Acta Mathematicae Applicatae Sinica, 19(2003), A global solution to a two-dimensional Riemann problem involving shocks as free boundaries

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1 Acta Mathematicae Alicatae Sinica, 19(003), A global solution to a two-dimensional Riemann roblem involving shocks as free boundaries Yuxi Zheng 1 Deartment of Mathematics The Pennsylvania State University University Park, PA 1680 Abstract We resent a global solution to a Riemann roblem for the ressure gradient system of equations. The Riemann roblem has initially two shock waves and two contact discontinuities. The angle between the two shock waves is set initially to be close to 180 degrees. The solution has a shock wave that is usually regarded as a free boundary in the self-similar variable lane. Our main contribution in methodology is handling the tangential oblique derivative boundary values. Keywords: Pressure gradient equation, free boundary, shock waves, tangential oblique derivative, -D Riemann roblem, gas dynamics. AMS subject classification: Primary: 35L65, 35J70, 35R35; Secondary: 35J65. 1 Introduction Recent imortant work of Canic, Keyfitz, Lieberman, and Kim on the roblem of existence of solutions to the transonic small disturbance system of conservation laws (TSD) [5][6][7] romise further develoment. In this aer, we try their aroach on the ressure gradient system of equations. The advantage of this system is that it has similar roblems as the -D steady or unsteady TSD equations but with bounded subsonic domains instead of unbounded subsonic domains. We are able to carry their aroach further to finally obtain global solutions. The new ingredient is the handling of loss of uniform obliqueness. 1 Research artially suorted by NSF-DMS , ,

2 We refer the reader to references Zheng [30], Dai and Zhang [11], Song [7], Kim and Song [16], Zheng s book [31], and Li, Zhang and Yang s book [17] for background information and recent rogress regarding the ressure gradient system. For recent related work on multi-dimensional shock free boundaries, we refer to Chen and Feldman [8][9] and Shuxing Chen [10] in addition to the work [5][6][7]. The roof of the result is based heavily on the aforementioned work [5][6][7]. A full and self-contained resentation of the roof will be lengthy and reetitive. We resent in this write-u the result and the ortion of the roof that is needed in addition to the work done in [5][6][7]. We aologize to the nonexert readers that the resentation is not self-contained. The gist The ressure gradient system takes the form u t + x = 0 v t + y = 0 E t + (u) x + (v) y = 0, (1) where E = (u +v )/+. Again, we refer the reader to the books [31][17] for background information. The system aeared first in Agarwal and Halt [1]. Roughly seaking, this system is the Euler system in gas dynamics without the inertial arts. y 1 θ 1 x 4 θ 3 Figure 1. The four sectors in the initial data. The Riemann roblem for (1) is an initial value roblem. The initial values are such that they are indeendent of the variable r, where (r, θ) denote the olar coordinate system in the (x, y) lane. This is very general. A secial case is when the initial data

3 are constant in each of the four quadrants. In this aer, however, we deal with a tye of data that are constant in four sectorial regions with sectorial angles θ 1 π, θ = π/, θ 3 0, θ 4 = π/. See Figure 1. As we will see, the condition θ = π/, θ 4 = π/ can be relaxed to other angles easily. We further require the following four conditions on the initial data. 1) A single (forward) shock will form from the one-dimensional interaction between the states 1 and 4. The shock is denoted S ) A single (backward) shock will form from the one-dimensional interaction between the states 1 and. The shock is denoted S 1. 3) A contact discontinuity forms from the one-dimensional interaction between the states and 3. The contact discontinuity is denoted J ) Another contact discontinuity forms from the one-dimensional interaction between the states 3 and 4. The contact discontinuity is denoted J 34. This tye of data exists, see Configuration H of the book of Li et al [17]. Thus, in the self-similar lane (ξ, η) = (x/t, y/t), the solution will look like Figure before we settle the interaction of these four waves within the sonic circle of the state 1: ξ + η = 1. η 1 θ 1 + S S 1 14 ξ 4 J + 3 θ 3 J 34 Figure. The far field solution at t =1. In this aer we rove that the solution will be like in Figure 3, where the two shocks bend slightly and meet to form a smooth curve, rovided that 1 > = 3 = 4 and the angle θ 1 is close to π. The two contact discontinuities remain contact discontinuities and vanish at the origin. The ressure is C smooth in the entire subsonic domain. The condition that θ 1 is close to π makes this roblem a erturbation of a single 1-d shock wave. 3

