Physics. A Boundary Value Problem for the Two Dimensional Broadwell Model* ^+f l f 2 =f 3 f\ f 1 (0,y) = φ 1 (y), (1.1) Communications in Mathematical

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1 Cmmun. Math. Phys. 114, (1988) Cmmunicatins in Mathematical Physics Springer-Verlagl988 A Bundary Value Prblem fr the Tw Dimensinal Bradwell Mdel* Carl Cercignani 1, Reinhard Illner 2 and Marvin Shinbrt Plitecnic di Milan, Milan, Italy University f Victria, Victria, Canada V8W 2Y2 Abstract. It is shwn that a certain bundary value prblem fr the steady twdimensinal Bradwell mdel n a rectangle has a slutin. The bundary cnditins specify the inging particle densities n each side f the rectangle. 1. Intrductin Very little is knwn abut bundary value prblems fr the Bltzmann equatin, even fr steady flws. The linearized equatin has been much studied [1, 2, 4-9, 13], and there are sme results fr nnlinear flws near t equilibrium, but that is all we knw f, in general. Recently, in an attempt t make further prgress n the prblem, we began t study bundary value prblems fr discrete velcity mdels f the Bltzmann equatin. In [11], we shwed that, in ne dimensin, the bundary value prblem assciated with discrete velcity prblems in a slab has slutins quite generally, althugh we were unable t prve any kind f uniqueness fr the slutins we fund. In [12], we extended the results f [11] t discrete velcity flws in a half-line. We btained the result, expected because f the physical analgy, that the slutin at infinity is a Maxwellian. Naturally, ne-dimensinal steady prblems are prblems invlving rdinary differential equatins. In tw dimensins, fr discrete velcity flws in a dmain, virtually nthing is knwn. In this paper, we present a nn-trivial example f the slutin f a bundary value prblem assciated with a natural 4-velcity mdel in a rectangle. This is the first example we knw f such a result. The mdel is easily described. We slve the fllwing prblem in the rectangle R = [0,ά]x[0,b]: ^+f l f 2 =f 3 f\ f 1 (0,y) = φ 1 (y), (1.1) * Research supprted by the Natural Science and Engineering Research Cuncil Canada under Grants A7847 and A8560

2 C. Cercignani, R. Illner, and M. Shinbrt -^+fψ=pp, P(a,y) = φ 2 (y), (1.2) δf3 +PP=PP, dy P(x,0) = φ 3 (x), (1.3) dp +PP=PP, p{x,b) = φ4 {x). (1.4) The underlying time-dependent mdel assciated with (1.1-4) is well knwn as the (tw-dimensinal) Bradwell mdel [13,15]. This is ne f the simplest mdels fr which the glbal existence prblem fr the Cauchy prblem is unslved fr large data 1 (see [16]). In spite f this, the slutin f the bundary value prblem (1.1-4) that we present here is, like the results f [11] and [12], cmpletely glbal, in n way depending n the lengths f the intervals (0, a) and (0, b), r the size f the data φ 1, φ 2, φ 3, r φ 4. A piece f ntatin befre we start. We dente the cntinuus functins n a set S by C (S); the cntinuus, nn-negative functins n S are dented by C + (S). The nn-negative functins with ne cntinuus derivative n S are dented by Cl^S). We dente the maximum nrm in C (S) by. 2. A Bundary Value Prblem fr Sme Ordinary Differential Equatins We begin this sectin by studying the fllwing bundary value prblem invlving nly rdinary differential peratrs: δ L- + f^p = h i, / 1 (0) = φ 1, (2.1) x where the functins h, as well as the bundary data φ are given. We prve Lemma 2.1. Let φ\φ 2 eέli, (h\h 2 )e{c + [0,a]} 2 :=C + l0,a']x C + l0,a2. Then, the prblem (2.1)-(2.2) has a unique slutin f=(f\f 2 )e{c ί +[O,a~]} 2. Prf. T slve (2.1) (2.2), we use the methd f Kaniel and Shinbrt [17], which is well suited t prblems f this type. We begin by defining fur sequences, {/*}, {u ι n}, {I 2 }, and {u 2 }, as fllws. Take /J = O = /Q, and let u$ and ul be the slutins f 1 Althugh the crrespnding ne-dimensinal Cauchy prblem has been slved [3]

