TY - JOUR
T1 - Time-decay estimates for the linearized water wave type equations
AU - Tesfahun, Achenef
N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2022/3
Y1 - 2022/3
N2 - Recently, A. Bulut showed that the free waves Sα(t) f: = exp (it| ∇ | α) f in 1D for α∈ (1 / 3 , 1 / 2] , which are known to be associated with the linearized gravity water wave equations, decay at time scale of order | t| - 1 / 2 for large t, provided that the Hx1(R)-norm of f and the Lx2(R)-norm of x∂xf are bounded. In this note we derive a decay estimate of order (1 - α) - 1 / 2(α| t|) -d/2 on Sα(t) f for all α∈ (0 , 1) and d≥ 1 , assuming a bound only on the B˙1,1d(1-α/2)(Rd)-norm of f. Our estimate extends to any dimension, a wider range of α and describes well the behaviour of the decay near α= 0 and α= 1 , without requiring a spatial-decay assumption on f or its derivative.
AB - Recently, A. Bulut showed that the free waves Sα(t) f: = exp (it| ∇ | α) f in 1D for α∈ (1 / 3 , 1 / 2] , which are known to be associated with the linearized gravity water wave equations, decay at time scale of order | t| - 1 / 2 for large t, provided that the Hx1(R)-norm of f and the Lx2(R)-norm of x∂xf are bounded. In this note we derive a decay estimate of order (1 - α) - 1 / 2(α| t|) -d/2 on Sα(t) f for all α∈ (0 , 1) and d≥ 1 , assuming a bound only on the B˙1,1d(1-α/2)(Rd)-norm of f. Our estimate extends to any dimension, a wider range of α and describes well the behaviour of the decay near α= 0 and α= 1 , without requiring a spatial-decay assumption on f or its derivative.
KW - Fractional wave equations
KW - Linearized water waves
KW - Time-decay estimates
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U2 - 10.1007/s00028-022-00766-x
DO - 10.1007/s00028-022-00766-x
M3 - Article
AN - SCOPUS:85125463743
SN - 1424-3199
VL - 22
JO - Journal of Evolution Equations
JF - Journal of Evolution Equations
IS - 1
M1 - 4
ER -