CUBO A Mathematical Journal Vol.16, No¯ 01, (01–07). March 2014 Stationary Boltzmann equation and the nonlinear alternative of Leray-Schauder type Rafael Galeano Andrades, Pedro Ortega Palencia and John Fredys Cantillo Palacio Institute of Applied Mathematics, Universidad de Cartagena, Cartagena, Colombia. rgaleanoa@unicartagena.edu.co, portegap@unicartagena.edu.co, jcantillop@unicartagena.edu.co ABSTRACT By applying a nonlinear alternative of Leray-Schauder type, a fixed point of an operator is found, which, in turn, comes to be a solution of stationary Boltzmann equation with boundary conditions of Maxwellian type. RESUMEN Aplicando una alternativa no lineal del tipo Leray-Schauder, se encontró un punto fijo de un operador, el cual corresponde a la solución de una ecuación de Boltzmann estacionaria con condiciones de frontera del tipo Maxwelliano. Keywords and Phrases: Nonlinear alternative of Leray-Schauder type, Fixed point, Solution of stationary Boltzmann equation. 2010 AMS Mathematics Subject Classification: 35Q20. 2 R. Galeano, P. Ortega & J. Cantillo CUBO 16, 1 (2014) 1 Introduction Let us consider the Banach space E = { u ∈ L1(Ω × B3R(0)) : vi ∂u ∂xi ∈ L1(Ω × B3R(0)) } with norm ‖u‖ E := ‖u‖ L1(Ω×B 3R (0)) + ∥ ∥ ∥ ∥ vi ∂u ∂xi ∥ ∥ ∥ ∥ L1(Ω×B 3R (0)) and we expect to find u(x, v) ≥ 0, such that { v.∇xu = Q(u, u), u ∈ BE(0, R) u(x, v) = M(v) = e−|v| 2 (Maxwellian), u ∈ ∂B E (0, R), (R > 0) (1.1) Here Q(u, u)(v) = ∫ B 3R (0) ∫ |p|=1 [p · (v − z)]p[u(x, z′)u(x, v′) − u(x, z)u(x, v)]dpdz, is the collision operator, Ω bounded and regular and the speeds related by the following relations: { v′ = v − [p · (v − z)]p z′ = z + [p · (v − z)]p (1.2) (z, v) and ( z′, v′) son the pre-collision and post-collision speeds, respectively. It can be noted that if z, v ∈ B R (0) in Rn, then z′, v′ ∈ B 3R (0). Here, the following problem will be proved. Theorem 1.1. Let us suppose that Q(u, u) ∈ B E (u, R/2); vi ∂u ∂xi ∈ B E (0, R/2n) y vi ∂un ∂xi ∈ B E ( vi ∂u ∂xi , R∗∗ ) , R∗∗ ≥ 0, for n = 1, 2, . . . , moreover 0 < ∫ B 3R (0) ∫ |p|=1 ‖v − z‖dpdz < ∞ and 0 < ∫ B 3R (0) ∫ |p|=1 ‖v − z‖dpdv < ∞, then there exits a solution for u ∈ B E (0, R) of problem (1.1). In these stationary problems, the flows of quantities as entropy control and compactness prop- erties are under control, but they do not imply, per se, the desired results. Anyway, energy control and similar