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Strain-gradient theory of hydroelastic travelling waves and Young measures of their singular limits. (English) Zbl 1243.35180

In this comprehensive paper the authors study two-dimensional, steady, periodic waves traveling with constant speed, without changing shape, on the surface of an inviscid, incompressible fluid, under gravity. The surface is in contact with a heavy, thin elastic sheet. Furthermore, it is supposed that there are no frictional forces between the fluid and the sheet, and shear forces in the sheet are neglected. The fluid beneath the sheet is at rest at infinite depth and, relative to a frame moving with the wave, the fluid’s velocity field is stationary and irrotational. The surface sheet is hyperelastic with a stored energy function that depends on the stretch, strain-gradient, and curvature.
The paper investigates the balance between these elastic and hydrodynamic effects to produce a steady hydroelastic wave. The main result consists in a statement of the free-boundary problem satisfied by generalized maximizers, and a detailed description of the Young measures that arise as limits of a sequence of maximizers of problems regularized by strain-gradient effects.

MSC:

35R35 Free boundary problems for PDEs
35Q35 PDEs in connection with fluid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
35C07 Traveling wave solutions
35B25 Singular perturbations in context of PDEs
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[1] Antman, S.S.: Nonlinear Problems of Elasticity, Second edn. Springer-Verlag, New York (2005) · Zbl 1098.74001
[2] Aubin J.-P., Frankowska H.: Set-Valued Analysis. Birkhäuser, Boston (1990)
[3] Baldi, P., Toland, J.F.: Steady periodic water waves under nonlinear elastic membranes. J. Reine Angew. Math. (2011). doi: 10.1515/CRELLE.2011.015 · Zbl 1277.35281
[4] Baldi P., Toland J.F.: Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves. Interfaces Free Bound. 12, 1–22 (2010) · Zbl 1191.35273 · doi:10.4171/IFB/224
[5] Ekeland I., Temam R.: Convex Analysis and Variational Problems. North Holland, Amsterdam (1976) · Zbl 0322.90046
[6] Ericksen J.L.: Equilibrium of bars. J. Elast. 5, 149–201 (1975) · Zbl 0306.73037 · doi:10.1007/BF01390075
[7] Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: L p -Theory. Springer Monographs in Mathemtics. Springer, New York (2007) · Zbl 1153.49001
[8] Martinez-Avendano A., Rosenthal P.: An Introduction to Operators in the Hardy-Hilbert Space. Springer, New York (2007)
[9] McShane E.J.: Generalized curves. Duke Math. J. 6, 513–536 (1940) · Zbl 0023.39801 · doi:10.1215/S0012-7094-40-00642-1
[10] McShane E.J.: Necessary conditions in generalized-curve problems of the calculus of variations. Duke Math. J. 7, 1–27 (1940) · Zbl 0024.32503 · doi:10.1215/S0012-7094-40-00701-3
[11] McShane E.J.: Existence theorems for Bolza problems in the calculus of variations. Duke Math. J. 7, 28–61 (1940) · Zbl 0024.32504 · doi:10.1215/S0012-7094-40-00702-5
[12] Plotnikov P.I., Toland J.F.: Phase transitions with a minimal number of jumps in the singular limits of higher order theories. Ann. Inst. H. Poincaré. 27, 655–691 (2010) · Zbl 1192.82034 · doi:10.1016/j.anihpc.2009.11.002
[13] Roubiček T.: Relaxation in Optimization Theory and Variational Calculus. de Gruyter, Berlin New-York (1997)
[14] Roubiček T.: Optimality conditions for nonconvex variational problems relaxed in terms of Young measures. Kybernetika 34, 335–347 (1998) · Zbl 1274.49040
[15] Shargorosky, E., Toland, J.F.: Bernoulli Free-Boundary Problems. Memoirs of AMS no. 914, American Mathematical Society, Providence, RI (2008). ISSN 0065-9266
[16] Tartar, L.: Compensated compactness and applications to partial differential equations. In: Knops R.J. (ed.) Nonlinear Analysis and Mechanics, Heriot-Watt Symposium IV. Pitman Research Notes in Mathematics, vol. 39. Pitman, San Fransisco (1979) · Zbl 0437.35004
[17] Toland, J.F.: Heavy hydroelastic travelling waves. Proc. R. Soc. Lond. A 463, 2371–2397 (2007). doi: 10.1098/rspa.2007.1883
[18] Toland, J.F.: Steady periodic hydroelastic waves. Arch. Ration. Mech. Anal. 189(2), 325–362 (2008). doi: 10.1007/s00205-007-0104-2 · Zbl 1147.76008
[19] Young L.C.: Generalized curves and the existence of an attained absolute minimum in the calculus of variations. Compt. Rendus Soc. Sci. Lett. Vars. Cl. III 30, 212–234 (1937) · Zbl 0019.21901
[20] Young L.C.: Necessary conditions in the calculus of variations. Acta Math. 69, 229–258 (1938) · Zbl 0019.26702 · doi:10.1007/BF02547714
[21] Zygmund, A.: Trigonometric Series I & II. Corrected reprint of 2nd edn (1959). Cambridge University Press, Cambridge (1968)
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