journal of Nonlinear Science, 30(2):645–656, April 2020. doi:10.1007/s00332-019-09585-w.
$\bullet $ Christian Seis. A note on the vanishing viscosity limit in the Yudovich class. Canadian Mathematical Bulletin, pages 1–11, April 2020. doi:10.4153/S0008439520000296.
$\bullet $ Christian Seis. Diffusion limited mixing rates in passive scalar advection. arXiv e-prints, March 2020. arXiv:2003.08794.
$\bullet $ Camilla Nobili and Christian Seis. Renormalization and energy conservation for axisymmetric fluid flows. arXiv e-prints, June 2019. arXiv:1906.07400.