Articulo 17.dvi CUBO A Mathematical Journal Vol.12, No¯ 02, (275–298). June 2010 The Maxwell problem and the Chapman projection1 V. V. Palin, E. V. Radkevich 2 Department of Mech.-Math., Moscow State University, Moscow 119899, Vorobievy Gory, Russia. email: evrad07@gmail.com ABSTRACT We study the large-time behavior of global smooth solutions to the Cauchy problem for hyperbolic regularization of conservation laws. An attracting manifold of special smooth global solutions is determined by the Chapman projection onto the phase space of consoli- dated variables. For small initial data we construct the Chapman projection and describe its properties in the case of the Cauchy problem for moment approximations of kinetic equations. The existence conditions for the Chapman projection are expressed in terms of the solvability of the Riccati matrix equations with parameter. RESUMEN Nosotros estudiamos el comportamiento temporal de soluciones globales suaves del prob- lema de Cauchy para regularización hiperbólica de leyes de conservación. Una variedad atractora de soluciones globales suaves es determinada por la proyección de Chapman so- bre el espacio de fase de las variables consolidadas. Para datos iniciales pequeños nosotros construimos la proyección de Chapman y descubrimos sus propiedades en el caso del prob- lema de Cauchy para aproximación de momentos en ecuaciones kineticas. Las condiciones de existencia para la proyección de Chapman son expresadas en términos de la solubilidad de las ecuaciones matriciales de Riccati con parámetros. 1This work was supported by the Russian Foundation of Basic Researches (grant no. 09-01-00288) 2Corresponding author 276 V. V. Palin and E. V. Radkevich CUBO 12, 2 (2010) Key words and phrases: closure, the state equation, the Chapman projection, matrix equation, dynamic separation, inertional manifold AMS (MOS) Subj. Class.: UDC 517.9 1 Introduction 1.1 The state equation. Closure This paper is devoted to mathamatical aspects of the Maxwell problem [2] about the derivation of the Navier-Stokes equation from kinetics. Following [1] we study the behavior of solutions to the Cauchy problem for hyperbolic regularizations of conservation laws or (in another terminology) for systems of conservation laws with relaxation. Consider m− conservation laws (1.1) and N − m conservation laws with relaxation (1.2): ∂tui + divx f i(u, v) = 0, i = 1, . . . , m, (1.1) ∂tvk + divx g k(u, v) + bk(u)v = 0, k = m + 1, . . . , N. (1.2) then we have m conservative variables u(x, t) : Rd × R+ → Rm and N − m co-called nonequilibrium variables v(x, t) : Rd × R+ → RN−m, where x ∈ Rd, b is the relaxation (N − m) × (N − m)-matrix, f i(u, v) ∈ Rd, i = 1, . . . , m; gk(u, v) ∈ Rd, k = 1, . . . , N − m, are currents. The leading part of the system (1.1) is nonstrictly hyperbolic in the sense of the following definition. Definition 1.1. A system is nonstrictly hyperbolic if the characteristic matrix τ E + ξ · ( fu(u, v) fv(u, v) gu(u, v) gv(u, v) ) (1.3) has only real (possibly multiple) roots τ = τj (ξ, u, v), j = 1, . . . , N . The condition in Definition 1.1 is satisfied if the system (1.1) is simmetrizable. Examples of such systems are the following: - moment approximations of kinetic equations and the Dirac-Schwinger extension of the Maxwell equations [3]. Hyperbolic regularizations of conservation laws (or systems of conservation laws with relaxation) were considered by many authors. First of all, this concerns the study of the relaxation phenomenon, in particular, the stability and singular limit as the relaxation time tends to zero (cf., for example, [4]-[7]). The so-called ”intermediate attractor” for (1.1-1.2) was studied in connection with the Maxwell problem (cf. [8,9]). To derive equations of hydrodynamics from the kinetic gas theory, it is important to find a simple functional dependence of the transport coefficients on the interaction potential and thereby to simplify the analysis of the equations under consideration. We intereste in the Chapman conjecture [1], [9], about the existence problem of state equation v = Qu, (1.4) (so-called the Chapman state equation or the Chapman projection) expressing the nonequillibrium variables in terms of the conservative variables (the projection into the phase space of conservative CUBO 12, 2 (2010) The Maxwell problem and the Chapman projection 277 variables), where Q is an operator with respect to space variables x. This equation completes the system of conservation laws ∂tw + ∂xf (w, Qu(w)) = 0. (1.5) so that the solutions w to the Caushy problem of the corresponding closer (1.5) define the set of the special solutions UChEns = {u = w, v = Qw} to the Cauchy problem for the system (1.1-1.2) form an invariant attracting manifold MChEns, called an intermediate attractor. In other words, for any solution U = (u, v) to the Cauchy problem for the system (1.1-1.2) with the initial data U|t=0 = (u0, v0) it is possible to choose initial data w0 = T (u0, v0) for the closure (1.5) in such a way that some norm of the difference U − UChEns between U and the special solution UChEns = (w, Qw) tends to zero as t → ∞. Moreover, if, in the phase space of conservative variables, w → 0, when t → ∞, then UH tends to zero faster than UChEns. We can say that in this case the influence of nonequilibrium veriables is inessential(we have the separation of dynamics) [9]. Now we can define the approximation of the state equation and corresponding closure(so-called Navier-Stokes approximation). Due to physical point of view [9] we assume that derivatives of nonequi- librium variables are small, then we find the following relation v = −b−1 divx g(u, 0) (1.6) and the corresponding closure ∂tu + ∂xf (u, −b−1 divx g(u, 0)) = 0 (1.7) (so-called Navier-Stokes approximation to (1.1-1.2)), where det b(u) 6= 0. For thirteen-moment Grad system to the Boltzman kinetic equation the Navier-Stokes approximation (1.7) is the Navier-Stokes equations exactly. Considering conservation laws with stiff relaxation ∂tu + divx f (u, v) = 0, ∂tv + divx g(u, v) + 1 ε b(u)v = 0, (1.8) we find that the Navier-Stokes approximation v = −εb−1 divx g(u, 0), ∂tu + divx f (u, −εb−1 divx g(u, 0)) = 0, (1.9) is the first approch to so-called local equilibrium approch(see [1]) ∂tu + divx f (u, 0) = 0 2 Linear analysis. Reduction to a quadratic matrix equation. 