Abstract:
With the aid of potential theory the classical solvability of initial-boundary value problems is proved for the equation
∂2∂t2(∂2u∂x21+∂2u∂x22+∂2u∂x23)+∂2u∂x23=0
in a bounded domain of the space Ω, and also in the complement of this domain. For the first boundary value problem a method of obtaining estimates of solutions in uniform norms is established, with an indication of the explicit dependence of the constants on the time exhibited.
Bibliography: 6 titles.
\Bibitem{Kap79}
\by B.~V.~Kapitonov
\paper Potential theory for the equation of small oscillations of a~rotating fluid
\jour Math. USSR-Sb.
\yr 1980
\vol 37
\issue 4
\pages 559--579
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\crossref{https://doi.org/10.1070/SM1980v037n04ABEH002095}
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https://doi.org/10.1070/SM1980v037n04ABEH002095
https://www.mathnet.ru/eng/sm/v151/i4/p607
This publication is cited in the following 26 articles:
Bruno Voisin, “Added mass of oscillating bodies in stratified fluids”, J. Fluid Mech., 987 (2024)
M. O. Korpusov, R. S. Shafir, A. K. Matveeva, “Numerical Diagnostics of Solution Blow-Up in a Thermoelectric Semiconductor Model”, Comput. Math. and Math. Phys., 64:7 (2024), 1595
M. O. Korpusov, A. K. Matveeva, “On critical exponents for weak solutions of the Cauchy problem for a $(2+1)$-dimensional nonlinear composite-type equation with gradient nonlinearity”, Comput. Math. Math. Phys., 63:6 (2023), 1070–1084
M. O. Korpusov, R. S. Shafir, “On Cauchy problems for nonlinear Sobolev equations in ferroelectricity theory”, Comput. Math. Math. Phys., 62:12 (2022), 2091–2111
R. S. Shafir, “Solvability and Blow-Up of Weak Solutions of Cauchy Problems for $(3+1)$-Dimensional Equations of Drift Waves in a Plasma”, Math. Notes, 111:3 (2022), 484–497
M. O. Korpusov, R. S. Shafir, “On the blowup of solutions of the Cauchy problem for nonlinear equations of ferroelectricity theory”, Theoret. and Math. Phys., 212:3 (2022), 1169–1180
M. O. Korpusov, R. S. Shafir, “Blow-up of weak solutions of the Cauchy problem for $(3+1)$-dimensional equation of plasma drift waves”, Comput. Math. Math. Phys., 62:1 (2022), 117–149
M. O. Korpusov, E. A. Ovsyannikov, “Local solvability, blow-up, and Hölder regularity of solutions to some Cauchy problems for nonlinear plasma wave equations: I. Green formulas”, Comput. Math. Math. Phys., 62:10 (2022), 1609–1631
M. O. Korpusov, A. K. Matveeva, “On critical exponents for weak solutions of the Cauchy problem for a non-linear equation of composite type”, Izv. Math., 85:4 (2021), 705–744
M. O. Korpusov, D. K. Yablochkin, “Potential theory and Schauder estimate in Hölder spaces for $(3 + 1)$-dimensional Benjamin–Bona–Mahoney–Burgers equation”, Comput. Math. Math. Phys., 61:8 (2021), 1289–1314
Bruno Voisin, “Boundary integrals for oscillating bodies in stratified fluids”, J. Fluid Mech., 927 (2021)
M.O. Korpusov, A.A. Panin, “On the blow-up of the solution and on the local and global solvability of the Cauchy problem for a nonlinear equation in Hölder spaces”, Journal of Mathematical Analysis and Applications, 504:2 (2021), 125469
M. O. Korpusov, E. A. Ovsyannikov, “Blow-up instability in non-linear wave models with distributed parameters”, Izv. Math., 84:3 (2020), 449–501
M. O. Korpusov, G. I. Shlyapugin, “O razrushenii reshenii zadach Koshi dlya odnogo klassa nelineinykh uravnenii teorii ferritov”, Materialy Vserossiiskoi nauchnoi konferentsii «Differentsialnye uravneniya i ikh prilozheniya», posvyaschennoi 85-letiyu professora M. T. Terekhina. Ryazanskii gosudarstvennyi universitet im. S.A. Esenina, Ryazan, 17–18 maya 2019 g. Chast 1, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 185, VINITI RAN, M., 2020, 79–131
Bruno Voisin, “Near-field internal wave beams in two dimensions”, J. Fluid Mech., 900 (2020)
M. O. Korpusov, D. K. Yablochkin, “Potential theory for a nonlinear equation of the Benjamin–Bona–Mahoney–Burgers type”, Comput. Math. Math. Phys., 59:11 (2019), 1848–1880
E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, “Large-time asymptotic behaviour of solutions of non-linear Sobolev-type equations”, Russian Math. Surveys, 64:3 (2009), 399–468
A. B. Alshin, M. A. Istomina, “Solvability of the Neumann problem for a Sobolev pseudoparabolic equation”, Comput. Math. Math. Phys., 46:7 (2006), 1207–1215
Al'shin A., Al'shina E., “Numerical Solution of Initial-Boundary-Value Problems for Sobolev Equations Using the Dynamic-Potential Method”, J. Commun. Technol. Electron., 50:2 (2005), 213–219
P.A. Krutitskii, “Initial–boundary value problem for an equation of internal gravity waves in a 3-D multiply connected domain with Dirichlet boundary condition”, Advances in Mathematics, 177:2 (2003), 208
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