Existence and Uniqueness for Boussinesq Type Equations on a Circle

We establish local and global existence results for Boussinesq type equations on a circle, employing Fourier series and a fixed point argument. 0. Introduction and Main Results. In the present work, we want to consider the question of existence and uniqueness of solutions for Boussinesq type equations (0.1) utt − uxx + uxxxx + ∂ xf(u) = 0, x ∈ T, t ∈ R, where T is the unit circle and f(u) is a polynomial of u and |u|, under minimal regularity assumptions on the initial data prescribed at time t = 0, (0.2) u(0, x) = u0(x), ut(0, x) = u1(x). Equations of this type, but with the opposite sign in the fourth derivative, were originally derived by Boussinesq [Bo] in the context of water waves. Zakharov [Z] proposed equation (0.1) as a model of a nonlinear string. Falk et al derived an equation which is equivalent to (0.1) in their study of shapememory alloys, see [FLS]. In fact, the equation studied in [FLS] is of the following type (0.3) ett − gexx + exxxx + ∂ xf(e) = 0, 1The first author was partially supported by Sloan Fellowship. 2The second author is supported by a PYI, DMS-9157512, and a Sloan Fellowship. Typeset by AMS-TEX 1 2 Y. FANG & M.G. GRILLAKIS where g is a constant, e = ux is the strain and f(e) = 4e − 6e. In general however f(e) contains a term of the form exp(γe). McKean studied the complete integrability of the good Boussinesq equation on a circle, see [M]. An interesting observation connecting the Kadomstev-Petviashvili equation with the Boussinesq equation is the following. For the KP equation (0.4) (ut + uxxx + uux)x + uyy = 0, consider waves that move in the x direction, i.e. u(t, x, y) = v(x− ct, y) and denote ξ = x− ct, thus the KP equation is reduced to (0.5) vyy − cvξξ + vξξξξ + ∂ ξ (u/2) = 0, which is (0.1) with f(u) = u/2, and the time variable is now played by the y direction, see [HP]. Equation (0.1) has certain features that are interesting, the linear equation (0.6) utt − uxx + uxxxx = 0 has solutions that are periodic in space but only aperiodic in time. By this we mean that the function is a linear combination of functions with different non integer periods. Also in contrast to the equation on the real line, i.e. x ∈ R, there is no dispersion and no decay in the time variable. On the other hand, equation (0.1) can be written as a Hamiltonian system as follows (0.7) { ut = vx, vt = ux − uxxx − ∂xf(u). The above equation conserves the energy, namely the integral