2.1 Reduction to a quadratic matrix equation We consider the Cauchy problem for the first order linear hyperbolic system with constant coefficients and with relaxation [1] ∂tu + Aj ∂xj u + Bu = 0 (2.1) 278 V. V. Palin and E. V. Radkevich CUBO 12, 2 (2010) where x ∈ Rn, u ∈ RN , Aj and B are constant matrices. In the case of the system (2.1), the Chapman conjecture [1,9] of the state equation existence asserts that u = Π uc = (uc, Π21uc), where uc = (u1, .., um, 0, .., 0) T and Π - is a zero order pseudodifferential matrix operator. Suppose Following to [1] that the matrix of operator Π corresponding to the Chapman-Enskog projection into m equations of the system (2.1) has the form Π = ( Π11 Π12 Π21 Π22 ) where Π11 = Em is the unit matrix of order m and Π22 = 0N−m is the zero square matrix of order N − m. We denote by Λ(ξ) the resolvent matrix ∑n j=1 Aiξj + B and represent it in the block form: Λ = ( Λ11 Λ12 Λ21 Λ22 ) . Since Π is a projection, Π∂tuc + AΠ∂xuc + BΠ uc = 0, (2.2) Since Π2 = Π, Π∂tuc + ΠAΠ∂xuc + Π BΠ uc = 0. (2.3) Subtracting (2.3) from (2.2), we find (E − Π)(A∂x + B)Π uc = 0 We denote by P the Fourier image of Π with respect to x. After the Fourier transform with respect to x, the last equality takes the form (E − P )Λ P vc = 0, i.e. Λ P vc ∈ Ker(E − P ). We note that for ∀v ∈ Ker(E − P ) admitting the representation vT = (vTm, vTN−m), with vk ∈ Ck the following equality holds: vN−m = P21vm. Hence we find the system of equations for P21 which completely determines the projection Π: P21(Λ11 + Λ12P21) = Λ21 + Λ22P21. After transformations this equation takes the form P21Λ12(ξ)P21 − Λ22(ξ)P21 + P21Λ11(ξ) − Λ21(ξ) = 0, (2.4) i.e., we obtain a Riccati type matrix equation. This object is nontrivial object. For example, we will consider two special 2 × 2 cases (2.4): X2 = 0, X2 = ( 0 1 0 0 ) There are infinitely many such matrices in the first case, and they form two-dimensional cone in C4(det X = 0, trX = 0). There are no solutions to the second equation, since a matrix has only CUBO 12, 2 (2010) The Maxwell problem and the Chapman projection 279 the zero eigenvalue if the squared matrix possesses this property, i.e. X is nilpotent and the squared nilpotent matrix of second order vanishes. Lemma 3.1. For any γ ∈ R the set of solutions to the matrix equation (2.4) with a matrix Λ coincides with the set of solutions to the same equation (2.4) with the matrix è Λ + γ E. Proof. Indeed, with Λ + γ E we associate the matrix equation P21Λ12P21 − (Λ22 + γ E)P21 + P21(Λ11 + γ E) − Λ21 = 0, where the left-hand side differs from the left-hand side of (2.4) by −γ EP21 + P21γ E = 0. It is obvious that the sets of solutions to these equations coincide. Thus, to study the matrix equation (2.4), we can assume without loss of generality that det(Λ) 6= 0. 2.2 Solutions to the Quadratic Matrix Equation in the Case |Λ| 6= 0 This section is devoted to the solvability condition for the matrix equation. Proposition 3.1. Assume that |Λ| 6= 0 and P = ( P11 P12 P21 P22 ) , (2.5) where P11 is the unit matrix of order m, P22 is the zero square matrix of order N − m, and P12 is the zero matrix. Then the quadratic matrix equation (2.4) is solvable if and only if there exists a matrix P of the form (2.5) such that P is a solution to the quadratic matrix equation (E − P )Λ P = 0. (2.6) Proof. We first assume that the matrix equation (2.4) is solvable. Taking P of the form (2.5) and representing the product (E − P )Λ P in the block form, we see that P is a solution to the matrix equation (2.6). Conversely, let P of the form (2.5) be a solution to the matrix equation (2.6). Representing M = (E − P )Λ P via blocks of the same size as the blocks of P , we see that the blocks M11, M12, and M22 are zero. The equation for M21 coincides with (2.4) up to a sign, i.e., the matrix equation (2.4) is solvable. As a consequence it follows Theorem 3.1. Let a matrix Π21 be a solution Π21Λ12Π21 − Λ22Π21 + Π21Λ11 − Λ21 = 0 (2.7) and X = Λ Π, where Π = ( Π11 Π12 Π21 Π22 ) is a quadratic matrix of order N , Π11 is the identity matrix of order m, and Π12, Π22 are zero matrices. Then X is a solution to the quadratic matrix equation X2 − Λ X = 0. (2.8) 280 V. V. Palin and E. V. Radkevich CUBO 12, 2 (2010) The matrix equation (2.8) is simpler than the general matrix equation and it’s not difficulty to describe one completely. Solutions of the matrix equation (2.4) correspond to a part of the set of solutions for the equation (2.8) only. So that we must to define the selection rule. Theorem 3.2. Let |Λ| 6= 0. Then the quadratic matrix equation (2.4) is solvable if and only if there are two solutions X1 and X2 to the quadratic matrix equation X2 − Λ X = 0, (2.9) such that 1. X1ej = 0 for all j > m. 2. eTj X2 = e T j Λ for all j ≤ m. 3. Λ X2 = X1Λ. Proof. Assume that the matrix equation (2.4) is solvable. Then the matrix equation (2.6) is also solvable. We note that a matrix P belongs to the above class if and only if P ej = 0 ∀j > m and eTj P = e T j ∀j ≤ m. Multiplying the matrix equation (2.6) by Λ from the left and making the change of variables X1 = Λ P , we see that the matrix X1 is