properties are available from momentum flows and mass control that can be imposed on the problem to replace entropy-bounding non-availability. There are controls based on involution of entropy dissipation.. Using such tools, in the last years Arkeryd L. and Noury A., [2] -[3] -[4]- [5], have made a de- velopment focused on the results of solutions existence in the L1 context for nonlinear equations Boltzmann type and also for those presenting maxwellian equilibrium. The case of perturbation on the global maxwellian equilibrium has been typically studied since the 60’s. Methods of general type as Hilbert spaces and contraction mapping have been used, being the pioneers [6] -[7] -[9] -[10]; in [11] exposed, generally discussed the problem of boundary value for the stationary equation, in [12] and [14] is proved the theorem for the stationary equation Povzner with certain spatial boundary conditions of type Maxwellian and in [13] there are applications to dynamics of fluids, we present the main result five lemmas. CUBO 16, 1 (2014) Stationary Boltzmann equation and the nonlinear alternative . . . 3 2 Development The problem (1.1), v.∇xu = Q(u, u) is equivalent to u + v.∇xu = u + Q(u, u), this implies that u = u + v.∇xu − Q(u, u) which suggests the following operator: J(u) := u + v.∇xu − Q(u, u) Defined on E = { u ∈ L1(Ω × B3R(0)) : vi ∂u ∂xi ∈ L1(Ω × B3R(0)) } , E is a Banach space with the norm ‖u‖ E := ‖u‖ L1(Ω×B 3R (0)) + ∥ ∥ ∥ ∥ vi ∂u ∂xi ∥ ∥ ∥ ∥ L1(Ω×B 3R (0)) Finding fixed points of J, coincides with finding solutions of (1.1). So we will work to find fixed points, via alternative Leray-Schauder type, in effect: Let C = B E (0, R), this is a convex and closed set in E and U := B E (0, R), open ball centered in 0 and radius R. Lemma 2.1. Let us suppose that Q(u, u) ∈ B E (u, R/2) and vi ∂u ∂xi ∈ B E (0, R/2n), u = 1, 2, . . . , then J sends C in C. Proof. Let u ∈ C = B E (0, R), as J(u) := u + v.∇xu − Q(u, u), then |J(u)| ≤ |u − Q(u, u)| + |v.∇xu| = |u − Q(u, u)| + n∑ i=1 ∣ ∣ ∣ ∣ vi ∂u ∂xi ∣ ∣ ∣ ∣ , i.e, ‖J(u)‖ L1(Ω×B 3R (0)) ≤ ‖u − Q(u, u)‖ L1(Ω×B 3R (0)) + n∑ i=1 ∥ ∥ ∥ ∥ vi ∂u ∂xi ∥ ∥ ∥ ∥ L1(Ω×B 3R (0)) (2.1) now: ∂J(u) ∂xi = ∂u ∂xi + ∂ ∂xi [ n∑ i=1 vi ∂u ∂xi ] + ∂Q(u, u) ∂xi , then ∣ ∣ ∣ ∣ ∂J(u) ∂xi ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ ∂ ∂xi ( u − Q(u, u) ) ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ ∂ ∂xi n∑ i=1 vi ∂u ∂xi ∣ ∣ ∣ ∣ ∣ , i.e., ∥ ∥ ∥ ∥ ∂J(u) ∂xi ∥ ∥ ∥ ∥ L1(Ω×B 3R (0)) ≤ ∥ ∥ ∥ ∥ ∂ ∂xi ( u − Q(u, u) ) ∥ ∥ ∥ ∥ L1(Ω×B 3R (0)) + n∑ i=1 ∥ ∥ ∥ ∥ ∂ ∂xi ( vi ∂u ∂xi ) ∥ ∥ ∥ ∥ L1(Ω×B 3R (0)) . (2.2) from (2.1) and (2.2) we conclude that: ‖J(u)‖ E ≤ ‖u − Q(u, u)‖ E + n∑ i=1 ∥ ∥ ∥ ∥ vi ∂u ∂xi ∥ ∥ ∥ ∥ E ≤ R 2 + R 2 = R, that is to say, J(u) ∈ C = B E (0, R). 