ON UNIQUENESS MORAWETZ PROBLEM FOR THE CHAPLYGIN EQUATION

For the equation Lz = K(y)zxx + zyy = 0, where yK(y) > 0 for y 6= 0) in D, bounded by a Jordan (non-selfintersecting) ”elliptic” arc Γ (for > 0) with endpoints A(0, 0)and B(l, 0), l > 0, and for y < 0 by a characteristic γ1 through A which meets the characteristic γ2 through B at the points C, the uniqueness of the Morawetz problem is proved without assuming that Γ is monotone. AMS Subject Classification: 35M12


Consider the equation
in an open domainD, where yK(y) > 0 for y = 0 and the domain D is bounded by curves: a piecewise smooth curve Γ in the half-plane y > 0, which intersects the line y = 0 at the points A(0, 0) and B(l, 0), l > 0; in y < 0, D is bounded by two charasteristics γ 1 and γ 2 of (1) issuing from A and B and meeting at Received: September 14, 2014 c 2014 Academic Publications, Ltd.
url: www.acadpubl.euthe point C: where In this paper using a variation of the energy-integral method (abc method) we obtain sufficient conditions for the uniqueness of solution of Morawetz problem for the Chaplygin equation.It arises in the study of transonic flow, and the proof of uniqueness in this case leads to a proof that continuous transonic flows past smooth profilies do not exist in general [1].
The Morawetz problem.Find function z(x, y) satisfying the following conditions: where ϕ and ψ are given functions.
The question uniqueness of solution of Morawetz problem for equation of mixed type has been dealt with in the literature by many authors.For an extensive bibliography we refer the reader to [2], [3], [4].

Reducing Morawetz Problem to the Tricomi Problem
To every solution z(x, y) of (1) there corresponds a function defined by the integral which is independent of the path of integration and the function v(x, y) is to satisfy the eqution Then the Morawetz problem is transformed into the analogue Tricomi problem for equation (3) Definition 1.We call a function v(x, y) quazi-regular solution of (3) if the following hold: i) v(x, y) satisfies (5); ii) we can to applicate Green's theorem to the integrals iii) the boundary integrals which arise exist in the sense that: the limits taken over corresponding interior curves exist as these interior curves approach the boundary.

Theorem of Uniqueness
We introduce Francl's function The following statement is a more general result than Theorem 6, given in [3].
The integrals I 3 and I 5 are non-positive if the following two conditions hold in D: Obviously, condition (12) holds for all a(x, y).If, now we substitute functions a(x, y), b(x, y) and c(x, y) into (13) we obtain Consider two cases.Case 1.If F (y) > 0 for y ≤ 0 then choose a(x, y) = const ≥ 0. In this case conditions (11) and ( 14) hold for y ≤ 0 and all the integrals I 1 , I 2 , ..., I 6 are non-positive.
the ordinate of point C. Let D + be subdomain of D with y > 0 and D − be subdomain of D with y < 0.