3 Bundary Value Prblem fr Tw Dimensinal Bradwell Mdel 689 Next, with Z*_ 1? M*_ l5 Z 2 _ 1? and u\_ γ given, we define Z*, u\, Z 2, and w 2 as the slutins f the initial value prblems dx iλ _L/ 2 I, 1 h 1 (2.3) and U _ ~ U L /2 L2 (2.4) A straightfrward inductin shws that Thus, the sequences {Z^} and {w^} are mntne and bunded, as are the sequences {11} and {ul}. All fur are therefre cnvergent fr each x e [0, a]. Let {ξ} cnverge t Z ι, {wjj} t u\ i= 1,2. Integrating (2.3) and (2.4), we can send n t infinity in the result t find that u\x)+]l\ξ)u\ξ)dξ = φ ι +] u 2 (x) + ί I\ξ)u 2 2 (ξ)dξ = Ψ +] h 2 (ξ)dξ, X I\x)+}u\ξ)l 2 {ξ)dξ = φ 2 +}h 2 (ξ)dξ. X The functins Z 1, Z 2, w 1, w 2 are bunded, and these frmulas shw that they are abslutely cntinuus in x, and they satisfy X X dx' d (2.5) dx' die l2 -uψ=-h 2, (2.6)

4 690 C. Cercignani, R. Illner, and M. Shinbrt Thus, subtracting (2.5b) frm (2.5a), and (2.6b) frm (2.6a), we find { 1 and 5 2 /2-1 I (U / (2.Ϊ Thus, Jχ (U ~ '~Ix Sϊnce(u ί -l ι )(0) = 0, then, {u 1 -l 1 )(x) = {u 2 -l 2 ){x)-(u 2 ~l 2 ){0). Setting x = a and using the fact that (u 2 I 2 ) (a) = 0, we find (u ί -l 1 )(a)=-(u 2 -l 2 )(0), which is impssible unless bth sides are zer. Thus, we have with bth sides being zer when x = 0 OΪ x = a. Equatin (2.7) nw gives Since u ι l ι =0 when x = 0, this means that u 1 = l 1. Similarly, we find u 2 = l 2. Writing f 1 fr the cmmn value f I 1 and w 1, and f 2 fr I 2 and u 2, we find frm (2.5) that (/\/ 2 ) is a slutin f (2.1) and (2.2). Since (f\f 2 ) satisfy f\x)+ ]f 2 (ξ)f 1 (ξ)dξ = φ 1 +] p{x)+ ]f 2 (ξ)f\ξ)dξ = ψ 2 + ]h 2 (ξ)dξ, they are abslutely cntinuus, and these equatins can be differentiated t prduce (2.1) and (2.2). This shws that Eq. (2.1) (2.2), integrated with respect t x, have nn-negative slutins / 1,/ 2 GL OO (0,a). Hwever, integratin f (2.1) (2.2) gives a representatin f/ 1 and f 2 as integrals; thus, f ι and f 2 are abslutely cntinuus. This fact used in (2.1) (2.2) again shws that (/ 1,/ 2 )e{c 1 +[0, α]} 2. This cmpletes the prf f existence. As fr uniqueness, ntice that any slutin f (2.1)-(2.2) in {C + [0, a]} 2 satisfies 0 = Z(x)^/ ί (x)^«(x), i=l,2. An inductin then shws that f^x)^f\x)^n{x\ 1 = l,2, ft = 0,1, Since the sequences {l ι } and n {uι n } cnverge t the slutin cnstructed abve, it fllws that any slutin is equal t the cnstructed slutin.

