Recent Advances in the Theory of Nonlinear Differential Equations (PDEs): A Review

Abstract

The main findings in existence, uniqueness, stability, blow-up phenomena, and

regularity in the theory of nonlinear partial differential equations (PDEs) over the past

decade are the subject of this paper. Important developments include the resolution of

Onsager's conjecture in fluid turbulence, the construction of finite-time blow-up

solutions in previously unexplored regimes, such as defocusing nonlinear Schrödinger

equations, and the use of convex integration to show that weak solutions for the NavierStokes equations are not unique. Stability analysis has improved our knowledge of

damping mechanisms in fluids and plasmas, while geometric PDEs have offered new

insights into elliptic equations and minimum surfaces. Thanks to innovative techniques

like concentration-compactness, convex integration, and probabilistic methods, the

arsenal for solving nonlinear PDEs has expanded. Notwithstanding these successes,

basic problems such as the global regularity of 3D Navier-Stokes are still unresolved.

The topic is still advancing because to the multidisciplinary integration of concepts,

which presents interesting directions for further study