TY - JOUR
T1 - On the pathwise solutions to the CAMASSA–HOLM equation with multiplicative noise
AU - Tang, Hao
PY - 2018
Y1 - 2018
N2 - In this paper we consider the Camassa–Holm (CH) equation with multiplicative noise, which can be obtained when the nonhydrostatic pressure in the deterministic equation is subject to a turbulent velocity field. For the periodic boundary value problem for this SPDE, we establish the local existence and pathwise uniqueness of the pathwise solution in Sobolev spaces H s with s > 3/2. For the linear noise case, conditions that lead to the global existence and the blow-up in finite time of the solution, and their associated probabilities, are also acquired. Finally, we study the pathwise dissipative effect of the linear noise on the periodic peakons to the deterministic CH equation.
AB - In this paper we consider the Camassa–Holm (CH) equation with multiplicative noise, which can be obtained when the nonhydrostatic pressure in the deterministic equation is subject to a turbulent velocity field. For the periodic boundary value problem for this SPDE, we establish the local existence and pathwise uniqueness of the pathwise solution in Sobolev spaces H s with s > 3/2. For the linear noise case, conditions that lead to the global existence and the blow-up in finite time of the solution, and their associated probabilities, are also acquired. Finally, we study the pathwise dissipative effect of the linear noise on the periodic peakons to the deterministic CH equation.
KW - Blow up
KW - Global existence
KW - Martingale solutions
KW - Pathwise solutions
KW - Periodic peakons
KW - Stochastic Camassa–Holm equation
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U2 - 10.1137/16M1080537
DO - 10.1137/16M1080537
M3 - RGC 21 - Publication in refereed journal
VL - 50
SP - 1322
EP - 1366
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
SN - 0036-1410
IS - 1
ER -