4 Main Theorem. There exists an (entroy) solution (, u, v) defined for all (ξ, η) IR, rovided that 1 > = 3 = 4 and the angle θ 1 is close to π. The shock curve is C 1 smooth. The ressure is C smooth in the subsonic domain. η 1 θ 1 ξ S S 1 14 subsonic J 3 θ 3 J 34 4 Figure 3. The solution. The difficulty in this aer is the degeneracy at the oint at which the two shocks meet, as the two shocks actually belong to two different classes, and the obliqueness fails (see later). In fact, the directions of the oblique derivatives are tangent to the boundary at the middle oint, see Figure 4, and the degeneracy tye of the boundary value is known as emergent tye in the linear case, see [3]. We use a cut-off to remove the degeneracy and later remove the cutoff, which has been attemted in Canic, Keyfitz, and Kim in [6][7]. The significance is that we can remove this cut-off and this solution is thus global. 4

5 η 1 ξ S1 subsonic S 14 Figure 4. Directions of the tangential oblique derivatives. 3 Technical formulation To be more recise, the Riemann roblem for (1) is an initial value roblem of the form (u, v, ) t=0 = (u i, v i, i ) in sector i, i = 1,, 3, 4. () In the self-similar lane, the system of equations are ξu ξ ηu η + ξ = 0, ξv ξ ηv η + η = 0, ξe ξ ηe η + (u) ξ + (v) η = 0. (3) The initial condition () becomes boundary condition lim(u, v, ) = (u i, v i, i ) in sector i, i = 1,, 3, 4 (4) as the self-similar radius (ξ + η ) 1/ aroaches infinity along a fixed ray. Across a shock, the Rankine-Hugoniot relation is dη dξ = σ ± = ξη ± (ξ + η ), (5) ξ 5

6 [u] = ξ ± η (ξ + η ) [], (6) (ξ + η ) [v] = η ξ (ξ + η ) []. (7) (ξ + η ) Again, we refer the reader to the books [31][17] for the derivation of these relations and other technical informations below. One eliminates the (u, v) variables in system (3) to derive a de-couled equation for in smooth regions: ( ξ ) ξξ ξη ξη + ( η ) ηη + (ξ ξ + η η ) (ξ ξ + η η ) = 0. (8) Once is found, we use the first two equations in (3) and the inner boundary values given by (6-7) to evaluate (u, v). So we shall focus on (8). A nontrivial issue is to make sure the two systems are equivalent. We will imose boundary conditions on (8) to guarantee the equivalence. That is, the oblique derivative boundary condition will suffice rovided that the free boundary does not contain erfect circular arcs. We take an angle θ 1 (0, π) for the angle of the first sector and a state (u 1, v 1, 1 ) for the first sector. We take a ressure in the interval (0, 1 ) for the ressure in the other three sectors. We will take 3 = 4 = in the remainder of the aer. Now we imagine an oen bounded set Ω IR, whose boundary Ω consists of two simle arts: σ and Σ, where σ is an arc of a sonic circle and Σ is a coule of shock waves: Σ + and Σ. See Figure 5, where and m is an internal ressure between 1 and. η m := ( m + )/ (9) We notice that the angle θ 1 (0, π) is to control the sloe of Σ. Alternatively, in lace of θ 1 (0, π), we can use the m (, 1 ) to do the same thing. Let Σ be a curve starting at the oint (ξ, η) = (0, η m ). When m is close to 1, we have θ 1 close to π (roved later). 6

7 η σ 1 S 1 (π θ )/ 1 A Ω Σ m G Σ + B η m ξ 1 + S 14 Figure 5. Mixed Dirichlet and oblique derivative boundary conditions. We will sometimes dro the suerscrit in Σ + since we will be dealing with symmetric data only in the current aer. On σ we imose the simle Dirichlet data. On Σ, we require that all three equations (3) (taken the limit from the inside) and all three Rankine-Hugoniot relations (5-7) hold. We differentiate the last two equations in the Rankine-Hugoniot relations along the shock wave so we have five differential equations for six derivatives (u ξ, v ξ, ξ, u η, v η, η ). We solve four of them (u ξ, v ξ, u η, v η ) from four equations in terms of the other two, and ut them into the fifth equation which has only the derivatives ( ξ, η ). This is the oblique derivative condition on the shock wave: ( σ ± ξ + η ) + (ξ ξ + η η ) ξσ ± η ( ṗ ± ξ + η [] 4 (ξ + η ) ) = 0 (ξ +η ) where [] =, and the term ṗ denotes the tangential differentiation along the shock: ṗ = ξ + σ ± η. The first equation in the R-H relation (5) is symbolically used to determine the location of Σ. The directions deicted in Figure 4 are given by l when the oblique derivative condition (10) is written as l = 0 in this way: ṗ ( [] 4 (ξ + η ) (ξ + η ) ) (ξ + η ) { } ( σ ± ξ + η ) + (ξ ξ + η η ) ξσ (11) ± η = 0. Note that the intersection oints A and B of σ with Σ are not redetermined, in articular the olar angles θ A and θ B are to be determined with Σ. 7 (10)

