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Global well-posedness and singularity propagation for the BBM-BBM system on a quarter plane. (English) Zbl 1382.35207

Summary: Nonlinear, dispersive wave equations arise as models of various physical phenomena. A major preoccupation on the mathematical side of the study of such equations has been to settle the fundamental issues of local and global well-posedness in Hadamard’s classical sense. The development so far has been mostly for the initial-value problem for single equations.
However, systems of such equations have also received consideration, and there is now theory for pure initial-value problems where data are given on the entire space or on the torus. Here, consideration is given to non-homogeneous initial-boundary-value problems for a class of BBM-type systems having the form \[ u_t + u_x -u_{xxt} + P(u,v)_x \, = \, 0, \]
\[ v_t + v_x - v_{xxt} + Q(u,v)_x \, = \, 0, \] where \(P\) and \(Q\) are homogeneous, quadratic polynomials, \(u\) and \(v\) are real-valued functions of a spatial variable \(x\) and a temporal variable \(t\), and subscripts connote partial differentiation. Local in time well-posedness is established in the quarter plane \(\{(x,t): x \geq 0, \, t \geq 0 \}\). Under certain restrictions on the coefficients of the nonlinearities \(P\) and \(Q\), global well posedness is also shown to obtain.

MSC:

35Q35 PDEs in connection with fluid mechanics
35M33 Initial-boundary value problems for mixed-type systems of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35G61 Initial-boundary value problems for systems of nonlinear higher-order PDEs
45G15 Systems of nonlinear integral equations
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B25 Solitary waves for incompressible inviscid fluids
76B55 Internal waves for incompressible inviscid fluids
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Full Text: Euclid