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.


-Introduction
Consider the simplified model of a gas whose particles can have only finitely many speeds, say c 1 , . . . , c N ∈ IR n . Call u i = u i (t, x) the density of particles with speed c i . The evolution of these densities can then be described by a semilinear system of the form Here the coefficient a ijk measures the rate at which new i-particles are created, as a result of collisions between j-and 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 For sufficiently small time intervals, the existence of a unique fixed point follows from the contraction mapping principle, without any assumption on the constants a ijk . 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 This means that the blow-up rate must be different from the natural growth rate u(t) L ∞ = O(1) · (T − t) −1 which would be obtained in case of a quadratic equationu = C u 2 .
In the final section of this paper we discuss a possible scenario for blow-up. The analysis highlights how carefully chosen should be the initial data, if blow-up is ever to happen. This suggests that finite time blow-up is a highly non-generic phenomenon, something one would not expect to encounter in numerical simulations.

-Coordinate rescaling
In the following, we say that We say that (t * , x * ) is a primary blow-up point if it is a blow-up point and the backward cone does not contain any other blow-up point. Proof. If u is not continuous, it must be unbounded in the neighborhood of some point. Hence some blow-up point exists. Call B the set of such blow-up points. Define the function By Ekeland's variational principle (see [1], p.254), there exists a point x * such that for all x ∈ IR 2 . Then P * . = ϕ(x * ), x * is a primary blow-up point. Let now (t * , x * ) be a primary blow-up point. One way to study the local asymptotic behavior of u is to rewrite the system in terms of the rescaled variables w i = w i (τ, η), defined by (2.1) The corresponding system of evolution equations is Any nontrivial stationary or periodic solution w of (2.2) would yield a solution u of (1.1) which blows up at (t * , x * ). On the other hand, the non-existence of such solutions for (2.2) would suggest that finite time blow-up for (1.1) is unlikely.

-The two-dimensional Broadwell model
Consider a system on IR 2 consisting of 4 types particles ( fig. 1), with speeds The evolution equations are After renaming variables, the corresponding rescaled system (2.2) takes the form Our first result rules out the possibility of asymptotically self-similar blow-up solutions. A sharper estimate will be proved later.
Theorem 1. The system (3.2) admits no nontrivial positive bounded solution which is constant or periodic in time.
Restricted to any horizontal moving line y = y(t) such thatẏ = y + 1 ( fig. 2), the equations Call The definition of ε and the bound (3.3) on w 1 imply From this, and a similar estimate for w 4 , we obtain Since ε < κ −1 , this yields d dt Observing that the Cauchy probleṁ has the solution , by a comparison argument from (3.4) we deduce and since a similar estimate can be performed for all components w i , we conclude The right hand side of (3.5) approaches zero as t → ∞. Therefore, nontrivial constant or timeperiodic L ∞ solutions of (3.2) cannot exist.

-Refined blow-up estimates
If (t * , x * ) is a blow-up point, our analysis has shown that in the rescaled coordinates τ, ξ the corresponding functions w i must become unbounded as τ → ∞. In this section we refine the previous result, establishing a lower bound for the rate at which such explosion takes place.
Since w i = (t * − t)u i and τ . = ln(t * − t) , the above implies Proof of Theorem 2.
Let w i = w i (t, x, y) provide a solution to the system (3.2), with for all t ≥ t 0 and x, y ∈ [−1, 1]. The proof will be given in two steps. First we show that the L 1 norm of the components w i approaches zero as t → ∞. Then we refine the estimates, and prove that also the L ∞ norm asymptotically vanishes.
STEP 1: Integral estimates. Consider the function with k(t) as in (4.3). As in the proof of Theorem 1, let t → y(t) be a solution toẏ = y + 1. Then To estimate the right hand side, we notice that Therefore, If k(t) ≥ 1/2, we claim that the following two inequalities hold: = e 1/(2θ) one has k(t) ≥ 1 2 and hence d dt Using the above, and a similar estimate for the integral of w 4 , we obtain (4.5) Calling Recalling that k(t) = θ ln t for some 0 < θ < 1/4, the previous differential inequality can be written as Notice that Q 14 t 0 , y(t 0 ) ≤ 2k(t 0 ), and define the constant Then the function z(t) .
From our previous estimates (4.9) it trivially follows The time derivative of A(t) is computed as x(t), y(t) = −2w 4 t, x(t), y(t) .

-A tentative blow-up scenario
For a solution of the rescaled equation (3.1), the total mass x, y) dxdy may well become unbounded as t → ∞. On the other hand, the one-dimensional integrals along horizontal or vertical segments decrease monotonically. Namely, if t → y(t) satisfiesẏ = y + 1, then d dt x, y(t)) + w 4 (t, x, y(t)) dx ≤ 0 .
Similarly, ifẋ = x − 1, then Analogous estimates hold for the sums w 1 + w 2 and w 2 + w 3 . Therefore, a bound on the initial data yields uniform integral bounds on the line integrals of all components: If finite time blow-up is to occur, the mass which is initially distributed along each horizontal or vertical segment must concentrate itself within a very small region, thus forming a narrow packet of particles with increasingly high density. A possible scenario is illustrated in fig. 3. A packet of 1-particles is initially located at P 1 . In order to contribute to blow-up, this packet must remain within the unit square Q. At P 2 these 1-particles interact with 3-particles and produce a packet of 4-particles. In turn, at P 3 these interact with 2-particles and produce again a packet of 1-particles. After repeated interactions, the packet of alternatively 1-and 4-particles eventually enters within the smaller square Q ′ . After this time, it interacts with a packet of 2-particles at P 5 (transforming it into a packet of 1-particles) and eventually exits from the domain Q.
To help intuition, it is convenient to describe a packet as being "young" until it enters the smaller square Q ′ , and "old" afterwards. To maintain a young packet inside Q, one needs the presence of old packets interacting with it near the points P 2 , P 3 , P 4 . . . On the other hand, after it enters Q ′ , our packet can in turn be used to hit another young packet, say at P 5 , and preventing it from leaving the domain Q.
As t → ∞, the density of the packets must approach infinity. One thus expects that most of the mass will be concentrated along a finite number of one-dimensional curves. Say, the packet of alternatively 1-and 4-particles should be located along a moving curve γ 14 (t, θ), where θ is a parameter along the curve. The time evolution of such a curve is of course governed by the equations depending on whether γ 14 (t, θ) consists of 1-or 4-particles. The presence of interactions impose highly nonlinear constraints on these curves. For example, the interaction occurring in P 5 at time t implies the crossing of the two curves γ 14 and γ 12 , namely γ 14 (t, θ) = γ 12 (t,θ) = P 5 for some parameter values θ,θ. The complicated geometry of these curves resulting from the above constraints has not been analyzed.