5 Bundary Value Prblem fr Tw Dimensinal Bradwell Mdel 691 We nw shw that the slutin f (2.1-2) delivered by Lemma 2.1 depends cntinuusly n the data. Fr this, let (/\ f 2 ) satisfy (2.1-2), and let (F\ F 2 ) be the slutin f the same prblem with different data, namely We prve dx _df^_ dx < 2 = H\ F 1 {0) = Φ\ (2.9) 7 2 = H 2, F 2 (a) = Φ 2. (2.10) Lemma 2.2. Let (f\f 2 ) dente the slutin f (2.1)-(2.2), (F\F 2 ) the slutin f (2.9)-(2.10). Take ε>0. Then, there exists a δ>0 such that if ll^-^ll + ll^-^ll+^-^l + IΦ 2 -^ 2!^. (2.11) ε can be chsen t depend nly n δ, a, and the quantity Prf. Let We nte first that the functins f 1, / 2, F 1, F 2 are all unifrmly bunded, belw by 0 and abve by the crrespnding functin u ι (see the prf f Lemma 2.1), which 0 depends nly n c and a. Define g 1 =F ι f 1, g 2 = F 2 f 2. g 1 and g 2 satisfy *~ +F 2 g 1 +f ί g 2 = η 1 9 g\0) = ψ1 9 (2.12) dx -^+F 2 g 1 +/ 1 g 2 = /?2, g^o)^^2, (2.13) dx where n ι =ll ι -h γ, η 2 = H 2 -h 2, ψ ι = Φ 1 -φ\ ψ 2 = Φ 2 -φ 2. Subtracting (2.12) frm (2.13) and integrating the result shws that there is a cnstant c 1 such that ], (2.14) 0 say. Slving (2.14) fr g 2 and substituting int (2.12), we see that ( ) is equivalent t the prblem :=η. (2.15) where g'(0) = φ 1 and c γ must be chsen such that g 1 (α) = c 1 ψ 2 + η(a). Frm (2.15), we btain

6 692 C. Cercignani, R. Illner, and M. Shinbrt g 1 (a) = ψ 1^ +fwx)-c 1 / 1 (x)]«- dx. [ The cnditin g 1 (ά) = c 1 ιp 2 a l + j 0 + η(a) results in the equatin f// i_ F 2) Since f ί (x)^0, J we read ff immediately that \a)\ + \\η(x)\e* 0 if (2.11) is satisfied. Here, c 2 is a cnstant depending nly n c and a. Frm (2.14), it fllws that where c 3 is anther cnstant depending nly n c and a. Since g 1 satisfies the lemma fllws. Π Definitin 23. Fr ρ > 0, let B dente the ball f radius ρ in C {R), B the set f all + nn-negative elements f B : Let C 01 (R) be the set f all functins cntinuus in x and differentiable in y, nrmed by Λ Λ I denting the nrm in C (R), as befre. Similarly, we define C 10 {R) as the set f functins differentiable in x and cntinuus in y, with the nrm We dente the balls f radius ρ in C ι {R) and C 10 ι 10 (K) by B ρ and B ρ, respectively. + JBJ 1 ι and +JB* dente the nn-negative functins in B ρ and B]. In Eqs. (2.1) and (2.2), we nw allw the functins h as well as the bundary data φ t depend n y, and we prve Lemma 2.4 Let (φ\ φ 2 )e {C + [0,6]} 2, (/z 1, /z 2 )e [C + (Λ)] 2. Γ/zβπ, ^s. (2.1-2) have a unique slutin (/ 1,/ 2 )e[c c [(#)] 2. With the bundary data {φ\φ 2 ) fixed, let S ρ dente the set f all slutins f (2.1-2) with (h\h 2 )e +B ρ K Then, the clsure f S ρ is a cmpact subset f [C+(R)] 2.