8 An alternative derivation of the oblique derivative condition is as follows. The secondorder equation for lus the two first-order equations for (u, v) imlies for (ξ ξ + η η )L + L = 0, (1) L := (ξ ξ + η η )/ u ξ v η. (13) This ODE (1) in the form r r L + L = 0 has the only solution L = 0 if L(r 0 ) = 0 at any oint r = r 0 > 0. Thus the third equation of (3) will hold if it holds on the boundary Σ. We then use integral reresentations to reresent (u, v) through and insert them into L = 0 on Σ and simlify the exression to yield (10). Free boundary roblem: Given three ressure values < m < 1. Find a curve Σ and a function such that satisfies equation (8) with Dirichlet condition and oblique derivative condition (10) on = 1, on σ : ξ + η = 1, (θ B θ θ A ) (14) Σ : η = η(ξ) defined by η (ξ) = σ ±, η(0) = η m (15) where σ ± are given by the first R-H relation (5). See Fig. 5. We aroach the above free boundary roblem through a modified roblem which we call a naturally modified roblem. See Figure 6, where η 1 := ( 1 + )/. Naturally modified roblem: Given ɛ e > 0 and ɛ D > 0 and three ressure values < m < 1. Solve (, Σ) from the equation ɛ e + ( ξ ) ξξ ξη ξη + ( η ) ηη + (ξ ξ + η η ) (ξ ξ + η η ) = 0. (16) modified from equation (8), with Dirichlet condition = 1, on σ : ξ + η = 1, (θ B θ θ A ) = (D) := m ɛ D on σ : η = η m, ξ [ ɛ D, ɛ D ] (17) and oblique derivative condition (10) on Σ : η = η(ξ) defined by η (ξ) = σ ±, η(±ɛ D ) = η m (18) where (ɛ e, ɛ D ) will be sent to zero, and σ ± are given by the first R-H relation (5). See Fig. 6. 8

9 η σ S 1 Σ m Ω D 1 m Σ B Bm B1 η = η m η = η 1 ξ 1 S + 14 Figure 6. Naturally modified mixed Dirichlet and oblique derivative boundary value roblem. We will work on a variation of the above Naturally Modified Problem and call it the Core Problem, see below. We realize that we have many otions in removing the degeneracy at the oint (ξ, η) = (0, η m ). We also have many otions for the smoothness of an aroximation of Σ near the oint D. The set-u in the above Naturally Modified Problem is only one intuitive way, which introduces a corner D. To use Lieberman s result on otimal regularity [0], we find out to our surrise that the regularity of the solution is better if the corner D has a sharer (more acute) angle so as to make the direction at the corner of the oblique vector in the oblique derivative condition oint from the exterior to the exterior of the domain. Thus, in lace of the straight line segment η = η m, ξ [ ɛ D, ɛ D ] we use a circular arc symmetric with resect to the η-axis. Let η = A(ξ) be a circular are centered at a oint (ξ, η) = (0, η ɛd ) with radius r ɛd > 0 for ξ [ ɛ D, ɛ D ]. We will take η ɛd = 0 although we can take it to be either negative or ositive with absolute value in the order of ɛ D or larger. We take r ɛd = ɛ D or accordingly. Core roblem: Given ɛ e > 0 and ɛ D > 0 and three ressure values < m < 1. Solve (, Σ) from the equation ɛ e + ( ξ ) ξξ ξη ξη + ( η ) ηη + (ξ ξ + η η ) modified from equation (8), with Dirichlet condition (ξ ξ + η η ) = 0. (19) = 1, on σ : ξ + η = 1, (θ B θ θ A ) = (D) := m ɛ D on σ : η = A(ξ), ξ [ ɛ D, ɛ D ] (0) 9

