Colombeau's theory and shock wave solutions for systems of PDEs 

Electronic Journal of Differential Equations, Vol. 2000(2000), No. 21, pp. 1{17. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login: ftp) Colombeau’s theory and shock wave solutions for systems of PDEs F. Villarreal Abstract In this article we study the existence of shock wave solutions for sys-tems of partial dierential equations of hydrodynamics with viscosity in one space dimension in the context of Colombeau’s theory of generalized functions. This study uses the equality in the strict sense and the associa-tion of generalized functions (that is the weak equality). The shock wave solutions are given in terms of generalized functions that have the classical Heaviside step function as macroscopic aspect. This means that solutions are sought in the form of sequences of regularizations to the Heaviside function that have to satisfy part of the equations in the strict sense and part of the equations in the sense of association. Introduction Let R+ := R+ [f+1g. Fix (;) in R+ R+ with < . Let be a function in C1 (R )3;[;[ satisfying some conditions to be introduced in x5. We consider two associated systems of hydrodynamic equations with viscosity in one space dimension. The system (S) consists of the equations t + (u)x 0 (u)t + (p+ u2)x f[ (;p;e) −]uxgx et + [(e+ p)u]x f[ (;p;e) −]uuxgx e p+ 1 u2 ; 2 R and (S) consists of the two last equations and t + (u)x = 0 (u)t + (p+ u2)x = f[ (;p;e) −]uxgx where is the density, u the velocity, p the pressure and e the total energy. The symbol denotes the association relation in Gs(R2;R) (see x2). The purpose of this paper is to study the existence of shock wave solutions (see x5) for the sys-tems (S) and (S). More precisely, solutions with two constant states separated Mathematics Subject Classications: 46F99, 35G20. Key words and phrases: Shock wave solution, Generalized function, Distribution. 2000 Southwest Texas State University and University of North Texas. Submitted January 13, 2000. Published March 12, 2000. 1 2 Colombeau’s theory and shock wave solutions EJDE{2000/21 by a jump along a straight line. We know that, in theory of distributions, there exists a unique (Heaviside) distribution Y such that YjR− = 0, YjR+ = 1 and whose derivative Y is the Dirac distribution. The shock wave solutions for the system (S) are given in terms of generalized functions that have the function Y as macroscopic aspect (see [1] or [3]). These functions are called Heaviside generalized functions in R. Also, recall that, for every Heaviside generalized function H we have HjR 0, HjR 1 and the derivative H0 is associated with all generalized function that have the Dirac distribution as macroscopic aspect. The main results of this work are theorems 5.2 and 5.3. We will briefly describe the content of this paper. Generally speaking we can arm that in the rst four sections we collect the results to be used in the last one. In this work, unless otherwise stated, E;F1;:::;Fm and G denote K− Banach spaces (where K denotes either R or C). F denotes a K− Banach algebra (as a K− Banach space) and Ω (resp. Ω0) denotes an open subset of E (resp. F). In x1 we x some basic denitions about the space of simplied generalized func- tions Gs(Ω;F). In x2 we introduce the association relation in Gs(Ω;K) (when E = Rn) and we present some basic properties about the Heaviside generalized functions, although we omit well known proofs. In x3 we introduce the notion of composite function and we present a result about inverse multiplicative, for a certain class of generalized functions (adequate to the requirements of this work). This composition (;p;e) is \very delicate", it is a special case on composition of generalized functions. We need the inverse multiplicative when we study the system (S). In x4 we discuss properties of functions of the form ’(a1H1+b1;:::;amHm+bm) where H1;:::;Hm are Heaviside