7 Bundary Value Prblem fr Tw Dimensinal Bradwell Mdel 693 Prf. Applying Lemma 2.1 with y e [0, b'] fixed, we find that (f 1, f 2 ) exists fr each y, and that {f\,y\ f 2 {-,y))e{c\[0,d\} 2. The cntinuity f the pair C/ 1,/ 2 ) fllws frm Lemma 2.2 and the assumed cntinuity f the data. Fr the cmpactness, we shw that the set S ρ is equicntinuus. By Lemma 2.1, f 1 and f 2 are bunded. Equatins (2.1) and (2.2) give, then, that the derivatives df 1 /dx and df 2 /dx are bunded. Lemma 2.2 shws that the mduli f cntinuity f f 1 and f 2 in the y-directin depend n the mduli f cntinuity f φ ι, φ 2, h 1, and h 2 in the y-directin. φ 1 and φ 2 are fixed, by hypthesis, while the mduli f cntinuity f h 1 and h 2 are cntrlled by their membership in B 1. Thus, the functins f 1 and f 2 vary ver equicntinuus subsets f C + {R) as h 1 and h 2 vary ver +Bg 1. This cmpletes the prf f the lemma. One has merely t make the lexicgraphic change f replacing x by y and the indices 1 and 2 by 3 and 4 t prve Lemma 2.5. Let (φ\ φ 4 )e {C [0,a]} 2, (/z 3,/2 4 )e[c (#)] 2. Then, the equatins ^+/ 3 / 4 = Λ 3, f\0) = φ\ (2.16) dy have a unique slutin (/ 3, / 4 ) e [C+(i^)] 2. With the bundary data (φ 3, φ 4 ) fixed, let S' ρ dente the set f all slutins f ( ) vwf/z (/ι 3, /ι 4 ) e + βρ 1. T/ie^z, ί/ze c/θ5t/re / 2 S ^ i s a cmpact subset f [ C 3. The Operatr T and Sme f its Prperties We nw prceed t cnsider (1.1) (1.4). Fr this, we assume nce and fr all that (φ 1, φ 2, φ 3, φ 4 ) is fixed, satisfying the hyptheses f Lemmas 2.4 and 2.5. We establish a mapping frm the cne [C+(JR)] 4 int itself, as fllws. Let (g\g 2,g\g 4 )elc + (R)Y. Slve the equatins fj+/ 1 / 2 =*V, f 1 (^y)=φ 1 (yh (3 i) - ^-+/ 2 / 2 =g 3 g 4, f 2 {a,y) = φ 2 (y), (3.2) f^ +fψ = gv, / 3 (*, 0) = φ\x), (3.3) - < ^ + / 3 / 4 = g 1 g 2, /V^) = <P 4 W- (3-4) dy We prve Lemma 3.1. Lei (g 1,g 2,g 3,g 4 )e[c < i(i?)] 4. TTien, <js. (3.1)-(3.4) Ziαi e a slutin

8 694 C. Cercignani, R. Illner, and M. Shinbrt Prf. Equatins (3.1) (3.2) are subject t Lemma 2.4, Eqs. (3.3) (3.4) t Lemma 2.5. The result fllws immediately. Definitin 3.2. As the prf f Lemma 3.1 shws, the pair f Eqs. (3.1) (3.2) can be slved independently f the pair (3.3) (3.4). Thus, slving (3.1) (3.2) defines a mapping f the pair (g 3,g 4 ) int the slutin pair C/* 1,/ 2 ). We write (/\/ 2 ) = T x (g 3, g 4 ); the subscript x is used t indicate that T x is smthing with respect t the variable x (since the image f [C+(.R)] 2 under T x is cntained in [C+ CR)] 2 ) In a similar way, and fr similar reasns, we write (/ 3,/ 4 ) = XJ,(g\ g 2 ) if (/ 3,/ 4 ) is the slutin f (3.3)-(3.4). Finally, we write T{g\ g 2, g 3, g 4 ) = (/\ / 2,/ 3,/ 4 ) fr the full slutin f (3.1) (3.4). Any slutin f (1.1) (1.4) is a fixed pint f T. We begin ur study f the peratr T with Lemma 3.3. The peratr T 2 is cmpact. Prf We have, in the bvius ntatin, Accrdingly, T 2 g = (T x T y {g\g 2 \ T y T x (g\g 4 )). (3.5) We shw that T x T y is cmpact n [C (#)] 2. T y maps [C%(R)~] 2 int [C 1 + (R)~] 2. Accrdingly, if (g\g 2 ) varies ver a bunded subset f [C + (,R)] 2, all the images T y (g\ g 2 ) He in a ball [ + B 1 '] 2 fr sme ρ > 0. Lemma 2.4 therefre shws that 7;7; 9 is cmpact. A similar argument, using Lemma 2.5, shws that T y T x is cmpact als, and the result fllws frm this. Next, we prve Lemma 3.3. The peratr T 2 has a fixed pint in [C + (JR)] 4. Prf. Accrding t the therem f Schaefer 2 [18,19], we have t shw that any slutin ϊf=λt 2 f with 0 < λ < 1 is bunded. Suppse /= λt 2 f Then, (3.5) gives and (f\f 2 ) = λt x T y (f\f 2 ) (3.6) We shw first that f 1 is bunded. Let Then, accrding t (3.6), (g 3,g 4 )= 2 Schaefer's therem refers t a mapping f an entire Banach space int itself. Hwever, the prf depends n a retractin f the peratr under cnsideratin. As a cnsequence, it is easy t extend the prf t apply t a mapping n a cne, like [C + (R)] 2