a solution to (2.9) and satisfies condition 1). Similarly, multiplying (2.6) by Λ from the right and making the change of variables X2 = P Λ, we find that the matrix X2 is a solution to (2.9) and satisfies condition 2). Since X1 = Λ P and X2 = P Λ condition 3) is also valid. Assume that there exist two solutions X1 and X2 to the matrix equation (2.9) satisfying conditions 1)-3). We set P = Λ−1X1 = X2Λ −1. Then the matrix P has the required form because of conditions 1) and 2). Substituting X1 = Λ P into (2.9) and multiplying by Λ −1 from the left, we find that P is a solution to (2.6). Theorem 3.3. Let |Λ| 6= 0. Then the quadratic matrix equation (2.4) is solvable if and only if there is a solution X1 to the quadratic matrix equation (2.9) such that 1. X1ej = 0 for all j > m. 2. eTj Λ −1X1 = e T j for all j ≤ m. Proof. We set X2 = Λ −1X1Λ. It is obvious that X2 is a solution to the matrix equation (2.9). Furthermore, X2 satisfies condition 2) of Theorem 3.2 because of condition 2) of Theorem 3.3. Since X2 = Λ −1X1Λ we have Λ X2 = X1Λ, i.e., condition 3) of Theorem 3.2 is also satisfied. The proof details of next results look for in [14,17] Lemma 3.1. Suppose that det(Λ) 6= 0, X is a solution to the matrix equation (2.9), and vectors h1, . . . , hN form the Jordan basis for X. Then there exists K ≥ 0 such that h1, . . . , hK belong to the Jordan basis for Λ(moreover, if Xhj = λ hj + hj−1, then Λ hj = λ hj + hj−1) and hK+1, . . . , hN are the eigenvectors corresponding to the eigenvalue 0. Lemma 3.2. Let det(Λ) 6= 0. For K ≥ 0 we denote by X a matrix with the Jordan basis h1, . . . , hN , where the vectors h1, . . . , hK form the Jordan basis for Λ. (listed in such as way that if Xhj = λ hj + hj−1, then Λ hj = λ hj + hj−1) and hK+1, . . . , hN are the eigenvectors corresponding to the eigenvalue 0. Then X is a solution to the matrix equation (2.9). Bring one more the geometrical formulation of the necessary and sufficient conditions of the solvability of the quadratic matrix equation (2.7) CUBO 12, 2 (2010) The Maxwell problem and the Chapman projection 281 Theorem 3.4. Let |Λ| 6= 0, and let vectors v1, . . . , vm satisfy the following conditions: 1. V = Lin{vj}m1 is an eigenspace of the matrix Λ, i.̊a. Λ V = V . 2. v1, .., vm, em+1, .., eN form a basis. Then the quadratic matrix equation (2.4) is solvable. The inverse assertion is also true. 2.3 Explicit Formula Now, we discuss a possible explicit formula for solutions to the Riccati matrix equation. Theorem 4.1. Suppose that vectors v1, . . . , vm form a basis for a linear Λ-invariant subspace V and v1, . . . , vm, em+1, . . . , en is a basis for R n. We regard these vectors as columns of a matrix ( C11 C21 ) . Then the solution to the matrix equation (2.4), associated with these vectors listed in the above order, is represented in the form P21 = C21C −1 11 (2.10) Proof. Since we can assume that det(Λ) 6= 0, for the solution to the matrix equation (2.4) we have ( E 0 P21 0 ) = Λ−1 ( C11 0 C21 E )( J1 0 0 0 )( C−111 0 −C21C−111 E ) , where J1 is a block from the Jordan form of the matrix Λ corresponding to the space V . Hence ( E 0 P21 0 ) = Λ−1 ( C11J1C −1 11 0 C21J1C −1 11 0 ) . Multiplying both sides of the last equality by Λ from the left, we find ( Λ11 + Λ12P21 0 Λ21 + Λ22P21 0 ) = ( C11J1C −1 11 0 C21J1C −1 11 0 ) , which implies P21C11J1C −1 11 − C21J1C −1 11 = 0. in view of (2.4). Since Λ is invertible, the matrix J1 is also invertible. Hence we can multiply the last equality by C11J −1 1 from the right. Then P21C11 = C21, which implies (2.10). 2.4 The Number of Solutions Corollary 5.1. With every m-dimensional eigenspace V of the matrix Λ at most one solution to the matrix equation (2.4) is associated. Proof. Indeed, either V does not provide any solution to (2.4) (if Lin{v1, . . . , vm, em+1, . . . , en} = Rn, where V = Lin{v1, . . . , vm}) or V can be associated with a solution to (2.4) by formula (2.10). In the second case, we show that the solution is independent of the choice of the basis for the space V . Let w1, . . . , wm be another basis for V . We write the vectors v1, . . . , vm as columns of a matrix W 0 282 V. V. Palin and E. V. Radkevich CUBO 12, 2 (2010) and the vectors w1, . . . , wm as columns of a matrix W 1. Since these bases generate the same linear space V , there exists a nonsingular matrix K such that W 1 = W 0K or, in the block form, ( W 11 W 12 ) = ( W 01 W 02 ) K, which implies W 1j = W 0 j K, j = 1, 2. Hence the solution of the form (2.10) corresponding to the basis for W 1 can be written as P21,W = W 1 2 (W 1 1 ) −1 = W 02 KK −1(W 01 ) −1 = P21,V . Thus, the solutions defined by the bases v1, . . . , vm and w1, . . . , wm coincide. Next results we bring for the information(details look in [19,22]) Theorem 5.2. Let the matrix equation (2.4) have infinite number of solutions. Then there exists λ ∈ C such that dim(Ker(Λ − λ E)) ≥ 2. Theorem 5.2. The set of solutions to the matrix equation (2.4) is infinite then and only then if there exists the eigenspaces V and W of the matrix Λ, satisfying the following conditions: 1. V defines the solution to the equation (2.4). 2. W is a eigenspace of the matrix Λ, corresponding to a eigenvalue λ. 3. W contain two incollinear eigenvectors. 4. V ∩ W 6= {0}. 5. W \ V 6= ∅. 