10 and oblique derivative condition (10) on Σ : η = η(ξ) defined by η (ξ) = σ ±, η(±ɛ D ) = η m (1) where (ɛ e, ɛ D ) will be sent to zero, and σ ± are given by the first R-H relation (5). See Fig. 7. η σ S 1 Σ m Ω D 1 m Σ B Bm B1 η = η m η = η 1 ξ 1 + S 14 Figure 7. Core roblem: Modified mixed Dirichlet and oblique derivative boundary conditions. One might think that an alternative of the circular symmetric arc would be a smooth iece of curve which connects the rest of Σ smoothly at D. In this scenario, we have a mixed Dirichlet and oblique derivative boundary condition on a smooth curve. Unfortunately, there is no written English work on this roblem, a big surrise, although we find that there are claims that the regularity can be established with classically known methods. To utilize the available English references (rather than hard-to-find former USSR aers) on the regularity of solutions at corners, we will use the circular arc as aroximation. For those who are interested in a smooth aroximation rather than this odd arc aroximation, we refer to a two-age announcement of Kerimov [18] (courtesy of Lieberman). Also, we will see that there are quite a few aers on tangential oblique derivative roblems on linear roblems, starting with Hörmander [15], Egorov and Kondrat ev [1], and many afterward ([][3][4][1][][3][4][5][6][8][9]), culminates in Guan and Sawyer [14], but none of them alies directly to our roblem, albeit Guan and Sawyer s [14] is extremely close. We oint out that the book of Gilbarg and Trudinger [13] does not deal with tangential oblique derivative roblems. We choose to work out 10

11 our roblem rather than fit our roblem into their frame in this aer. Future work may well utilize these aers. It can be roved that a solution (, Σ) to the Free Boundary Problem yields a solution to the original Riemann roblem (1)() rovided that the shock wave Σ does not contain any arc of circle, which is the case for our current roblem in the limits ɛ e 0, ɛ D 0. We remark that in the work [5][6][7] this tye of roblem is aroached through working at the oint B. Our aroach is to work at the oint D. We oint out that we have checked that contact discontinuities can be handled easily. 4 Proof We resent the novel art of the roof in detail. The basic stes of the roof which are similar to [5][6][7] are sketched only. 4.1 A riori estimates We modify the main equation (19) in the Core Problem. First the denominator can be relaced by N() where N is C 3 smooth in IR 1 bounded from above and below by two ositive numbers and is the identity function in the interval [ m, 1 ]. Second, relace the two simle s in (19) by K(ξ, η, ) where K(ξ, η, z) = { z, if z ξ + η, ξ + η, if z < ξ + η. () This K is Lischitz in IR 3. We modify the oblique derivative boundary condition. We use the new form σ ± = η ξ( η) ± (ξ + η ) (3) for σ ± from the first R-H formula. All the s in σ ± and the oblique derivative condition are relaced by N(), excet the terms ( ξ, η ). This is the third cut-off: Relace all aearances of the term ξ + η by ϕ(ξ + η ) where ϕ(z) is smooth in IR 1, equals to the identity for z > 1 ɛ D, and is otherwise smooth and bounded below by 1 4 ɛ D. The cut-off level 1 ɛ D is lower than the value of ξ + η at oint D, which is 7 8 ɛ D. We list some hard facts: ξ B 1 = 1 ( 1 ), ξ B m = 1 ( 1 ) + 1 ( 1 m ). (4) 11

12 The arameters are ordered to be ɛ D < ɛ e < 1 m. (5) Lemma 4.1 (Maximum rincile) Any C solution of the core roblem satisfies the bounds rovided that σ ± < ( m )/( 1 ). m < (D) 1 (6) Proof. We need be C(Ω), C 1 (Ω Σ), and C (Ω), where Ω is oen. We need Σ be in H 1+α. These smoothness will be justified. The equation is uniformly ellitic, and the coefficients are at least Lischitz. It is well-known that there is no extreme values in the interior. We show that there is no extreme values on Σ. This is done by Hof s lemma and the oblique derivative condition. Obliqueness: An interior normal of Σ is ( σ ±, 1). The obliqueness is defined by the inner roduct of a unit (interior) normal with the oblique direction ( σ ±, 1)+ ξσ ± η (ξ, η). We ignore the tangential direction. We ignore the normalization of the interior normal for now as well. Thus Obliqueness = ( σ ±, 1) {( σ ±, 1) + ξσ ± η (ξ, η)}. (7) N() We do some simle algebra to yield, where the modification N( ) is omitted, Obliqueness = ξ { (σ ± + ξη ξ ) + ( ξ η } ) ( ξ ). (8) The obliqueness is obvious rovided that ξ η > 0 on Σ. We will need σ ± 0 to get the obliqueness before we establish that ξ η > 0 on Σ. We can easily see that there can be no global maximum on Σ, since a global maximum on Σ would mean that 1 and that the tangential derivative of is zero, thus an oblique derivative would be zero due to the oblique derivative condition, but this would contradict Hof s lemma. To avoid a global minimum on Σ, we need σ ± small to guarantee the obliqueness. The smallness deends only on m, 1, and. We have a different form Obliqueness = 1 { η + ξησ ± + ( ξ )σ ±}. (9) 1