9 Bundary Value Prblem fr Tw Dimensinal Bradwell Mdel 695 The definitins f T x and T y give =λg 3 g 4 f 1 / 2, f ι (0,y) = λφ 1 (y), (3.7) x λ ~ = \f l f 2 - λg 3 g\f 2 (a,y) = λ Ψ 2 (y), (3.8) Therefre, δg 3,, w 4-=fψ-gY, g 3 (x,0) = φ 3 (x), (3.9) ^-= g 3g+_/l/2 g4( X> i,)=φ4( χ)- (3. 10 ) gθ, (3.11) since 0<l<l, while the functins/ and g are all nn-negative. Integrating (3.11) ver R and using the divergence therem, we find I (Γ-P)dy- I (f 1 ~f 2 )dy+ f ( g 3 -g 4 )^- ί (g 3 - x a x=0 y b y = 0 Using the bundary cnditins n f 1, f 2, g 3, and g 4, we find ί f ι dy+ j / 2 dy+ ί g"dx+ j g 4^x x = α JC = O y = b y = 0 ^ j λφ ί (y)dy+ J 2φ 2 (3/)Jj;+ J φ 3 rfx+ J ώc x=0 x=α y=0 y=b (3.12) again using the fact that 0 < λ < 1. Here, s dentes arclength n dr, and φ: 5^^^+ is φ 1 n the left side f R, φ 2 n the right, φ 3 n the bttm and φ 4 n the tp. Equatin (3.12) gives L 1 bunds n the functins f 1, f 2, g 3, g 4 n the part f the bundary f R ppsite t that f the data. The inequality (3.12) fllws frm mass cnservatin. We nw return t (3.7) (3.10) and use mmentum cnservatin in the frm (3.13) Equatin (3.13) shws that f 1 +f 2 is a functin f y alne and g 3 + g 4 is a functin f x alne. Nw, chse xe[0,α] arbitrarily and chse an interval /C[0, b]. Integrating (3.13a) ver [0,x] x /, we find =$l<p l (y)+f 2 (0,y) y I (3.14) ^2 j φds, (3.15) er

10 696 C. Cercignani, R. Illner, and M. Shinbrt by (3.12). Let Γ be any psitive cnstant. Since the left side f (3.14) is independent f x, it fllws that we can partitin the interval [0,fr] by w + 1 pints 0 = y <y ί <,.,y n = b, als independent f x, in such a way that and n ^ 2Γ J φds + 1. ~ψ, (3.16) The same argument shws that the interval [0, a] can be partitined by m + 1 pints 0 = x <x ί <... <x m = a such that fr any where m^2γ J φds+1. We nw use these estimates t btain pintwise bunds n f ι. Frm (3.7), we have Next, we use (3.9) t estimate g 3, and insert the result in this last inequality. We find f\x, y) ύ V(3>) + λ J g 4 (σ, y)φ\σ)dσ + λ] ]g\σ,y)(pf 2 )(σ,τ)dτdσ. (3.18) Let 0 0 K(x) = We shw that K(x) is bunded by a cnstant depending nly n the data. In the rest f the prf, we reserve the letter c t dente such a cnstant; c may have different values in different frmulas. As λ<l, we have λφ 1 (y), λφ 3^c. By (3.17), then, Als, (3.14) shws that λ]g 4 (σ,y)φ 3 (σ)dσ^c. ]f 2 (σ,τ)dτ^c fr all σ. We apply the definitin f K and these estimates t the right side f (3.18) t find (3.19) Nw, take Γ equal t this last cnstant c, and chse the partitin in (3.17) crrespnding t this value f Γ. We see then that if O^xgx,,

