Source-Solutions for the Multi-dimensional Burgers Equation
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Source-Solutions for the Multi-dimensional Burgers Equation Denis Serre Communicated by I. Fonseca
Abstract We have shown in a recent collaboration that the Cauchy problem for the multidimensional Burgers equation is well-posed when the initial data u(0) is taken in the Lebesgue space L 1 (Rn ), and more generally in L p (Rn ). We investigate here the situation where u(0) is a bounded measure instead, focusing on the case n = 2. This is motivated by the description of the asymptotic behaviour of solutions with integrable data, as t → +∞. Notations We denote as · p the norm in Lebesgue L p (Rn ). The space of bounded measure over Rm is M (Rm ) and its norm is denoted ·M . The Dirac mass at X ∈ Rn is δ X or δx=X . If ν ∈ M (Rm ) and μ ∈ M (Rq ), then ν ⊗μ is the measure over Rm+q uniquely defined by ν ⊗ μ, ψ = ν, f μ, g whenever ψ(x, y) ≡ f (x)g(y). The closed halves of the real line are denoted R+ and R− .
1. Introduction The title of the present manuscript refers to the seminal paper [8] by Tai-Ping Liu and Michel Pierre, where the authors studied a general one-dimensional conservation law ∂t u + ∂x f (u) = 0, x ∈ R, t > 0,
(1)
when the initial data a = u(0) is a bounded measure instead of a bounded or an integrable function. Here we continue this exploration, though in a multi-dimensional context, with an equation ∂t u + divx f (u) := ∂t u + ∂1 f 1 (u) + · · · + ∂n f n (u) = 0.
(2)
D. Serre
As in [8] we are interested, for natural reasons, in data a whose total mass is finite. Whenever (2) is not linear, it is expected that u is damped out because of dispersion. The behaviour of f (s) at s = 0 is thus of great importance and we make the generic assumption (non-degeneracy) that f (0), . . . , f (n+1) (0) are linearly independent. Up to a change of coordinates, this amounts so saying that these vectors are parallel to those of the canonical basis. The paradigm of such conservation laws is therefore the (multi-dimensional) Burgers equation ∂t u + ∂1
u2 u n+1 + · · · + ∂n = 0. 2 n+1
(3)
In collaboration with Silvestre [12], we recently proved that the Cauchy problem is well-posed in every L p (Rn ), the solution of (3) being instantaneously damped out as an L ∞ -function. Following Liu and Pierre, we are concerned with the Cauchy problem for (3) when the data a is a bounded measure, and more precisely when a ∈ M \L 1 (Rn ). We show below that if n 2, this problem is not well-posed in the sense of Hadamard; in particular, the Cauchy problem behaves badly at the datum a = δ0 , see Corollary 5.1. This negative result seems to be caused by the extreme lack of regularity of the Dirac data. When the data instead display some mild regularity in n − 1 directions, we prove on the contrary the following compactness result. For the sake of simplicity, we focus here on the case n = 2. Let M, M < ∞ and a unit vector ξ ∈ S 1 be given. Let K ⊂ L 1 (R2 ) be the set of functions a such that a1 M and ξ · ∇a1 M . Then the set K of solutions u associated with data a ∈ K is relatively compact in (L 1
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