13 At σ ± = 0, the obliqueness is (N() η )/N() which is ositive along the now flat Σ. From this obliqueness and Hof s lemma, we conclude that there is no global minimum on Σ either. The smallness can be found as follows. We require that ξησ ± + N() η > 0. The minimum of N() is taken aroximately as m, η η m, and ξη ξ + η 1. Thus we only need σ ± m 1. (30) Note that the difference is indeed m rather than m 1. The sign of ξ needs attention, too. Our N() has min m. Our ξ has aroximately an uer bound equal to ξ Bm. So it is ositive. Thus there is no global minimum or maximum of on Σ. Hence the roof is comlete. QED We can now omit the cut-off N(). Lemma 4. (Monotonicity) Any C solution of the core roblem and any Σ yield σ + > 0 on Σ +. (31) Proof. We use the revious lemma to obtain (D). Then we have ( η ) D 1 ( m ɛ D) 1 ( m + ) = 1 8 ɛ D > 0. (3) Using the formula (3) for σ + we find that the derivative η (ξ) = σ + at ξ = ɛ D is ositive, thus Σ + is increasing, so η(ξ) is increasing, thus η remains ositive. This comletes the roof. QED Lemma 4.3 (Interior elliticity) The elliticity holds in the interior ξ η > 0, in Ω. (33) Proof. See Zheng [30], Lemma 4.4 (Edge elliticity) The elliticity holds on the free boundary ξ η > 0, on Σ +. (34) 13

14 Proof. We use the notation w := ξ η. Suose w has a nonositive minimum on Σ + at a oint (ξ 0, η 0 ). By the revious lemma, we have w = 0, w ξ + σ + w η = 0, w ν 0, at (ξ 0, η 0 ) Σ + (35) Using = w + (ξ, η) at (ξ 0, η 0 ), we rewrite the oblique derivative in terms of w: ( σ + w ξ + w η ) + (ξw ξ + ηw η ) ξσ + η +( ξσ + + η) + (ξ + η ) ξσ + η { + ξ+ησ + ξ + η 4 (ξ + η ) } = 0. ϕ(ξ +η ) (36) The first two terms add to be nonnegative since the direction of obliqueness is ointing inward. The next two terms add to be (ξσ + η)( ξ + η 1) (37) which we show to be nonnegative. First ξ + η at (ξ 0, η 0 ). Second, the factor ξσ + η is ositive since σ + 0 from the revious lemma. Now look at the last term in (36). The factor ξ + ησ + can be written as ξ + ησ + = η(ξ + η ) + ξ ϕ ξ( η) + ϕ(ξ + η ). (38) Using the assumtion that ξ + η, we see that it is ositive. The other factor can be maniulated as ξ + η 4 (ξ + η ) = (ξ + η ){1 } 4 {1 } 4 = ( ) (+ > 0. ) Thus there is contradiction. So the roof is comlete. (39) QED Lemma 4.5 (Interior uniform elliticity) The elliticity holds in the interior ξ η > ɛ δ, in Ω δ (40) where Ω δ = Ω B r where B r is a ball centered at (0, 0) with radius r = 1 δ for small δ > 0 and ɛ δ > 0 is indeendent of ɛ e or ɛ D. Proof. See Zheng [30],