2.5 Continuity of Solutions to the Quadratic Matrix Equation In this section, we study the continuity of the constructed solutions with respect to the parameter ξ. We begin with auxiliary assertions(see [17]). Lemma 6.1. Let a matrix A(x) be a continuous function of the parameter x in some neighbor- hood U (x0) of a point x0. Denote by λ(x) an eigenvalue of A(x) that continuously depends on x in U (x0) and is simple in a punctured neighborhood of x0. Then the corresponding eigenvector vλ(x) is also continuous with respect to x in the same neighborhood. Lemma 6.2. Suppose that a matrix-valued function A(x) is continuous with respect to x in some neighborhood U (x0) of a point x0 and its kernel has constant dimension k in U (x0). Let vectors v1(x), . . . , vm(x) . Ker(A(x)) be continuous in U (x0) and linearly independent in the cor- responding punctured neighborhood. Then there exists a basis for the invariant subspace V (x) = Lin{v1(x), . . . , vm(x)}, i.e., w1(x), . . . , wm(x), that is continuous in U (x0) and is linearly independent in U (x0). Theorem 6.1. Suppose that a matrix A(x) continuously depends on x in some neighborhood U (x0) of a point x0 and λ1(x), . . . , λk(x) are continuous eigenvalues of A(x) such that each of them is simple in a punctured neighborhood of x0, λ1(x0) = ... = λk(x0) = λ0. Let V (x) be the eigenspace of A(x) corresponding to λ1(x), ..., λk(x). If there are no eigenvalue λ(x) of A(x), different from CUBO 12, 2 (2010) The Maxwell problem and the Chapman projection 283 λj (x), j = 1, . . . , k, that is continuous and λ(x0) = λ0, then there exists a basis for V (x) continuously depending on x in U (x0). Proof. In a punctured neighborhood of x0, for a basis for V (x) we take the eigenvectors vj (x) corresponding to the eigenvalues λj (x) of the matrix A(x). By the above lemma, vj (x) are continuous in U (x0). Further, let us introduce the matrix M (x) = Π k j=1(A(x)−λj (x)E).. Under the assumptions of the theorem, M (x) is continuous in U (x0), dim(Ker(M (x))) = k and V (x) = Ker(M (x)). Hence the matrix M (x) and subspace V (x) satisfy the assumptions of Lemma 5.2 with m = k, which implies the required assertion. Theorems 5.1 and 4.1 lead to the following assertion concerning the continuity of solutions to the quadratic matrix equation (2.4) with respect to the parameter ξ Theorem 6.2. Assume that a matrix Λ is continuously depends on the parameter ξ and is invertible for all ξ ∈ Ξ0; moreover, the eigenvalues of Λ are simple for all ξ /∈ Ξ∗, where the set Ξ∗ is finite. Then the matrix equation (2.4) with the matrix Λ has a solution continuously depending on the parameter ξ if and only if there exists an m-dimensional eigenspace V satisfying the assumptions of Theorem 4.1 for all ξ ∈ Ξ0. Proof. Indeed, by Theorem 4.1 and its consequences, the subspace V determines a solution to the quadratic matrix equation (2.4) in the form P21 = C21C −1 11 . Since the matrix Λ satisfies the assumptions of Theorem 5.1, the basis for the space V continuously depends on ξ. Therefore, the invertibility of C11 immediately implies the required assertion. 2.6 The Lyapunov Equation. Separation of Dynamics The Lyapunov matrix equation −M11Q12 + Q12M22 − M12 = 0 (2.11) is a special case of the quadratic matrix equation (2.4) with vanishing quadratic term. The following assertion is proved in [20]. Theorem 7.1. Suppose that det(M11) 6= 0 and det(M22) 6= 0. Assume that the matrix M = ( M11 M12 0 M22 ) has no eigenvalues λ such that, in the block form, the corresponding eigenvector has the form v0 = ( v0,1 0 ) and the corresponding associated eigenvector has the form v1 = ( v1,1 v1,2 ) , where v1,2 6= 0. Then there exists a solution Q12 to the Lyapunov matrix equation (2.11) with the matrices M11, M12, and M22. We construct the canonical form of (2.1). Lemma 7.1. Suppose that a matrix S is invertible and can be written in the block form with blocks Sij , i, j = 1, 2, where S11 and S22 are square matrices. Assume also that F S = SF = E, 284 V. V. Palin and E. V. Radkevich CUBO 12, 2 (2010) where the matrix F can be represented by blocks of the same size. In this case, if S11 = E then the matrix F22 is invertible. Proof. Assume the contrary. Since F S = E, we have F21 + F22S21 = 0. (2.12) Since F22 is noninvertible, there is a row h 6= 0 such that hF22 = 0. Using (2.12), we find hF21 = 0. But, in this case, the last rows of the matrix F are linearly dependent: there is a row v such that v 6= 0 and vF = 0. Thus, the matrix F is noninvertible. On the other hand, the matrix F is the inverse of S. We arrive at a contradiction. Theorem 7.2. Suppose that Λ is divided into blocks Λij , i, j, = 1, 2. Then the quadratic matrix equation (2.4) is solvable if and only if there exists a matrix S satisfying the following conditions: 1. S is invertible, 2. S11 = E. 3. (S−1Λ S)21 = 0. Proof. Assume that there exists a matrix S satisfying conditions 1) -3). For F = S−1 we have F21 + F22S21 = 0, F21(Λ11 + Λ12S21) + F22(Λ21 + Λ22S21) = 0. Expressing F21 from the first equation and substituting into the second equation, we find F22(−S21(Λ11 + Λ12S21)) + F22(Λ21 + Λ22S21) = 0. We note that the matrix S satisfies the assumptions of Lemma 6.1. Hence (−S21(Λ11 + Λ12S21)) + (Λ21 + Λ22S21) = 0, i.e., the matrix S21 satisfies the quadratic matrix equation (2.4). Assume that the quadratic matrix equation (2.4) is solvable. We set S11 = E, S12 = 0, S21 = P21, S22 = E. It is easy to verify that the inverse matrix exists: S−1 = 2E − S. We see that the matrix S satisfies conditions 1) and 2). Computing (S−1Λ S)21, we find (S−1Λ S)21 = F21(Λ11 + Λ12S21) + F22(Λ21 + Λ22S21) = = (−P21)(Λ11 + Λ12P21) + (Λ21 + Λ22P21) = 0, since P21 is a solution to the quadratic matrix equation (2.4). Thus, the matrix S also satisfies condition 3). The theorem is proved. Thus, the existence of a Chapman-Enskog projection is equiv- alent to the possibility to represent the original system in the block form such that (S−1Λ S)21 = 0, which allows us to separate dynamics. The following theorem (cf. the proof in [16]) provides us with conditions under which a matrix . can be reduced to the block-diagonal form. Theorem 7.3. Assume that a matrix Λ is invertible