My research interests are mainly the study of PDEs, in particular those arising in fluid mechanics - I use tools from Harmonic /Functional Analysis.

email: x.y@aum.edu.kw, where x=omar and y=lazar

Publications :


  1. Global existence for the critical dissipative surface quasi-geostrophic equation.

Communication in Mathematical Physics, 322, no. 1, pp. 73-93 (2013).

  1. Global and local existence for the dissipative critical SQG equation with small oscillations.

Journal of Mathematical Fluid Mechanics, 17-3, pp. 533-549 (2015)

  1. Infinite energy solutions for a 1D transport equation with nonlocal velocity (with Pierre-Gilles Lemarié-Rieusset).

Dynamics of PDEs Vol. 13, No. 2, pp. 107-131 (2016)

  1. On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion,

Journal of Differential Equations, Volume 261, Issue 9, pp 4974-4996 (2016).

  1. Global existence of weak solutions to dissipative transport equations with nonlocal velocity (with Hantaek Bae and Rafael Granero-Belinchon).

Nonlinearity, Vol. 31, No. 4 (2018), pp. 1484-1515.

  1. On the local and global existence of solutions to 1D transport equations with nonlocal velocity (with Hantaek Bae and Rafael Granero-Belinchon).

Networks & Heterogeneous Media, 14 (3) : 471-487. (2019)

  1. Regularity results for a class of generalized surface quasi-geostrophic equations (with Liutang Xue).

Journal de Mathématiques Pures et Appliquées, 130, 200-250 (2019).

  1. Growth in the Muskat problem (with Rafael Granero-Belinchon).

Math. Model. Nat. Phenom., Volume 15, 1-23, (2020).

  1. Paralinearization of the Muskat equation and application to the Cauchy problem (with Thomas Alazard).

Arch. Ration. Mech. Anal., 237, 545-583 (2020).

  1. Global well-posedness for the 2D stable Muskat problem in H^3/2 (with Diego Cordoba).

Ann. Sci. Éc. Norm. Supér., 54(4): 1315-1351, (2021).

  1. On the dynamics of the roots of polynomials under differentiation (with Thomas Alazard and Quoc-Hung Nguyen).

Journal de Mathématiques Pures et Appliquées, 162, 1-22 (2022)

  1. Global well-posedness for the three dimensional Muskat problem in the critical Sobolev space (with Francisco Gancedo)

Arch. Ration. Mech. Anal., 246, 141-207 (2022)

Submitted/To appear articles :



  1. On the regularity of temperature fronts for the 3D viscous Boussinesq system (with Yatao Li and Liutang Xue), 39p

Indiana Univ. Math. J (to appear)


Other publications (lecture notes, conference proceedings)


  1. [lecture notes] Small scale creation in inviscd fluids - in Transport Fluid and Mixing (with Mikhail Chernobay, Alexander Kiselev and Chao Li), Partial Differ. Equ. Meas. Theory, De Gruyter Open, Warsaw, (2017), pp 95-117.