Protter-Morawetz multidimensional problems
Tóm tắt
About 50 years ago M.H. Protter introduced boundary value problems that are multidimensional analogues of the classical plane Morawetz problems for equations of mixed hyperbolic-elliptic type that model transonic fluid flows. Up to now there are no general existence results for the Protter-Morawetz multidimensional problems, and an understanding of the situation is not at hand. At the same time, Protter also formulated boundary value problems in the hyperbolic part of the domain—the nonhomogeneous wave equation is studied in a (3+1)-D domain bounded by two characteristic cones and a non-characteristic ball. These problems could be considered as multidimensional variants of the Darboux problem in ℝ2. In the frame of classical solvability the hyperbolic Protter problem is not Fredholm, because it has an infinite-dimensional cokernel. On the other hand, it is known that the unique generalized solution of a Protter problem may have a strong power-type singularity even for some very smooth right-hand side functions. This singularity is isolated at the vertex O of the boundary light cone and does not propagate along the characteristic cone. In the general case of smooth right-hand side function, some necessary and sufficient conditions for the existence of a bounded solution are given and a priori estimates for the solution are found. The semi-Fredholm solvability of the problem is proved.
Tài liệu tham khảo
S. A. Aldashev, “Darboux-Protter Spectral Problems for One Class of Multidimensional Hyperbolic Equations,” Ukr. Mat. Zh. 55(1), 100–107 (2003) [Ukr. Math. J. 55, 126–135 (2003)].
A. K. Aziz and M. Schneider, “Frankl-Morawetz Problem in ℝ3,” SIAM J. Math. Anal. 10, 913–921 (1979).
Ar. B. Bazarbekov and Ak. B. Bazarbekov, “The Goursat and Darboux Problems for the Three-Dimensional Wave Equation,” Diff. Uravn. 38(5), 660–665 (2002) [Diff. Eqns. 38, 695–701 (2002)].
A. V. Bitsadze, Some Classes of Partial Differential Equations (Nauka, Moscow, 1981; Gordon & Breach Sci. Publ., New York, 1988).
E. T. Copson, “On the Riemann-Green Function,” Arch. Ration. Mech. Anal. 1, 324–348 (1958).
L. Dechevski and N. Popivanov, “Morawetz-Protter 3D Problem for Quasilinear Equations of Elliptic-Hyperbolic Type. Critical and Supercritical Cases,” C. R. Acad. Bulg. Sci. 61(12), 1501–1508 (2008).
D. E. Edmunds and N. I. Popivanov, “A Nonlocal Regularization of Some Over-determined Boundary-Value Problems. I,” SIAM J. Math. Anal. 29(1), 85–105 (1998).
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. I.
P. R. Garabedian, “Partial Differential Equations with More than Two Independent Variables in the Complex Domain,” J. Math. Mech. 9, 241–271 (1960).
M. K. Grammatikopoulos, T. D. Hristov, and N. I. Popivanov, “Singular Solutions to Protter’s Problem for the 3-D Wave Equation Involving Lower Order Terms,” Electron. J. Diff. Eqns., 03 (2003).
L. Hörmander, The Analysis of Linear Partial Differential Operators. III (Springer, Berlin, 1985).
M. N. Jones, Spherical Harmonics and Tensors for Classical Field Theory (Res. Stud. Press, Letchworth, 1985).
Jong Bae Choi and Jong Yeoul Park, “On the Conjugate Darboux-Protter Problems for the Two Dimensional Wave Equations in the Special Case,” J. Korean Math. Soc. 39(5), 681–692 (2002).
Jong Duek Jeon, Kan Cher Khe, Ji Hyun Park, Yong Hee Jeon, and Jong Bae Choi, “Protter’s Conjugate Boundary Value Problems for the Two Dimensional Wave Equation,” J. Korean. Math. Soc. 33(4), 857–863 (1996).
G. D. Karatoprakliev, “Uniqueness of Solutions of Certain Boundary-Value Problems for Equations of Mixed Type and Hyperbolic Equations in Space,” Diff. Uravn. 18(1), 59–63 (1982) [Diff. Eqns. 18, 49–53 (1982)].
S. Kharibegashvili, “On the Solvability of a Spatial Problem of Darboux Type for the Wave Equation,” Georgian Math. J. 2, 385–394 (1995).
K. Ch. Khe, “Nontrivial Solutions of Some Homogeneous Boundary Value Problems for a Many-Dimensional Hyperbolic Euler-Poisson-Darboux Equation in an Unbounded Domain,” Diff. Uravn. 34(1), 133–135 (1998) [Diff. Eqns. 34, 139–142 (1998)].
P. D. Lax and R. S. Phillips, “Local Boundary Conditions for Dissipative Symmetric Linear Differential Operators,” Commun. Pure Appl. Math. 13, 427–455 (1960).
D. Lupo, K. R. Payne, and N. I. Popivanov, “Nonexistence of Nontrivial Solutions for Supercritical Equations of Mixed Elliptic-Hyperbolic Type,” in Contributions to Nonlinear Analysis (Birkhäuser, Basel, 2005), Prog. Nonlinear Diff. Eqns. Appl. 66, pp. 371–390.
C. S. Morawetz, “Mixed Equations and Transonic Flow,” J. Hyperbolic Diff. Eqns. 1(1), 1–26 (2004).
N. Popivanov and T. Popov, “Exact Behavior of Singularities of Protter’s Problem for the 3-D Wave Equation,” in Inclusion Methods for Nonlinear Problems. With Applications in Engineering, Economics and Physics, Ed. by J. Herzberger (Springer, Wien, 2003), Comput. Suppl. 16, pp. 213–236.
N. Popivanov and T. Popov, “Singular Solutions of Protter’s Problem for the 3 + 1-D Wave Equation,” Integral Transforms Spec. Funct. 15(1), 73–91 (2004).
N. Popivanov, T. Popov, and R. Scherer, “Asymptotic Expansions of Singular Solutions for (3 + 1)-D Protter Problems,” J. Math. Anal. Appl. 331(2), 1093–1112 (2007).
N. I. Popivanov and M. Schneider, “The Darboux Problems in R 3 for a class of Degenerated Hyperbolic Equations,” J. Math. Anal. Appl. 175(2), 537–578 (1993).
N. I. Popivanov and M. Schneider, “On M.H. Protter Problems for the Wave Equation in ℝ3,” J. Math. Anal. Appl. 194(1), 50–77 (1995).
M. H. Protter, “New Boundary Value Problems for theWave Equation and Equations of Mixed Type,” J. Ration. Mech. Anal. 3, 435–446 (1954).
H. Skhiri, “On the Topological Boundary of Semi-Fredholm Operators,” Proc. Am. Math. Soc. 126, 1381–1389 (1998).
W. I. Smirnow, Lehrbuch der höheren Mathematik (Harri Deutsch, Frankfurt/Main, 1995), Teil III, 2.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, Princeton, NJ, 1970).
Tong Kwang-Chang., “On a Boundary-Value Problem for the Wave Equation,” Sci. Rec., New Ser. 1, 277–278 (1957).