11 Bundary Value Prblem fr Tw Dimensinal Bradwell Mdel 697 and frm (3.19), we cnclude that K that is, This is a bund fr f 1 n the rectangle [0, xjx [0, b]. Repeating the argument in the rectangle [^i,x 2 ] x [0, b], we find ( ) ^, fr 1 ^ ^ 2, and, inductively, we prve K(x)^2 m Γ fr O^x^α. Since m depends nly n Γ and Γ depends nly n the data, the estimate n f 1 is cmplete. / 2, g 3, and g 4 are estimated in a similar way. Schaefer's therem thus implies the result. 4. The Main Result A slutin f the prblem (1.1) (1.4) is a fixed pint f T. We shw in this sectin that such a fixed pint exists. Therem 4.1. Let (φ\φ 2 )e{c + [0,b]} 2, (φ 3,φ 4 )e{c + [0,α]} 2. Then, the prblem (1.1)-(1.4) has a slutin f=(f\λ/ 3,/ 4 )e[c + ( )] 4. Prf. Let h = (h 1,h 2,h 3,h 4 ') be the fixed pint f T 2 guaranteed by Lemma 3.3. Then, we have (h\h 2 )=T x T y (h\h 2 ). Set Then, (g\g*)=t y (h\h 2 ). (h\h 2 )=T x (g\g 4 ). Since T(f\f 2 J 3 J 4 ) = (T x (f 3 J 4 \ T y {f\f 2 )\ as we saw in Lemma 3.1, the functin /=(h ι, h 2, g 3, g 4 ) is a fixed pint f T and a slutin f the prblem. References 1. Arthur, M.D., Cercignani, C: Nnexistence f a steady rarefied supersnic flw in a halfspace. Z. Angew. Math. Phys. 31, 634 (1980) 2. Bards, C, Caίlisch, R.E., Niclaenk, B.: The Milne and Kramers prblems fr the Bltzmann equatin f a hard sphere gas. Cmmun. Pure Appl. Math. 39, 322 (1986) 3. Beale, J.T.: Large-time behavir f the Bradwell mdel f a discrete velcity gas. Cmmun. Math. Phys. 102, (1985)

12 698 C. Cercignani, R. Illner, and M. Shinbrt 4. Beals, R.: An abstract treatment f sme frward-backward prblems f transprt and scattering. J. Funct. Anal. 34, 1 (1979) 5. Cercignani, C: Mathematical methds in kinetic thery. New Yrk: Plenum Press Cercignani, C: Thery and applicatin f the Bltzmann equatin. New Yrk: Elsevier Cercignani, G: Elementary slutins f the linearized gas dynamics Bltzmann equatin and their applicatin t the slip flw prblem. Ann. Phys. (NY) 20, 219 (1962) 8. Cercignani, C: Existence and uniqueness in the large fr bundary value prblems in kinetic thery. J. Math. Phys. 8, (1967) 9. Cercignani, C: On the general slutin f the steady linearized Bltzmann equatin. In: Rarefied gas dynamics. Becker, M., Fiebig, M. (eds.). Prz-Wahn: DFVLR Press Cercignani, C: Half-space prblems in the kinetic thery f gases. In: Trends in applicatins f pure mathematics t mechanics. Krner, E., Kirchgasser, K. (eds.), Lecture Ntes in Physics, Vl. 249, p. 35. Berlin, Heidelberg, New Yrk: Springer Cercignani, C, Illner, R., Shinbrt, M.: A bundary value prblem fr discrete velcity mdels. Duke Math. J. (t appear) 12. Cercignani, C, Illner, R., Pulvirenti, M., Shinbrt, M.: On nnlinear statinary half-space prblems in discrete kinetic thery (t appear) 13. Gatignl, R.: Therie Cinetique des gaz a repartitin discrete des vitesses. Lecture Ntes in Physics, Vl. 36. Berlin, Heidelberg, New Yrk: Springer Greenberg, W., van der Mee, C: An abstract apprach t evapratin mdels in rarefied gas dynamics. Z. Angew. Math. Phys. 35, 166 (1984) 15. Illner, R.: Glbal existence results fr discrete velcity mdels f the Bltzmann equatin in several dimensins. J. Mec. Ther. Appl. 1, (1982) 16. Illner, R.: Examples f nn-bunded slutins in discrete kinetic thery. J. Mec. Ther. Appl. 5, (1986) 17. Kaniel, S., Shinbrt, M.: The Bltzmann equatin. I. Uniqueness and lcal existence. Cmmun. Math. Phys. 58, (1978) 18. Schaefer, H.: Uber die Methde der a priri Schranken. Math. Ann. 129, (1955) 19. Smart, D.R.: Fixed pint therems. New Yrk: Cambridge University Press 1974 Cmmunicated by J. L. Lebwitz Received July 14, 1987