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4, 283318. 6. F. Chazel, Influence of topography on water waves, M2AN. [To appear] 7. D. Coutand and S. AP/0511236. 8. W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Commun. Partial Differ. Equations 10 (1985), no. 8, 787-1003. 9. W. Craig, Nonstrictly hyperbolic nonlinear systems, Math. Ann. 277 (1987), no. 2, 213-232. 10. Craig, U. Schanz, and C. Sulem, The modulational regime of threedimensional water waves and the Davey–Stewartson system, Ann.

C 1 (Ω) ˜ and, respectively, h ∈ C01 (Ω) can be arbitrary. 12), h 0 ˜ (Au)t + divx Bu = 0 in D′ (Ω) if and only if u˜τ = 0 in D′ (Ω). , u(t, x) is constant along each characteristic t = t(τ ; y), x = x(τ ; y). , the function p ◦ u is also a generalized solution for any p (z) ∈ C(R). 4. 1. Existence and nonuniqueness in multi-dimensional case. We prove the existence of a generalized solution in the case of an arbitrary dimension n by using the regularization of coefficients. 34 Evgenii Panov Theorem 1.

26 Evgenii Panov Definition 1. , there exists a set E ⊂ (0, +∞) of full Lebesgue measure such that for t ∈ E u(t, ·) ∈ L1loc (R, Rn ), and u(t, ·) → u0 in L1loc (R, Rn ) as t → 0, t ∈ E. The notion of a strong generalized entropy solution coincides with that of a renormalized solution introduced in [3]. 1) (these entropies are completely described in [15]). 2). 4). Note that the flux function f (r) is here only continuous, and the unconditional uniqueness of a generalized entropy solution is valid only in the case of one spatial variable (see [11, 12, 16] for details).

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