On the Blow-up for a Discrete Boltzmann Equation in the Plane

We study the possibility of finite-time blow-up for a two dimensional Broadwell model. In a set of rescaled variables, we prove that no self-similar blow-up solution exists, and derive some a priori bounds on the blow-up rate. In the final section, a possible blow-up scenario is discussed. 1 Introduction Consider the simplified model of a gas whose particles can have only finitely many speeds, say c1, . . . , cN ∈ IR . Call ui = ui(t, x) the density of particles with speed ci. The evolution of these densities can then be described by a semilinear system of the form ∂tui + ci · ∇ui = ∑ j,k aijk ujuk i = 1, . . . ,N. (1.1) Here the coefficient aijk measures the rate at which new i-particles are created, as a result of collisions between jand k-particles. In a realistic model, these coefficients must satisfy a set of identities, accounting for the conservation of mass, momentum and energy. Given a continuous, bounded initial data ui(0, x) = ūi(x), (1.2) on a small time interval t ∈ [0, T ] a solution of the Cauchy problem can be constructed by the method of characteristics. Indeed, since the system is semilinear, this solution is obtained as the 1 fixed point of the integral transformation ui(t, x) = ūi(x− cit) + ∫ t 0 ∑ j,k aijk ujuk ( s, x− ci(t− s) ) ds . (1.3) For sufficiently small time intervals, the existence of a unique fixed point follows from the contraction mapping principle, without any assumption on the constants aijk. If the initial data is suitably small, the solution remains uniformly bounded for all times [3]. For large initial data, on the other hand, the global existence and stability of solutions is known only in the one-dimensional case [2, 6, 10]. Since the right hand side has quadratic growth, it might happen that the solution blows up in finite time. Examples where the L norm of the solution becomes arbitrarily large as t → ∞ are easy to construct [7]. In the present paper we focus on the two-dimensional Broadwell model and examine the possibility that blow-up actually occurs in finite time. Since the equations (1.1) admit a natural symmetry group, one can perform an asymptotic rescaling of variables and ask whether there is a blow-up solution which, in the rescaled variables, converges to a steady state. This technique has been widely used to study blow-up singularities of reaction-diffusion equations with superlinear forcing terms [4, 5]. See also [9] for an example of self-similar blow-up for hyperbolic conservation laws. Our results show, however, that for the two-dimensional Broadwell model no such self-similar blow-up solution exists. If blow-up occurs at a time T , our results imply that for times t → T− one has