TY - JOUR

T1 - Non-viscous regularization of the Davey-Stewartson equations

T2 - Analysis and modulation theory

AU - Guo, Yanqiu

AU - Hacinliyan, Irma

AU - Titi, Edriss S.

PY - 2016/8/1

Y1 - 2016/8/1

N2 - In the present study, we are interested in the Davey-Stewartson equations (DSE) that model packets of surface and capillary-gravity waves. We focus on the elliptic-elliptic case, for which it is known that DSE may develop a finite-time singularity. We propose three systems of non-viscous regularization to the DSE in a variety of parameter regimes under which the finite-time blow-up of solutions to the DSE occurs. We establish the global well-posedness of the regularized systems for all initial data. The regularized systems, which are inspired by the α-models of turbulence and therefore are called the α-regularized DSE, are also viewed as unbounded, singularly perturbed DSE. Therefore, we also derive reduced systems of ordinary differential equations for the α-regularized DSE by using the modulation theory to investigate the mechanism with which the proposed non-viscous regularization prevents the formation of the singularities in the regularized DSE. This is a follow-up of the work [Cao et al., Nonlinearity 21, 879-898 (2008); Cao et al., Numer. Funct. Anal. Optim. 30, 46-69 (2009)] on the non-viscous α-regularization of the nonlinear Schrödinger equation.

AB - In the present study, we are interested in the Davey-Stewartson equations (DSE) that model packets of surface and capillary-gravity waves. We focus on the elliptic-elliptic case, for which it is known that DSE may develop a finite-time singularity. We propose three systems of non-viscous regularization to the DSE in a variety of parameter regimes under which the finite-time blow-up of solutions to the DSE occurs. We establish the global well-posedness of the regularized systems for all initial data. The regularized systems, which are inspired by the α-models of turbulence and therefore are called the α-regularized DSE, are also viewed as unbounded, singularly perturbed DSE. Therefore, we also derive reduced systems of ordinary differential equations for the α-regularized DSE by using the modulation theory to investigate the mechanism with which the proposed non-viscous regularization prevents the formation of the singularities in the regularized DSE. This is a follow-up of the work [Cao et al., Nonlinearity 21, 879-898 (2008); Cao et al., Numer. Funct. Anal. Optim. 30, 46-69 (2009)] on the non-viscous α-regularization of the nonlinear Schrödinger equation.

UR - http://www.scopus.com/inward/record.url?scp=84982665586&partnerID=8YFLogxK

U2 - 10.1063/1.4960047

DO - 10.1063/1.4960047

M3 - Article

AN - SCOPUS:84982665586

SN - 0022-2488

VL - 57

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 8

M1 - 081502

ER -