and v1, . . . , vm is a basis for its eigenspace V such that Lin{v1, . . . , vm, em+1, . . . , eN } = RN . We also assume that V cannot be extended to an CUBO 12, 2 (2010) The Maxwell problem and the Chapman projection 285 m + 1-dimensional eigenspace of the matrix Λ by extending the basis v1, . . . , vm with an associated eigenvector of Λ. Then there exist matrices P21 and Q12 such that ( E −Q12 0 E )( E 0 −P21 E ) Λ ( E 0 P21 E )( E Q12 0 E ) = ( M11 0 0 M22 ) . Now, we consider the representation of the solution as the sum of three terms and introduce the notion of the L2-well-posedness in the sense of Chapman-Enskog. Suppose that a matrix Λ satisfies the assumptions of Theorem 7.3. We make the change of variables U = S−1u. Then a solution to the Cauchy problem (2.1) with the initial data U|t=0 = ( U0 V0 ) can be written in terms of the Fourier images as follows: U = e−Mt ( U0 V0 ) , where M = S−1Λ S. By Theorem 7.3, the matrix M takes the form M = ( E Q12 0 E )( M11 0 0 M22 )( E −Q12 0 E ) , which implies U = ( E Q12 0 E ) exp ( − ( M11 0 0 M22 ) t )( E −Q12 0 E )( U0 V0 ) = UCh + UCor + UH , where each of the terms is a solution to the system (2.1) with some initial data: UCh = e −Mt ( U0 0 ) , UCor = e −Mt ( −Q12V0 0 ) , UH = ( Q12e −M22tV0 e−M22tV0 ) . The first term UCh corresponds to the projection onto the phase space of consolidated variables, the second term UCor is a corrector describing the influence of the initial data relative to nonequillibrium variables, and the third term UH is a remainder. Definition 7.1. We say that a projection P satisfies the Chapman L2-well-posedness condition for a class of initial data H = {(U0, V0)} if for any initial data (U0, V0) ∈ H there is a constant T0 > 0 such that for all t > T0 ||UH||(t) ||UCh||(t) ≤ Ke−δ t, t > T0, (2.13) where K and δ > 0 are constants. 286 V. V. Palin and E. V. Radkevich CUBO 12, 2 (2010) 2.7 Crack Condition and the Existence of an Attracting Manifold We find conditions that guarantee the validity of the estimate ||UH|| = o(||UCh||), t → ∞, where ‖f‖ denotes the norm of f in the space L2. For this purpose, we prove several technical auxiliary assertions(see [21]): Lemma 8.1. Suppose that a matrix Λ polynomially depends on ξ and there exists k0 > 0 such that for all ξ : |ξ| > k0, all the eigenvalues λ(ξ) of Λ are algebraically simple and |λ(ξ)| ≤ C1(1+|ξ|)d1 , where C1 and d1 are constants. Let v be an eigenvector of Λ. Then for |ξ| > k0 : max{|eTi v|} min{|eTi v| 6= 0} ≤ C2(1 + |ξ|)d2 , (2.14) where C2 and d2 are constants. Lemma 8.2. Suppose that a matrix Λ is defined for all ξ ∈ R and satisfies the assumptions of Theorem 6.3 for all ξ ∈ Ξ, where Ξ = R\Ξ− and the set Ξ− is finite. Then P21 and Q12 are defined on Ξ. Assume that the matrices P21(ξ) and Q12(ξ) can be defined by continuity on the set Ξ−. We also assume that the matrix Λ. polynomially depends on ξ and there is k0 > 0 such that for all ξ : |ξ| > k0, all the eigenvalues of the matrix Λ are algebraically simple and satisfy the following estimate: |λ(ξ)| ≤ C1(1 + |ξ|)d1 , where C1 and d1 are constants. Then there is d ∈ N such that for all ξ ∈ R |P21| ≤ K1(1 + |ξ|)d, |Q12| ≤ K2(1 + |ξ|)3d, where K1 and K2 are constants and |A| is the matrix norm of A in L∞. Notation 8.1. The minimal number d ∈ N satisfying the assumptions of Lemma 8.2 is denoted by dΛ. We also need a two-sided estimate for |e−Mtv|, where |.| — denotes the L∞(R). For the sake of brevity, we introduce the following notation. Notation 8.2. Suppose that a square matrix M continuously depends on the parameter ξ. Let λj , j = 1, ..., s, be eigenvalues of M . We denote by dj the maximal size of the Jordan cell corresponding to the eigenvalue λj . Let the eigenvalues λj be listed in ascending order of the real part. Let l(M ) and L(M ) denote the minimal and maximal eigenvalues respectively, i.e. l(M ) = Re λ1 ≤ Re λ2 ≤ · · · ≤ Re λs = L(M ). We set d(M ) = d1. We will use one technical lemma still(see [19]: Lemma 8.3. Let a square matrix M continuously depend on the parameter ξ. Then for any ε > 0 there is T0 > 0 such that for all t > T0 the following estimate holds: e−L(M)t|v| ≤ |e−Mtv| ≤ 1 + ε (d(M ) − 1)!|M| d(M)−1e−l(M)ttd(M)−1|v|, (2.15) where |A| denotes the matrix norm of A in L∞(R). CUBO 12, 2 (2010) The Maxwell problem and the Chapman projection 287 Notation 8.3. Let Γ(ξ) be a finite set of continuous functions γ1(ξ), . . . , γs(ξ) of the parameter ξ. Introduce the notation l(ξ, Γ(ξ)) = infs{Re γs(ξ) | γs(ξ) ∈ Γ(ξ)}, l0(Γ) = infξ l(ξ, Γ(ξ)), L(ξ, Γ(ξ)) = sups{Re γs(ξ) | γs(ξ) ∈ Γ(ξ)}, L0(Γ) = supξ L(ξ, Γ(ξ))). Condition 8.1. A pair of sets Γ1(ξ) and Γ2(ξ) satisfy the strong crack condition if ∃γ > 0 : l0(Γ2) − L0(Γ1) ≥ γ. (2.16) Now, we formulate the conditions for the existence of an attracting manifold. Theorem 8.1. Let the matrix Λ in the problem (2.1) satisfy the assumptions of Lemma 8.2. Suppose that Γ1 is the set of all those eigenvalues of Λ that determine the separation of dynamics for the eigenspace V and Γ2 is the set of all the remaining eigenvalues of Λ. Assume that Γ1 and Γ2 satisfy the strong crack condition. Let the Fourier images of initial data (U0, V0) belong to the set H = {(U0, V0) : ||U0|| 6= 0, (1 + |ξ|)3dΛ |M22|d(M22)−1V0 ∈ L2(R)}, Then the projection P corresponding to the separation of dynamics satisfies the Chapman-Enskog L2-well-posedness condition (Definition 6.1) for the class of initial data H with constants K and δ such that (i) K depends on ||U0||, ||V0||, (ii) δ depends on δ and some properties of the matrix M . Proof. Indeed, ||UH (t)|| =   ∫ R ∣ ∣ ∣ ∣ ∣ ( Q12e −M22tV0 e−M22tV0 )∣ ∣ ∣ ∣ ∣ 2 dξ   1 2 ≤ ( ∫ R |1 + |Q12|2||e−M22tV0|2dξ ) 1 2 . Using Lemmas 8.2 and 8.3, we find ||UH (t)||2 ≤ ∫ R (1 + K22 (1 + |ξ|)10dΛ )( 1 + ε (d(M22) − 1)! )2|M22|2d(M22)−2e−2l(M22)tt2d(M22)−2|V0|2dξ. From (2.16) it follows that e−l(M22)t ≤ e−l0(Γ2)t ≤ e−γte−L0(Γ1)t; e−L(M11)t ≥ e−L0(Γ1)t. By Lemma 8.3, ||UCh(t)|| ≥ ( ∫ R e−2L(M11)t|U0|2dξ ) 1 2 . Combining the last four inequalities, we find ( ||UH||(t) ||UCh||(t) )2 ≤ e−2γtt2d(M22)−2 ∫ R e−2L0(Γ1)t(1 + K22 (1 + |ξ|)10dΛ ) ( 1+ε (d(M22)−1)! )2 |V0|2dξ ∫ R e−2L0(Γ1)t|U0|2dξ , which implies the required estimate (2.13) because L0(Γ1) is independent of ξ. 288 V. V. Palin and E. V. Radkevich CUBO 12, 2 (2010) 3 Nonlinear analysis. Chapman projection 3.1 Statement of the Problem and Auxiliaries We consider the nonlinear system of equations ∂tu + n ∑ j=1 Aj ∂xj u + Bu = f (u), (3.1) with the initial condition u|t=0 = φ, where u is an N -dimensional vector, Aj and B are constant matrices, n ≤ 3, and f (u) is a vector-valued polynomial, i.e. f (u) = ∑N j=1 ( ∑ σ∈Θj K(j, σ)uσ ) ej , where σ ∈ (N ∪ {0})N , uσ = ∏N j=1 u σj j . We set α = min{|σ| : σ ∈ ∪Nj=1Θj}, α + β = max{|σ| : σ ∈ ∪Nj=1Θj}. Assume that f (u) contains no terms of zero or first order, i.e. α ≥ 2. We denote by || · || the norm in L2 with respect to the variable x. Let |u| = √ uT u and let |u|0 denote the norm of u in C. Following [10], we denote by ∂ the vector consisting of all first order derivatives and by ∂x the vector consisting of first order derivatives with respect to the spatial variables, i.e. ∂ = (∂x, ∂t). For the sake of brevity, we write ∂j instead of ∂xj . We begin with the following auxiliary assertion generalizing Lemma 8.3. Lemma 9.1. Let M be a square matrix. Then there are constants CM ∈ R and dM ∈ Z such that for any vector v and a number t ≥ 0 |e−Mtv| ≤ CM (1 + tdM )e−l(M)t|v|. (3.2) The following assertion concerns estimates for the norms of f (u) and its derivatives is tru: Lemma 9.2. For a vector-valued function u(x, t) ∈ C([0, T ), H2) ∩ C1([0, T ), H1) with T > 0 and s ∈ {1, 2}, j ∈ {1, 2, 3} the following estimates hold: ||f (u)|| ≤ C0,0|u|α−10 (1 + |u| β 0 )||u||, (3.3) ||∂sj f (u)|| ≤ Cs,0|u|α−10 (1 + |u| β 0 )||∂sj u||, (3.4) The following assertion concerning the norm of a vector-valued polynomial is a consequence of the above lemma. Lemma 9.3. Consider a vector-valued function u(x, t) ∈ C([0, T ), H2) ∩ C1([0, T ), H1) with T > 0, x ∈ Rn, n ≤ 3. Let g(u) be a vector-valued polynomial with α ≥ 1. Then there are κ ∈ (0, 1) and CG > 0 such that for all u(x, t) such that ||g(u)||H2 ≤ 2CG||u||αH2 . (3.5) CUBO 12, 2 (2010) The Maxwell problem and the Chapman projection 289 Proof. Indeed, from the inequalities (3.3) and (3.4) it follows that ||g(u)||H2 ≤ const |u|α−10 (1 + |u| β 0 )||u||H2 . By the embedding theorem, ||g(u)||H2 ≤ CG||u||αH2 (1 + ||u|| β H2 ). Hence the required inequality (3.5) holds for sufficiently small κ. Whence we obtain Lemma 9.4. Let u(x, t) and v(x, t) be vector-valued functions such that u(x, t), v(x, t) ∈ C([0, T ), H2) ∩ C1([0, T ), H1) for some T > 0. Assume that x ∈ Rn and n ≤ 3, f (u) is a vector-valued polynomial with α ≥ 2. Then there are κ ∈ (0, 1) and C∗ > 0 such that for all u(x, t), v(x, t) the inequalities ||u||H2 < κ, ||v||H2 < κ imply the inequality ||f (u) − f (v)||H2 ≤ C∗(||u||α−1H2 + ||v|| α−1 H2 )||u − v||H2 . (3.6) Lemma 9.5. Suppose that t > 0 and P (τ ) is a continuous function such that the inequality P (τ ) ≥ 0 for all τ ∈ [0, t]. Let d > 0. Then there is a constant CP > 0 such that ∫ t 0 (1 + (t − τ )d)2P (τ )dτ ≤ CP (1 + td)2 ∫ t 0 (1 + τ d)2P (τ )dτ. (3.7) Proof. We have ∫ t 0 (1 + (t − τ )d)2P (τ )dτ ≤ C1 ∫ t 0 (1 + τ 2d + t2d)P (τ )dτ ≤ ≤ C1 ( t2d ∫ t 0 P (τ )dτ + ∫ t 0 (1 + τ 2d)P (τ )dτ ) ≤ C1(1 + t2d) ∫ t 0 (1 + τ 2d)P (τ )dτ ≤ ≤ C1(1 + td)2 ∫ t 0 (1 + τ d)2P (τ )dτ. 3.2 Method of Successive Approximations We look for a solution to the system (3.1) with the initial data u|t=0 = φ(x) for small φ(x) by the method of successive approximations. We set u0 = 0, ∂tuk + n ∑ j=1 Aj ∂j uk + Buk = f (uk−1), uk|t=0 = φ(x). (3.8) Introduce the notation Λ = ∑n j=1 Aj iξj + B, l1 = infξ minλ∈σ(Λ) Re λ. We denote by F(·) the Fourier transform with respect to the spatial variables. We estimate from above the solution uk to the problem (3.8). 290 V. V. Palin and E. V. Radkevich CUBO 12, 2 (2010) Lemma 10.1. Let l1 > 0. Then there exist constants κ ∈ (0, 1) and C∗1 > 0, C∗2 > 0 such that the solution uk to the problem (3.8) with the initial data φ such that ||φ||H2 < κ satisfies the following inequality for any t ≥ 0 : ||uk||H2 ≤ C∗1 (1 + tdΛ )e−l1t(||φ||H2 + C∗2 √ t||φ||αH2 ). (3.9) Proof. We first prove that for sufficiently small initial data ||uk||H2 ≤ CΛ(1 + tdΛ )e−l1t(||φ||H2 + Ck √ t||φ||αH2 ), (3.10) where the constants Ck depend on k. We write an explicit expression for Ck. For this purpose, we use the method of mathematical induction. Let k = 1. Then the problem (3.8) takes the form ∂tu1 + n ∑ j=1 Aj ∂j u1 + Bu1 = 0, u1|t=0 = φ. The solution to this problem is written in terms of the Fourier images as follows: F(u1) = e−ΛtF(φ). By Lemma 10.1 ||u1||2 = ||F(u1)||2 = ∫ Rn |e−ΛtF(φ)|2dξ ≤ C2Λ ∫ Rn (1 + tdΛ )2e−2l1t|F(φ)|2dξ = = C2Λ(1 + t dΛ )2e−2l1t||F(φ)||2 = C2Λ(1 + tdΛ )2e−2l1t||φ||2. A similar inequality holds for the derivatives of u1. Thus, ||u1||H2 ≤ CΛ(1 + tdΛ )e−l1t||φ||H2 , and the inequality (3.10) is true with C1 = 0. Now, we write an explicit formula for the solution to the problem (3.8) in terms of the Fourier images: F(uk) = e−ΛtF(φ) + ∫ t 0 eΛ(τ −t)F(f (uk−1(τ )))dτ. (3.11) We set Iσ,k = (iξ) σ ∫ t 0 eΛ(τ −t)F(f (uk−1(τ )))dτ and find ||Iσ,k|| ≤ CΛCk √ t(1 + tdΛ )e−l1t||φ||αH2 , where the constant Ck is independent of σ. The proof of this assertion is similar to that of the inequality (3.10) for all k. Indeed, we have the auxiliary estimates ||Iσ,k||2 ≤ t ∫ t 0 ||(iξ)σeΛ(τ −t)F(f (uk−1(τ )))||2dτ ≤ ≤ tC2Λ ∫ t 0 (1 + (t − τ )dΛ )2e2l1(τ −t) ∫ Rn |(iξ)σF(f (uk−1(τ )))|2dξdτ ≤ ≤ tC2Λ ∫ t 0 (1 + (t − τ )dΛ )2e2l1(τ −t)||f (uk−1(τ ))||2H2 dτ. CUBO 12, 2 (2010) The Maxwell problem and the Chapman projection 291 Using Lemmas 9.3 and 9.5, we find ||Iσ,k||2 ≤ 4C2ΛC2F CP t(1 + tdΛ )2e−2l1t ∫ t 0 (1 + τ dΛ )2e2l1τ ||uk−1(τ )||2αH2 dτ. (3.12) Let k = 2. Using the estimate (3.10) for k = 1, we find ||Iσ,2||2 ≤ 4C2+2αΛ C 2 F CP t(1 + t dΛ )2e−2l1t ∫ t 0 e2l1(1−α)τ (1 + τ dΛ )2+2α||φ||2αH2 dτ ≤ ≤ 4C2+2αΛ C 2 F CP t(1 + t dΛ )2e−2l1t ∫ +∞ 0 e2l1(1−α)τ (1 + τ dΛ )2+2α||φ||2αH2 dτ. Note that for α ≥ 2 the integral is convergent. Setting C22 = 4C 2α Λ C 2 F CP ∫ +∞ 0 e2l1(1−α)τ (1 + τ dΛ )2+2αdτ, we obtain an inequality of the required form for ||Iσ,2||. Assume that the inequality (3.10) is valid for all k ≤ r, where r ≥ 2. Then for k = r + 1, by the inequality (3.12) ||Iσ,r+1||2 ≤ C2Λt(1 + tdΛ )2e−2l1t||φ||2αH2 ( C′C22 + +4C2αΛ C 2 F CP C ′C2αr ∫ +∞ 0 e2l1(1−α)τ (1 + τ dΛ )2+2ατ α||φ||2α(α−1) H2 dτ ) . Setting J = ∫ +∞ 0 e2l1(1−α)τ (1 + τ dΛ )2+2ατ αdτ ), we obtain the required estimate (3.10) with C2r+1 = C 2 2 C ′ + 4C2αΛ C 2 F CP C ′J||φ||2α(α−1) H2 C2αr We note that C′ > 1. We choose κ > 0 such that for ||φ|| < κ 4C2αΛ C 2 F CP C ′J||φ||2α(α−1) H2 < 1 (C22 C ′ + 1)α . Let qr = C 2 r . Then q2 < C 2 2 C ′ + 1. We note that for κ, as above, qr < C 2 2 C ′ + 1 for all r ≥ 2. Indeed, qr+1 = C 2 2 C ′ + 4C2αΛ C 2 F CP C ′J||φ||2α(α−1) H2 qαr < C 2 2 C ′ + 1 (C22 C ′ + 1)α qαr < C 2 2 + 1. Thus, Cr < √ C22 C ′ + 1 and the inequality (3.10) with small κ implies (3.9). From this result it follows Lemma 10.2. Suppose that l1 > 0 and ||φ||H2 < κ in (3.8) with sufficiently small κ. Then the solutions uk to the system (3.8) converge in C((0, +∞); H2). 292 V. V. Palin and E. V. Radkevich CUBO 12, 2 (2010) 3.3 Construction of a Nonlinear Chapman Projection 11.1. Weak nonlinearity. We consider the system ∂tu + A11∂xu + A12∂xv + B11u + B12v = 0, ∂tv + A21∂xu + A22∂xv + B21u + B22v = G(u)v (3.13) with the initial data u|t=0 = φ1(x), v|t=0 = φ2(x). We set φ = ( φ1 φ2 ) . Suppose that u(x, t) : R × R+ → Rm and v(x, t) : R × R+ → RN−m. Assume that the data of the problem (3.13) satisfy all the assumptions of Lemma 10.2. We also assume that the following condition is satisfied. Condition 11.1. The linearized part of the problem (3.1) and the initial data satisfy all the assumptions of Theorem 7.1. Moreover, lj = infξ minλ∈Γj Re λ and l1 > 0, l2 − αl1 < 0. We denote by P21 the symbol of the Chapman-Enskog projection for the linearized problem (3.13). If the initial data φ are sufficiently smooth and ||φ||2H2 + ||P21(∂x)φ||2H2 < κ ≪ 1 then, according to the method of successive approximations, there exists a solution ( w z ) to the problem (3.13) with the initial data w|t=0 = Υ(φ1, φ2), z|t=0 = P21(∂x)Υ(φ1, φ2), where Υ is the operator of the initial data corresponding to the sum of UCh and UCor in the linear case. The goal of this section is to construct a nonlinear operator P21(w, ∂x) such that z = P21(w, ∂x)w. Let M = SΛS−1, where S = ( E 0 −P21 E ) . We write the system (3.13) in terms of the Fourier images and use the fact that P21 is the symbol of the Chapman projection for the linearized problem. Then ∂tF(w) + M11F(w) + M12F(v′) = 0, ∂tF(v′) + M22F(v′) = F(G(w)z), where z = P21w + v ′. Based on this fact, we look for z in the form z = P21w + ∞ ∑ j=1 vj , (3.14) where vj is a solution to the equation ∂tF(vj ) + M22F(vj ) = F(G(w)vj−1) (3.15) with the initial data vj|t=0 = 0 and v0 = P21w. Using the method of variation of constants, we find vj = F−1 ( e−M22t ∫ t 0 eM22τ F(G(w(τ ))vj−1 (τ ))dτ ) . CUBO 12, 2 (2010) The Maxwell problem and the Chapman projection 293 This representation shows that vj = Πj (w, ∂x)w. It remains to prove that for small φ the series (3.14) is convergent. From Lemma 10.1 and the method of successive approximations it follows that ||w||H2 ≤ C0e−l1t(1 + td)(||φ||H2 + ||P21φ||H2 ). Furthermore, for P21w we have the similar estimate ||P21w|| ≤ C1e−l1t(1 + td)||φ||Hs with some s. Based on these two inequalities and the embedding theorem, we find the following estimate for v1: ||v1||2 ≤ t ∫ t 0 ||eM22(τ −t)F(G(w(τ ))P21 w(τ ))||2dτ ≤ te−2l2t ∫ t 0 e2l2τ q0(t, τ )|w|2α−2∞ ||P21w(τ )||2dτ, where q0(t, τ ) is a polynomial depending only on the structure of M . Further, ||v1||2 ≤ C2te−2l2t ∫ t 0 e2(l2−αl1)τ q0(t, τ )(1 + τ d)2α(||φ||H2 + ||P21φ||H2 )2α−2||φ||2Hs dτ. Hence, under the above conditions on the system (3.13), there are constants d1 ≥ 0 and K1 > 0 such that ||v1||2 ≤ K1t(1 + td1 )e−2l2t||φ||2αHs . (3.16) Moreover, d1 depends only on the structure of the matrix M and K1 is independent of φ. Similarly, for v2 we find ||v2||2 ≤ t ∫ t 0 ||eM22(τ −t)F(G(w(τ ))v1 (τ ))||2dτ ≤ ≤ C3te−2l2t ∫ t 0 e2l2τ q0(t, τ )(1 + τ d)2α−2e−(2α−2)l1τ ||v1(τ )||2(||φ||H2 + ||P21φ||H2 )2α−2dτ. Using the above estimate for v1, we find ||v2||2 ≤ K2te−2l2t(1 + td1 )||φ||4α−2Hs , where the constant K2 is independent of φ, because ∫ t 0 τ re−γτ dτ ≤ ∫ ∞ 0 τ re−γτ dτ = const. Arguing in the same way, it is easy to obtain the inequality ||vj||2 ≤ Kjt(1 + td1 )e−2l2t||φ||2jα−2j+2Hs , where the constants Kj are independent of φ and Kj ≤ Kj0, K0 = const. Hence for sufficiently small φ the series (3.14) is convergent. We note that the smallness of the norm of the initial data in some space Hs and the estimate |G(w)|∞ ≤ CG|w|α−1∞ (1 + |w|β∞) 294 V. V. Palin and E. V. Radkevich CUBO 12, 2 (2010) imply |G(w)|∞ ≤ C′G|w|α−1∞ , where C′G > CG. Furthermore, d1 is the degree of the polynomial q0(t, τ ) in the variable t. Since q0 is a polynomial, there exist constants I1 and I2 such that ∫ +∞ 0 e2(l2−αl1)τ q0(t, τ )(1 + τ d)2αdτ ≤ I1(1 + td1 ), and ∫ +∞ 0 e−(2α−2)l1τ q0(t, τ )(1 + τ d)2α−2(1 + τ d1 )τ dτ ≤ I2(1 + td1 ). Indeed, both integrals on the left-hand sides of these inequalities are polynomials in t of degree d1, which implies the required estimates. To prove the assertions concerning the constants Kj , we need the following lemma. Lemma 11.1. Assume that all the assumptions of Lemma 10.2 and Condition 11.1 are satisfied. Then the solution vj to the problem (3.15) with the initial data vj|t=0 = 0 for j ≥ 1 satisfies the inequality ||vj||2 ≤ Kjt(1 + td1)e−2l2t||φ|| 2jα−2j+2 Hs , where Kj ≤ (C′G)j C 2jα−2j+2 W I j−1 2 I1 and CW = max{C0, C1}. Proof. We use the method of mathematical induction. As was already shown, the required estimate is valid for ||v1||2. Furthermore, it is easy to see that K1(1 + t d1 ) = C2 ∫ +∞ 0 e2(l2−αl1)τ q0(t, τ )(1 + τ d)2αdτ ≤ C2I1(1 + td1 ), where C2 = C ′ GC 2α W . Thus, the corresponding inequality for K1 also holds. Assume that the assertion holds for j ≤ k. Then for j = k + 1 we have ||vk+1||2 ≤ t ∫ t 0 ||eM22(τ −t)F(G(w(τ ))vk (τ ))||2dτ ≤ ≤ te−2l2t ∫ t 0 e2l2τ q0(t, τ )|w(τ )|2α−2∞ ||vk(τ )||2dτ ≤ Kk+1t(1 + td1 )e−2l2t||φ|| 2(k+1)α−2k Hs , where C3 = C ′ GC 2α−2 W , and from the inequality ∫ t 0 q0(t, τ )(1 + τ d)2α−2e−(2α−2)l1τ (1 + τ d1 )τ dτ ≤ ≤ ∫ +∞ 0 q0(t, τ )(1 + τ d)2α−2e−(2α−2)l1τ (1 + τ d1 )τ dτ ≤ I2(1 + td1 ) we obtain the required estimate for Kk+1. 11.2. General case. We consider the system ∂tu + A11∂xu + A12∂xv + B11u + B12v = G11(u)u, ∂tv + A21∂xu + A22∂xv + B21u + B22v = G21(u)u + G22(u)v (3.17) CUBO 12, 2 (2010) The Maxwell problem and the Chapman projection 295 with the same initial data, as above. Then we construct a nonlinear operator P21(∂x, w) that deter- mines the solution ( w z ) . We look for z in the form (3.14), where vj are solutions to the problem ∂tF(v1) + M22F(v1) = F(P21(G11(w)w) + G21(w)w + G22(w)P21w), F(v1)|t=0 = 0, ∂tF(vj ) + M22F(vj ) = F(G22(w)vj−1), F(vj )|t=0 = 0, j ≥ 2. Then we can estimate ||v1|| as follows: ||v1||2 ≤ t ∫ t 0 C2M (1 + (t − τ )dM )2e2l2(τ −t)||P21(G11(w)w) + G21(w)w + G22(w)P21w||2(τ )dτ ≤ ≤ const t(1 + tdM )2e−2l2t||φ||2αH2 ∫ +∞ 0 e2(l2−αl1)τ (1 + τ dM )2(1 + τ dΛ )2α(1 + √ τ )2αdτ = = K′1t(1 + t dM )2e−2l2t||φ||2αH2 . Thus, for v1 we have an estimate of the form (3.16). We note that the equation for vj , j > 1, is the same as in the previous subsection. Furthermore, for estimating from above vj , j > 1, we used the estimate (3.16), but not an explicit form of v1. Consequently, Lemma 11.1 remains valid. Therefore, the series (3.14) converges in the L2-norm for small initial data, which means the existence of a nonlinear projection P21. 3.4 Properties of Nonlinear Projections We study properties of the nonlinear operator P21 constructed in the previous section. Lemma 12.1. Let the data of the problem (3.17) satisfy all the assumptions of Lemma 10.2 and Condition 11.1. Assume that φ ∈ H3, |P21|0 ≤ const(1 + |ξ|s), s ≤ 2. Then for every term vj of the series (3.14) the following inequality holds: ||vj||2H1 ≤ K j 0 t(1 + t dM )2e−2l2t||φ||2jα−2j+2 H3 . (3.18) Proof. We estimate each ||∂kvj||. For this purpose, we note that ||∂kvj|| satisfies the problem ∂tF(∂kvj ) + M22F(∂kvj ) = F(∂k(Fj (w, vj−1))), F(∂kvj )|t=0 = 0, where F1(w, v0) = F1(w) = −P21(G11(w)w) + G12(w)w + G22(w)P21w, Fj (w, vj−1) = G22(w)vj−1, j ≥ 2. Using Lemmas 9.1 and 9.5, we find ||∂kv1||2 ≤ const t(1 + tdM )2e−2l2t ∫ +∞ 0 e2l2τ (1 + τ dM )2||F1(w)||2H1 dτ, Using Lemma 9.2 and the inequality |P21| ≤ const(1 + |ξ|s), s ≤ 2 we finally find ||∂kv1||2 ≤ const t(1 + tdM )2e−2l2t||φ||2αH3 . 296 V. V. Palin and E. V. Radkevich CUBO 12, 2 (2010) Futher ||∂kvj||2 ≤ CM t(1 + tdM )2e−2l2t ∫ +∞ 0 e2l2τ (1 + τ dM )2||∂k(G22(w)vj−1)||2dτ, Taking into account that G22 is a matrix polynomial and arguing as in Lemma 11.1, we obtain the required estimates. For the sake of brevity, we introduce the notation L1 = supξ maxλ∈Γ1 Re λ. Theorem 12.1. Let ( w z ) be a solution to the system (3.17) with the initial data w|t=0 = φ1, z|t=0 = P21φ1. Let φ ∈ H3, and let all the assumptions of Lemma 12.1 be satisfied. Denote by ( w0 z0 ) the solution to the linearized problem (3.17) with the same initial data. If α > L1 l1 , ||φ||H3 < κ ≪ 1, then the following estimate holds: ||eM11tF(w − w0)||2 ≤ 1 γ const(||φ||2αH2 + ||φ|| α+1 H3 ), where 0 < γ < min{ l2 − L1 2 , 2αl1 − 2L1}. Proof. We note that the Fourier images satisfy the equality ∂tF(w) + M11F(w) + M12F(z − P21w) = F(G11(w)w). Hence F(w) = e−M11t ( F(φ1) + ∫ t 0 eM11τ (F(G11(w)w) − M12F(z − P21w))dτ ) . Thus, ||eM11t(w − w0)|| ≤ || ∫ t 0 eM11τ F(G11(w)w)dτ|| + || ∫ t 0 eM11τ M12F(z − P21w)dτ||. Further, || ∫ t 0 eM11τ F(G11(w)w)dτ||2 ≤ const γ ∫ +∞ 0 e(2L1+γ)τ (1 + τ d)2||w||2αH2 dτ. Using Lemma 10.1, we find || ∫ t 0 eM11τ F(G11(w)w)dτ||2 ≤ const γ ∫ +∞ 0 e(2L1+γ−2αl1)τ (1+τ d)2(1+τ dΛ )2α(||φ||H2 +||φ||αH2 √ τ )2αdτ, By the conditions on α, γ, and φ, it follows that || ∫ t 0 eM11τ F(G11(w)w)dτ||2 ≤ constγ ||φ|| 2α H2 . Estimating the second term, we find || ∫ t 0 eM11τ M12F(z − P21w)dτ||2 ≤ ∫ +∞ 0 e−γτ dτ ∫ +∞ 0 e(2L1+γ)τ (1 + τ d)2||M12F(z − P21w)||2dτ. CUBO 12, 2 (2010) The Maxwell problem and the Chapman projection 297 We note that M12 = Λ12. Thus, ||M12F(z − P21w)|| = ||z − P21w||H1 . Using Lemma 14.1 and taking φ with sufficiently small H3-norm, we find || ∫ t 0 eM11τ M12F(z − P21w)dτ||2 ≤ const γ ||φ||α+1 H3 . which implies the required assertion. Remark 12.1. Applications of the obtained results to models of continuum mechanics can be found in [11, 13, 14, 16] Acknowledgments. Received: July 2009. Revised: August 2009. References [1] G. Q. Chen, C. D. Levermore, and T.-P. Lui, Hyperbolic conservation laws with stiff relax- ation terms and entropy, Commun. 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