Main results:

•  Development of the homogenization theory for boundary value problems for  domains with fine-grained boundaries, for weekly connected  domains, and for domains with traps.
•  Development of the homogenization models for dynamics of fluids with microstructure and for harmonic fields on Riemannian manifolds of increasing type.
•  Description of the long-time asymptotic behaviour of solutions of non-linear  evolution equations; description of decay of nondecreasing (step-like)  solutions of nonlinear evolution equations into asymptotic solitons.
•  Construction of a new class of asymptotical solutions of Kadomtsev-Petviashvili equations (curved solitons).
•  Development of methods for solving one-dimensional inverse problems of electromagnetic sounding theory.
•  Development of models of motion of quantized vortices in the rotating superfluids and in the Bose-Einstein condensates.

Pricez and awards:

The State Prize of Ukraine (V.A.Marchenko, E.Ya.Khruslov, 1989): for the series of works "Boundary value problems in domains with fine-grained boundaries".

The Krylov Prize of the National Academy of Sciences of Ukraine (E.Ya.Khruslov, V.P.Kotlyarov, 1995): for the series of works "The decay of the solutions of nonlinear evolution equations into asymptotic solitons".

The Lavrentiev Prize of the National Academy of Sciences of Ukraine  (V.A.Marchenko, B.I.Ptashnik, E. Ya.Khruslov, 2007) for the series of works "Analytic and asymptotic methods of the investigation of non-standard  boundary value problems".

Invited lecture of E.Khruslov at the International Congress of Mathematicians, Zurich, 1994: “Homogenized models of strongly non-homogeneous media".

Lecture of D.Shepelsky at the International Congress of Mathematicians, Beiging, Chine, 2002.