TY - JOUR

T1 - On behaviour of free-surface profiles for bounded steady water waves

AU - Kozlov, V.

AU - Kuznetsov, N.

N1 - Funding Information: V.K. was supported by the Swedish Research Council (VR). N.K. acknowledges the financial support from the Royal Swedish Academy of Sciences.

PY - 2008/7

Y1 - 2008/7

N2 - The paper deals with the classical non-linear problem of steady two-dimensional waves on water of finite depth. The problem is formulated so that it describes all waves without stagnation points on the free-surface profiles that are bounded themselves and have bounded slopes. By virtue of reducing the problem to an integro-differential equation the following three results are proved. First, there are no waves when the flow is critical. Second, there are no waves having profiles totally above the upper boundary of the uniform subcritical stream. Finally, only two types of the free-surface behaviour are possible at positive (or/and negative) infinity: the profile either oscillates infinitely many times around the upper boundary of the subcritical uniform stream or asymptotes the upper level of a uniform stream (subcritical or supercritical). The latter assertion is proved under additional assumption that the slope of the free surface is a uniformly continuous function.

AB - The paper deals with the classical non-linear problem of steady two-dimensional waves on water of finite depth. The problem is formulated so that it describes all waves without stagnation points on the free-surface profiles that are bounded themselves and have bounded slopes. By virtue of reducing the problem to an integro-differential equation the following three results are proved. First, there are no waves when the flow is critical. Second, there are no waves having profiles totally above the upper boundary of the uniform subcritical stream. Finally, only two types of the free-surface behaviour are possible at positive (or/and negative) infinity: the profile either oscillates infinitely many times around the upper boundary of the subcritical uniform stream or asymptotes the upper level of a uniform stream (subcritical or supercritical). The latter assertion is proved under additional assumption that the slope of the free surface is a uniformly continuous function.

KW - Finite depth

KW - Non-existence

KW - Steady water waves

KW - Subcritical flow

UR - http://www.scopus.com/inward/record.url?scp=50349092088&partnerID=8YFLogxK

U2 - 10.1016/j.matpur.2008.02.013

DO - 10.1016/j.matpur.2008.02.013

M3 - Article

AN - SCOPUS:50349092088

VL - 90

SP - 1

EP - 14

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

SN - 0021-7824

IS - 1

ER -