15 Lemma 4.6 (Real roerty) The ξ +η is bounded below by 3 4 ɛ D along Σ +, rovided that and 0 < 1 m c 0, (41) c 1 1 m (4) for some ositive constants c 0, c 1, indeendent of ɛ e, ɛ D, or ϕ. Proof. We can calculate the term s value at ξ = ɛ D, which yields 7 8 ɛ D; and its value on σ, which yields ( 1 )/ which is larger than 7 8 ɛ D. So the term must be increasing at some oint along Σ +. It therefore may be increasing all the time. We rove that it will not di below 6 8 ɛ D, which is sufficient for us. Suose it reached 6 8 ɛ D at the first oint (ξ 1, η 1 ) Σ +. Then the derivative along Σ + of the term would be nonositive at the oint. We show that is imossible. Here s the outline. We see d d Σ ξ (ξ + η )/ = ξ + ησ + = η(ξ + η ) + ξ ϕ ξ( η) + ϕ(ξ + η ). (43) We have maniulated that term before, see (38). At (ξ 1, η 1 ), we estimate d ϕ d Σ ξ (ξ + η )/ min {, ξ 1 } O (ɛ D ). (44) η 1 Examining the oblique derivative condition, we obtain aroximately [] 4 where µ is some bounded direction. Once we have (4), then ṗ + µ = 0 (45) ϕ ṗ ( 1 m ) ϕ ( 1 m )ɛ D. (46) We let 1 m be small indeendent of ɛ e or ɛ D. This would contradict the diing of ξ + η at (ξ 1, η 1 ). This comletes the roof. QED Thus the modifier ϕ can be taken away! This is a crucial ste, not achieved in revious aers. We use the now standard norm v H ( γ) 1+α = su{d 1+α γ B ( v 0 + v 0 + [ v] α ) ΩdB } (47) where d B stands for distance to the corners B and its counter art A, while Ω db is Ω B c d B (B) where B c d B (B) is the comlement of the ball centered at B with radius d B. 15

16 Lemma 4.7 (Smallness of sloe) The sloe of the free shock boundary Σ has a bound rovided that σ ± c 1 m (48) 1 H ( γ) 1+α Σ c 1 m (49) for some constant c, indeendent of ɛ e, ɛ D, or ϕ. Assume α Σ < γ. Proof. We have We have η (ξ) = η (ɛ D ) + ξ ɛ D η (ξ) dξ η (ɛ D ) + c ξ ɛ D ṗ(ξ) dξ. (50) ṗ c 1 m d (1+α Σ γ) B. (51) Since α Σ < γ, we see that ṗ is integrable. This comletes the roof. QED Now estimate (49) will start to come from linear theory. We comment that we have two obvious corners D and B. The corner B will be bona fide, while corner D is subject to our choice. With the symmetric circular arc, we find that the oblique derivative direction vector at both oints oint from the exterior of Ω to the exterior. We then use Lieberman s otimal regularity result [0], Theorem 4, to start the regularity estimate with γ 1. Thus will be in H 1+αΣ (Σ) for fixed ɛ e > 0, and estimate (4) will follow. 4. Solution to the core roblem Linear with fixed boundary: See Lieberman [19][0], Guan and Sawyer [14], and Winzell [8], or Canic et al [6]. Nonlinear with fixed boundary: See Poivanov and Kutev [4], or Canic et al [6]. Free boundary: Use Schauder fixed oint theorem (Corollary 11. of [13]), see Canic, Keyfitz and Kim [6]. 4.3 Final limits: Solution to the free boundary roblem The limits are easy to take due to the uniform estimates we have obtained. 16

17 5 Further observations If the angle θ 1 is not close to π, then the solution can be different. In terms of the arameter m, we see that m can be as small as. When m =, the free boundary shock Σ may degenerates artially to a ortion of the sonic circle of the state. See Figure 8. This situation resembles Configuration H in Li et al [17]. We lan to give a roof of this in the near future. η θ 1 1/ 1 1/ S 1 + S 14 ξ A C D Σ B 4 = + J J 3 34 Figure 8. Another tye of solution. Acknowledgment: Discussions with L. C. Evans, G. Lieberman, Jiequan Li, and Tong Zhang are very helful. References [1] Agarwal, R. K., and Halt, D. W.: A modified CUSP scheme in wave/article slit form for unstructured grid Euler flows, Frontiers of Comutational Fluid Dynamics (D. A. Caughey and M. M. Hafez, eds.), [] Bitsadze, A.V., Boundary value roblems for second order ellitic equations. Translated from the Russian by Scrita Technica, ltd. Translation edited by M. J. Laird. [3] Borrelli, R. L., The singular, second order oblique derivative roblem, J. Math. Mech., 16(1966), [4] Borrelli, R. L., The I-Fredholm roerty for oblique derivative roblems. Math. Z. 101(1967),

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