Communications in Mathematical Physics - Volume 211 by A. Jaffe (Chief Editor)

By A. Jaffe (Chief Editor)

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Global Weak Solutions for Shallow Water Equation 47 2. Proof of the Theorem Before proceeding with the proof, let us first present some lemmas that will be of use in our approach. Lemma 1 ([6,7,11]). Let u0 ∈ H 3 (R) and assume that y0 := u0 − u0,xx is nonnegative and belongs to L1 (R). 1) has a unique solution u ∈ C(R+ ; H 3 (R))∩C 1 (R+ ; H 2 (R)). Moreover, E(u) := R (u2 +u2x ) dx and F (u) := 3 2 R (u +uux ) dx are conservation laws and if y(t, ·) := u(t, ·)−uxx (t, ·), then for every t ≥ 0 we have (a) u(t, ·) − uxx (t, ·) ≥ 0 and |ux (t, ·)| ≤ u(t, ·) on R, (a) u(t, ·) L1 (R) = y(t, ·) L1 (R) = y0 L1 (R) , (a) ux (t, ·) L∞ (R) ≤ y0 L1 (R) and u(t, ·) H 1 (R) = u0 H 1 (R) .

E. the solution remains bounded while its slope becomes unbounded in finite time [5]. 1). 1) will blow up in finite time, [5]. Let us now recall a partial integration result for Bochner spaces (below ·, · is the H −1 (R), H 1 (R) duality bracket). Lemma 2 ([17]). Let T > 0. e. equal to a function continuous from [0, T ] into L2 (R) and f (t), g(t) − f (s), g(s) = for all s, t ∈ [0, T ]. t s d f (τ ) , g(τ ) dτ + dτ t s d g(τ ) , f (τ ) dτ dτ 48 A. Constantin, L. Molinet Throughout this paper, we will denote by {ρn }n≥1 the mollifiers ρn (x) := ρ(ξ ) dξ R −1 n ρ(nx), x ∈ R, n ≥ 1, where ρ ∈ Cc∞ (R) is defined by e1/(x 0 ρ(x) := 2 −1) for |x| < 1, for |x| ≥ 1.

Let θ1 , θ2 ∈ Sect(M). 5 of [L3], dθ1 +θ2 = dθ1 + dθ2 , and by Cor. 2 of [L6], dθ1 θ2 = dθ1 dθ2 . These two properties are usually referred to as the additivity and multiplicativity of statistical dimensions. Also note by Prop. 12 of [L4] dθ = dθ¯ . If a sector does not have finite statistical dimension in any of the above three equations, then the equation is understood as the statement that both sides of the equation are ∞. Assume λ, µ, and ν ∈ End(M) have finite statistical dimensions. e. a ∈ Hom(λ, µ) iff aλ(p) = µ(p)a for any p ∈ M.

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