functions under certain conditions. The propositions 4.2, 4.3 and 4.4 are fundamental for the study of existence of shock wave solutions for the systems (S) and (S). In x5 we study the initially proposed problem. We show that a necessary and sucient condition in order that the system (S) has a solution (;u;p;e) is that the jump conditions and two relations of technical nature are held (see theorem 5.1). By using the proposition 4.3 and the theorem 5.1 we get a result on existence of solutions. According to which the system (S) has shock wave solutions if and only if the jump conditions hold (see theorem 5.2). Also we exhibit that the rst two equations of the system (S) can not have shock wave solutions, which takes us to a result on nonexistence of solutions (see theorem 5.3). In [2] it was studied an analogous problem for systems when the viscosity function depends only on . More precisely, is a strictly increasing function (under certain conditions) when the system is considered with all equations in the sense of association and (x) = x2 when we consider the equality in the rst equation. This work constitutes a considerable advancement of the results contained in [2]. In the present paper an additional complication arises from the expression (;p;e), which requires the hard study of composition and inverse multiplicative of generalized functions in the sense of Colombeau’s theory. A considerable amount of technical computations is developed to deal with the proposed problem. The basic references for Colombeau’s theory are [1], [3], [4] and [5]. The general notations not mentioned in this work are those of [1]. EJDE{2000/21 F. Villarreal 3 1 The algebra Gs(Ω;F) We denote by (the same symbol) j j the norms in the considered spaces. The symbol L(F1;:::;Fm;G) denotes the space of continuous m - linear mappings from product space Fm := F1 Fm into G endowed with the norm j j :A 2 L(F1;:::;Fm;G) ! sup jA(y1;:::;ym)j 2 R+ : j j=1 1im If F1 = = Fm = F this space is denoted by L(mF;G) and L(0F;G) =: G. Let Es[Ω;F] := fu 2 F]0;1]Ω j u(";) 2 C1(Ω;F) for all " 2]0;1]g be. If p 2 N and x 2 Ω we set u(p)(";x) := [u(";)](p)(x). The notation K Ω means that K is a compact subset of Ω and ju(p)(";)jp;K := supx2K ju(p)(";x)jp. Let Es;M[Ω;F] denote the algebra of all u 2 Es[Ω;F] such that for each K Ω and each p 2 N there is N 2 N such that ju(p)(";)jp;K = o("−N) as " # 0. By Ns[Ω;F] we denote the set of all u 2 Es[Ω;F] such that for each K Ω and each (p;q) 2 N N we have ju(p)(";)jp;K = o("q) as " # 0. The Colombeau algebra of generalized mappings on Ω with values in F is dened by Gs(Ω;F) := Ns[Ω;F]] : If F = K we write Gs(Ω) instead of Gs(Ω;K) and a similar notation is used for sets that generate (as well as for subsets of) Gs(Ω;K). We indicate by Gs;‘b(Ω;F) the set of maps f 2 Gs(Ω;F) which have a representative f such that for each K Ω there are C > 0 and 2]0;1] satisfying jf(";x)j C for all (";x) in ]0;[K. If fi 2 Gs(Ω;Fi);1 i m, we denote by (f1;:::;fm) the class of (f1;:::;fm):(";x) 2]0;1] Ω ! f1(";x);:::;fm(";x) 2 Fm where fi is an arbitrary representative of fi. In this case, f1;:::;fm are called the components of (f1;:::;fm). We denote by Es;M[Ω;Ω0] the set of all u in Es;M[Ω;F] such that u(]0;1] Ω) Ω0. By Es;M;[Ω;Ω0] we denote the set of all u 2 Es;M[Ω;Ω0] such that for each K Ω there are K0 Ω0 and 2]0;1] such that u(]0;[K) K0. We indicate by Gs;(Ω;Ω0) the set of all elements of Gs(Ω;F) which have at least a representative in Es;M;[Ω;Ω0]. If (u;w) 2 Es[Ω;Ω0] Es[Ω0;G], let w u 2 Es[Ω;G] be dened by (w u)(";x) := w(";u(";x)); ((";x) 2]0;1] Ω): If dimF < +1 and (f;g) 2 Gs;(Ω;Ω0) Gs(Ω0;G) we dene the composite function g f := bg f + Ns[Ω;G] where f 2 Es;M[Ω;Ω0] and gb are arbitrary representatives of f and g, respec-tively. 