4 R. Galeano, P. Ortega & J. Cantillo CUBO 16, 1 (2014) Lemma 2.2. If un ∈ BE(u, R), such that |un(x, v)| ≤ R, |u(x, v)| ≤ R, for every n, x ∈ Ω, v ∈ B3R(0), moreover 0 < ∫ B 3R ∫ |p|=1 ‖(v − z)‖dpdz < ∞ y 0 < ∫ B 3R ∫ |p|=1 ‖(v − z)‖dpdv < ∞, then Q(un, un) ∈ BE(Q(u, u), r ∗R) with r∗ = 4R max { ∫ B 3R ∫ |p|=1 ‖(v − z)‖dpdz, ∫ B 3R ∫ |p|=1 ‖(v − z)‖dpdv } . (r∗ ≥ 0) Proof. By definition of Q(u, u), leads to ∣ ∣ ∣ Q(un, un)(v)−Q(u, u)(v) ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∫ B 3R (0) ∫ |p|=1 [p · (v − z)]p[un(x, z ′)un(x, v ′) − un(x, z)un(x, v)]dpdz − ∫ B 3R (0) ∫ |p|=1 [p · (v − z)]p[u(x, z′)u(x, v′) − u(x, z)u(x, v)]dpdz ∣ ∣ ∣ ∣ , That is to say: ∣ ∣ ∣ Q(un, un)(v)−Q(u, u)(v) ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∫ B 3R (0) ∫ |p|=1 [p · (v − z)]p [ un(x, z ′)un(x, v ′) − u(x, z′)u(x, v′) − un(x, z)un(x, v) + u(x, z)u(x, v) ] dpdz ∣ ∣ ∣ ∣ ≤ ∫ B 3R (0) ∫ |p|=1 |p · (v − z)| ∣ ∣ ∣ un(x, z ′)un(x, v ′) − un(x, z ′)u(x, v′) + un(x, z ′)u(x, v′) − u(x, z′)u(x, v′) + u(x, z)u(x, v) − u(x, z)un(x, v) + u(x, z)un(x, v) − un(x, z)un(x, v) ∣ ∣ ∣ dpdz then ∣ ∣ ∣ Q(un, un)(v)−Q(u, u)(v) ∣ ∣ ∣ ≤ ∫ B 3R (0) ∫ |p|=1 |p · (v − z)| [ ∣ ∣un(x, z ′) ∣ ∣ ∣ ∣un(x, v ′) − u(x, v′) ∣ ∣ + ∣ ∣u(x, v′) ∣ ∣ ∣ ∣un(x, z ′ ) − u(x, z′) ∣ ∣ + ∣ ∣u(x, z) ∣ ∣ ∣ ∣un(x, v) − u(x, v) ∣ ∣ + ∣ ∣un(x, v) ∣ ∣ ∣ ∣un(x, z) − u(x, z) ∣ ∣ ] dpdz CUBO 16, 1 (2014) Stationary Boltzmann equation and the nonlinear alternative . . . 5 so, ∥ ∥ ∥ Q(un, un) − Q(u, u) ∥ ∥ ∥ L1(Ω×B 3R (0)) ≤ ∫ Ω ∫ B 3R (0) ∣ ∣ ∣ Q(un, un)(v) − Q(u, u)(v) ∣ ∣ ∣ dxdv ≤ ∫ Ω ∫ B 3R (0) ∫ B 3R (0) ∫ |p|=1 |p · (v − z)| |un(x, z ′)| |un(x, v ′) − u(x, v′)|dpdzdvdx + ∫ Ω ∫ B 3R (0) ∫ B 3R (0) ∫ |p|=1 ‖v − z‖ |u(x, v′)| |un(x, z ′) − u(x, z′)|dpdzdvdx + ∫ Ω ∫ B 3R (0) ∫ B 3R (0) ∫ |p|=1 ‖v − z‖ |u(x, z)| |un(x, v) − u(x, v)|dpdzdvdx + ∫ Ω ∫ B 3R (0) ∫ B 3R (0) ∫ |p|=1 ‖v − z‖ |un(x, z) − u(x, z)|dpdzdvdx ≤ ∫ B 3R (0) ∫ |p|=1 R‖v − z‖dpdz ∫ Ω ∫ B 3R (0) |un(x, v ′) − u(x, v′)|dvdx + ∫ B 3R (0) ∫ |p|=1 R‖v − z‖dpdv ∫ Ω ∫ B 3R (0) |un(x, z ′) − u(x, z′)|dzdx + ∫ B 3R (0) ∫ |p|=1 R‖v − z‖dpdz ∫ Ω ∫ B 3R (0) |un(x, v) − u(x, v)|dvdx + ∫ B 3R (0) ∫ |p|=1 R‖v − z‖dpdv ∫ Ω ∫ B 3R (0) |un(x, z) − u(x, z)|dzdx Making the change of variables v′ → v y z′ → z, whose Jacobians are 1, then: ∥ ∥ ∥ Q(un, un) − Q(u, u) ∥ ∥ ∥ L1(Ω×B 3R (0)) ≤ 2R [ ∫ B 3R (0) ∫ |p|=1 ‖v − z‖dpdz + ∫ B 3R (0) ∫ |p|=1 ‖v − z‖dpdv ] ‖un − u‖ L1(Ω×B 3R (0)) ≤ 2R [ ∫ B 3R (0) ∫ |p|=1 ‖v − z‖dpdz + ∫ B 3R (0) ∫ |p|=1 ‖v − z‖dpdv ] ‖un − u‖E ≤ 4R max { ∫ B 3R ∫ |p|=1 ‖(v − z)‖dpdz, ∫ B 3R ∫ |p|=1 ‖(v − z)‖dpdv } ‖un − u‖E. Hence Q(un, un) ∈ BE(Q(u, u), r ∗R). Lemma 2.3. If un ∈ BE(0, R) ∩ BE(u, R), u ∈ BE(0, R), and vi ∂un ∂xi ∈ B E ( vi ∂u ∂xi , R∗∗ ) , with r∗∗ = r∗R + R + nR∗∗ ≥ 0 such that 0 < ∫ B 3R ∫ |p|=1 ‖(v − z)‖dpdz < ∞ and 0 < ∫ B 3R ∫ |p|=1 ‖(v − z)‖dpdv < ∞, then J(un) ∈ BE(J(u), r ∗∗) 6 R. Galeano, P. Ortega & J. Cantillo CUBO 16, 1 (2014) Proof. |J(un) − J(u)| = |un + v.∇xun − Q(un, un) − u − v.∇xu + Q(u, u)| ≤ |un − u| + |Q(un, un) − Q(u, u)| + |v.∇xun − v.∇xu|, luego: ∥ ∥ ∥ J(un) − J(u) ∥ ∥ ∥ L1(Ω×B 3R (0)) ≤ ∥ ∥ ∥ un − u ∥ ∥ ∥ L1(Ω×B 3R (0)) + ∥ ∥ ∥ Q(un, un) − Q(u, u) ∥ ∥ ∥ L1(Ω×B 3R (0)) + ∥ ∥ ∥ ∥ ∥ n∑ i=1 vi ∂ ∂xi (un − u) ∥ ∥ ∥ ∥ ∥ L1(Ω×B 3R (0)) (2.3) now calculating ∂J(u) ∂xi , we obtain that: ‖J(un) − J(u)‖E ≤ ‖un − u‖E + ‖Q(un, un) − Q(u, u)‖E + n∑ i=1 ∥ ∥ ∥ ∥ vi ∂ ∂xi (un − u) ∥ ∥ ∥ ∥ E ≤ r∗R + R + nR∗∗ = r∗∗ therefore J(un) ∈ BE(J(u), r ∗∗). Lemma 2.4. The operator J : U −→ C is compact. Dunford-Pettis Criterion will be applied, see [8], in fact: i) ∫ Ω ∣ ∣ ∣ J(u) ∣ ∣ ∣ du ≤ ∫ Ω 2R du = 2R m(Ω) ≤ 2Rδ, defining ε = 2Rδ, then the existence of δ, such that if m(Ω) ≤ δ, then ∫ Ω ∣ ∣ ∣ J(u) ∣ ∣ ∣ du ≤ ε. ii) Given ε∗ > 0, exists a closed, F ⊂ Ω such that if m(F) < ∞, then ∫ Ω−F ∣ ∣ ∣ J(u) ∣ ∣ ∣ du ≤ 2R m(Ω − F) ≤ 2Rε∗, if we defined 2Rε∗ ≤ ε, then ∫ Ω−F ∣ ∣ ∣ J(u) ∣ ∣ ∣ du ≤ ε. Lemma 2.5. For every u ∈ ∂U you have u = J(u). In fact, if u ∈ ∂U, then u = e−|v| 2 , and: J(u) = u + v.∇xu − Q(u, u) = u = e −|v| 2 then for every u ∈ ∂U y λ ∈ (0, 1) must be u 6= λJ(u). Therefore the nonlinear alternative Leray-Schauder type, see [1], page 48, we conclude that there is a fixed point of the operator J a solution resulting from (1.1). CUBO 16, 1 (2014) Stationary Boltzmann equation and the nonlinear alternative . . . 7 Received: October 2012. Accepted: September 2013. 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