Research

My research is focused on the mathematical analysis of Partial Differential Equations, particularly those connected with fluid dynamics and kinetic theory. I am primarily interested in understanding the long-time behavior of models arising in these fields. In fluids, this falls within the realm of hydrodynamic stability theory, a foundational topic and an extremely active area of research today.I am recently focusing on the evolution of highly concentrated vortices in viscous fluids. The goal is to understand the dynamics on large time-scales where spectacular phenomena such as vortex merging are observed to start. In kinetic theory, I focus on the time-decay properties of models like the Boltzmann or the Wave Kinetic Equation.

You can find my PhD thesis here, whose title is "Linear stability analysis of stationary Euler flows for passive scalars and inhomogeneous fluids".

Here is a list of my papers:

  1. Dolce, M. & Mescolini, G. Self-similar instability and forced nonuniqueness: an application to the 2D Euler equations (2024).
    arXiv:2411.18452
  2. Dolce, M., Knobel, N. & Zillinger, C. Large norm inflation of the current in the viscous, non-resistive magnetohydrodynamics equations (2024).
    arXiv:2410.22804
  3. Dolce, M. & Gallay, T. The long way of a viscous vortex dipole (2024).
    arXiv:2407.13562
  4. Dolce, M., Johansson, C.J.P., Sorella, M. Dissipation enhancing properties for a class of Hamiltonian flows with closed streamlines, Communications in Partial Differential Equations 1-52 (2025).
    arXiv:2407.06884
  5. Dolce, M. & Grande, R. On the convergence rates of discrete solutions to the Wave Kinetic Equation, Mathematics in Engineering 6, 4 (2024).
    arxiv:2404.14400
  6. Bianchini, R., Coti Zelati, M. & Dolce, M. Symmetrization and asymptotic stability in non-homogeneous fluids around stratified shear flows, Séminaire Laurent Schwartz — EDP et applications (2023).,
    slsedp.160
    arXiv:2309.12738
  7. Dolce, M. Stability threshold of the 2D Couette flow in a homogeneous magnetic field using symmetric variables, Communications in Mathematical Physics 405, 94 (2024).
    arXiv:2308.12589
  8. Coti Zelati, M., Dolce, M. & Lo, C-C Diffusion enhancement and Taylor dispersion for rotationally symmetric flows in discs and pipes, Journal of Mathematical Fluid Mechanics 26, 12 (2023).
    arXiv:2305.18162
  9. Bedrossian, J., Coti Zelati, M. & Dolce, M. Taylor dispersion and phase mixing in the non-cutoff Boltzmann equation on the whole space, Proceedings of the London Mathematical Society 129, 1 (2024).
    arXiv:2211.05079
  10. Dolce, M. & Drivas, T. D. On maximally mixed equilibria of two-dimensional perfect fluids, Archive for Rational Mechanics and Analysis (2022).,
    arXiv:2204.03587
  11. Bedrossian, J., Bianchini, R., Coti Zelati, M. & Dolce, M. Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations, Communications on Pure and Applied Mathematics (2022).
    arXiv:2103.13713
  12. Coti Zelati, M., Dolce, M., Feng, Y. & Mazzucato, A.L. Global Existence for the Two-dimensional Kuramoto-Sivashinsky equation with a Shear Flow, Journal of Evolution Equations 21 5079–5099 (2021).
    arXiv:2103.02971
  13. Antonelli, P., Dolce, M., & Marcati, P. Linear stability analysis of the homogeneous Couette flow in a 2D isentropic compressible fluid, Annals of PDE 7 24 (2021).
    arXiv:2101.01696
  14. Bianchini, R., Coti Zelati, M. & Dolce, M. Linear inviscid damping for shear flows near Couette in the 2D stably stratified regime, Indiana University Mathematics Journal 71, 4 1467–1504 (2022).
    arXiv:2005.09058
  15. Coti Zelati, M., & Dolce, M. Separation of time-scales in drift-diffusion equations on R2, Journal de Mathématiques Pures et Appliquées 142 58-75 (2020).
    arXiv:1907.04012
  16. Dolce, M., & Donatelli, D. Artificial compressibility method for the Navier–Stokes–Maxwell–Stefan system, Journal of Dynamics and Differential Equations 33 35-62 (2021).
    arXiv:1805.06815