The Carleman Model: Asymptotic Behaviour
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I apply now what I have found about oscillating solutions for the Carleman model to the study of the asymptotic behaviour, as t tends to ∞, of the solution of the system for fixed nonnegative initial data with finite total mass, i.e. ut + ux + u2 − v 2 = 0 in R × (0, ∞); u(·, 0) = a in R (20.1) vt − vx − u2 + v 2 = 0 in R × (0, ∞); v(·, 0) = b in R, with a, b ∈ L∞ (R) ∩ L1 (R), a, b ≥ 0 a.e. in R,
R
(a + b) dx = m < ∞.
(20.2)
Of course, my analysis uses the uniform Illner–Reed bound 0 ≤ u(x, t), v(x, t) ≤
C(m) a.e. in R × (0, ∞), t
(20.3)
but a good estimate of C(m) is not necessary. In order to analyse what is going on for large t, I consider the sequence (un , vn ) defined by un (x, t) = n u(n x, n t), vn (x, t) = n v(n x, n t) in R × (0, ∞),
(20.4)
and because one has 0 ≤ un , vn ≤ C(m) t , the solutions are uniformly bounded for t ≥ ε > 0. The first result relies on the finite speed of propagation. Lemma 20.1. For every ε, η > 0, the sequences un and vn converge to 0 in L∞ weak and Lploc strong for 1 ≤ p < ∞ on the subsets {(x, t) | x ≤ −t − η, t ≥ ε} and {(x, t) | x ≥ t + η, t ≥ ε}. Proof : Because (un + vn )t + (un − vn )x = 0, which expresses conservation of mass, one sees by integrating on the subset {(x, s) | x ≤ x0 − s, 0 ≤ s ≤ t} that
164
20 The Carleman Model: Asymptotic Behaviour
x0 −t
−∞
t
un (x0 −s, s) ds =
(un +vn )(·, t) dx+2 0
and therefore, using un ≥ 0, one has x0 −t (un + vn )(·, t) ≤ −∞
x0
−∞
(un +vn )(·, 0) dx, (20.5)
x0 −∞
(an + bn ) dx,
(20.6)
where an (x) = n a(n x) and bn (x) = n b(n x) on R; similarly, one has
+∞
+∞
(un + vn )(·, t) ≤ x1 +t
(an + bn ) dx,
(20.7)
x1
and this is valid for x0 = −η and x1 = +η, and because all the mass of an and bn concentrates at 0 and eventually enters the interval (−η, +η), one deduces that un + vn converges to 0 in L1loc (Ω) strong, where Ω is either {(x, t) | x ≤ −t − η} or {(x, t) | x ≥ t + η}. Adding the constraint t ≥ ε > 0 permits one to use the uniform L∞ estimate and the bound in L∞ together with the convergence in L1loc strong implies the convergence in Lploc strong for every p ∈ [1, ∞) (by using H¨ older inequality),1 and in L∞ weak . Lemma 20.2. Some subsequence (un , vn ) converges, and any limit (u∗ , v∗ ) of a subsequence is automatically a solution of the Carleman model for t > 0, having support in {(x, t) | −t ≤ x ≤ t}, and having total mass m. Proof : Because of the uniform L∞ bound for t ≥ ε > 0, and using the diagonal argument of CANTOR,2 one may extract a subsequence such that every power of un or vn converges in L∞ weak for any set {(x, t) | t ≥ ε} (and one only needs that un , vn , (un )2 , (vn )2 converge in L∞ weak ). Then one observes that σu = 0 for x < −t (and for x > t) by applying Lemma 20.1, and the inequality (σu )t + (σu )x ≤ 0 implies that σu = 0 for x < t, and therefore σu = 0 almost everywhere; this implies strong convergence of un in L2loc , and therefore strong convergence in Lploc for 1 ≤ p < ∞ because of the uniform bound in L∞ . A similar argument app
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