On unique solvability and regularity in the linearized two-dimensional wave resistance problem
Author:
Dario Pierotti
Journal:
Quart. Appl. Math. 61 (2003), 639-655
MSC:
Primary 35J25; Secondary 35B65, 76B15
DOI:
https://doi.org/10.1090/qam/2019616
MathSciNet review:
MR2019616
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Abstract: We discuss existence, uniqueness, and regularity of the solutions of a boundary value problem in a strip, which is obtained by linearization of the equations of the wave-resistance problem for a cylinder semisubmerged in a heavy fluid of constant depth $H$ and moving at uniform velocity $c$ in the direction orthogonal to its generators. We show that the problem has a unique solution, rapidly decreasing at infinity, for every $c > \sqrt {gH}$, where $g$ is the acceleration of gravity. For $c < \sqrt {gH}$, we prove unique solvability provided $c \ne {c_k}$, where ${c_k}$ is a known sequence monotonically decreasing to zero. In this case, the related flow has in general nontrivial oscillations at infinity downstream.
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C. D. Pagani and D. Pierotti, The Neumann-Kelvin problem for a beam, J. Math. Anal. Appl. 240 (1999), 60–79.
D. Pierotti, The subcritical motion of a surface-piercing cylinder: existence and regularity of waveless solutions of the linearized problem, Advances in Differential Equations 7 (2002), 385–418.
C. D. Pagani and D. Pierotti, Exact solution of the wave resistance problem for a submerged cylinder. I. Linearized theory, Arch. Rational Mech. Anal. 149 (1999), 271–288.
N. G. Kuznetsov and V. G. Maz’ya, On unique solvability of the plane Neumann-Kelvin problem, Math. USSR-Sb., 63 (1989), 425–446.
N. G. Kuznetsov, On uniqueness and solvability in the linearized two-dimensional problem of a supercritical stream about a surface-piercing body, Proc. Roy. Soc. London Ser. A, 450 (1995), 233–253.
C. D. Pagani and D. Pierotti, The Neumann-Kelvin problem in a bounded domain, J. Math. Analysis and Appl. 192 (1995), 41–62.
C. D. Pagani and D. Pierotti, On solvability of the nonlinear wave resistance problem for a surface-piercing symmetric cylinder, SIAM J. Math. Anal. 32 (2000), 214–233.
C. D. Pagani and D. Pierotti, The forward motion of an unsymmetric surface-piercing cylinder: the solvability of the nonlinear problem in the supercritical case, Q. Jl Mecc. Appl. Math. 54 (2000), 1–22.
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E. Zeidler, “Nonlinear functional analysis and its applications IV.” Springer-Verlag, New York (1988).
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© Copyright 2003
American Mathematical Society