4 Colombeau’s theory and shock wave solutions EJDE{2000/21 Let Es;M(F) be the set of all 2 F]0;1] such that there is N 2 N satisfying j(")j = o("−N) as " # 0 and let Ns(F) be the set of all functions 2 Es;M(F) such that for each q 2 N we have j(")j = o("q) as " # 0. The algebra of the Colombeau generalized vectors in F is dened by Es;M(F) Ns(F) We can identify F with a subspace of Fs and Fswith a subspace of Gs(Ω;F). The elements of the image of Fs in Gs(Ω;F) are called generalized constants. 2 Association and Heaviside GFs In this section will be considered the cases E = Rn and F = K. We say that an element f in Gs(Ω) is associated with 0 (indicated by f 0) if for some representative f of f we have f(";) ! 0 in D0(Ω) as " # 0. We say that two elements f and g in Gs(Ω) are associated with each other if f −g 0. The next result follows from dominated convergence theorem. Proposition 2.1 Let (;f) 2 KΩ Gs;‘b(Ω) be such that f(";) ! a.e. in Ω as " # 0 for some representative f of f. Then, f(";) ! in D0(Ω) as " # 0. In particular, if 2 C1(Ω) then f . We indicate by Y the classical Heaviside step function: Y() = 0 if < 0 and Y() = 1 if > 0. An element H 2 Gs(R) is said to be a Heaviside generalized function in R if there is a representative H of H such that H(";) ! Y in D0(R) as " # 0. We indicate by H(R) the set of all Heaviside generalized functions in R. We denote by Hp(R) the set of all elements H in Gs;‘b(R) which have a representative H such that H(";) ! Y in R as " # 0. We denote by Z := f’ 2 D(R) j ’ 0; ’(0) > 0; supp(’) [−1;1] and ’ = 1g: Lemma 2.1 There exists u 2 Es;M[R] such that 0 u(";) 1 in R, u(";) 1 in ] − "; "[ and supp[u(";)] [−3"; 3"], for all " 2]0;1]. Then, if v is dened by v := 1 in ]0;1] R− and v := u in ]0;1] R+, we have 0 v(";) 1 in R, supp[v(";)] ] −1;"[ and v(";) 1 in ] −1; 4[, for all " 2]0;1]. Proof If ’ 2 the function u:(";x) ! [(";) ’b(";)](x) satises the re-quired properties, where ’b:(";x) ! "−1’("−1x) and (";) is the characteristic function of [−"; "]. We say that a function H 2 Es;M[R] veries the property (Hr) if there is 2 R ]0;1] such that lim"#0 (") = 0 and H(";) = 0 (resp. H(";) = 1) for < −(") (resp. > (")), (" 2]0;1]). We indicate by Hr(R) the set of all elements of Gs;‘b(R) which have a representative verifying the property (Hr). EJDE{2000/21 F. Villarreal 5 Proposition 2.2 Let 2 Es;M(R) be such that 1 in ]0;1]. Then, there exists V 2 Es;M[R;R] verifying the following properties: (Hr), 0 V in R]0;1], V (";) (") in [−";"] for all " 2]0;1], and for each K R there is 2]0;1] such that 0 V 1 in ]0;[K. Proof If ’ 2 we consider the function u:(";x) ! [(";) ’b(";)](x) where ’b:(";x) ! "−1’("−1x) and (";) is the characteristicfunction of [−3"; 3"]. The function V : (";x) ! 1+v(";x)[(")u(";x)−1] satises the required properties (where v is as in lemma 2.1). Proposition 2.3 If ’ 2 and H’:]0;1] R ! R is dened by Z H’(";) := ’ dt; ((";) 2]0;1] R) −1 we have 0 H’ 1 in ]0;1] R and H’(";) = 0( resp. H’(";) = 1) for −"( resp. "), for all " 2]0;1]. Furthermore, if H’ is the class of H’ then H’ 2 Hr(R). Remark 2.1 Denoting by H(R) := fH’ 2 Gs(R) j ’ 2 g, where H’ is dened as in previous proposition, we have H(R) Hr(R) Hp(R) H(R): Remark 2.2 If H;K 2 Hp(R), we do not necessarily have HK0jR 0. Indeed, if v as in lemma 2.1, we consider H and K represented respectively by h i h i H:(";x) ! 1+v(";x) "4 cos(") −1 ; K:(";x) ! 1+v(";x) "sen(") −1 : The presented result as follows is a useful tool for the study of solvability of the systems (S) and (S). Proposition 2.4 If f;g 2 Gs(Rn) and S is a C1 - dieomorphism of Rn onto itself such that JS(x) > 0 for all x 2 Rn (JS denotes the jacobian of S) the following statements are held. (a) (gS)jΩ 0(resp. (gS)jΩ = 0) if and only if gjS(Ω) 0(resp. gjS(Ω) = 0) and (f S−1)jS(Ω) 0(resp. (f S−1)jS(Ω) = 0) if and only if fjΩ 0 (resp. fjΩ = 0). (b) If W is an open subset of Rn and :(;t) 2 Rn Rm ! 2 Rn then (f )jWRm 0 if and only if fjW 0 and (f )jWRm = 0 if and only if fjW = 0. Proof The result follows by a minor modication in the proof of the Proposition 2.11 in [2]. ... - tailieumienphi.vn