2012/03/09 by Marco Cappiello, Cappiello, Marco, Todor Gramchev +3
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #math.AP
paper · pdf · doi:10.48550/arxiv.1203.2075
arxiv created 2012/03/09 · openalex publication_date 2012/03/09 · arxiv updated 2012/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the decay for |x|→ ∞ of weak Sobolev type solutions of semilinear nonlocal equations Pu=F(u). We consider the case when P=p(D) is an elliptic Fourier multiplier with polyhomogeneous symbol p(ξ) and derive sharp algebraic decay estimates in terms of weighted Sobolev norms. In particular, we state a precise relation between the singularity of the symbol at the origin and the rate of decay of the corresponding solutions. Our basic example is the celebrated Benjamin-Ono equation equation (|D|+c)u=u2, c>0,equation for internal solitary waves of deep stratified fluids. Their profile presents algebraic decay, in strong contrast with the exponential decay for KdV shallow water waves.