THE CAUCHY PROBLEM VIA THE METHOD OF CHARACTERISTICS

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1 THE CAUCHY PROBLEM VIA THE METHOD OF CHARACTERISTICS ARICK SHAO In this short note, we solve the Cauchy, or initial value, problem or general ully nonlinear irst-order PDE. Throughout, our PDE will be deined by the unction F : R 2 x,y R z R 2 p,q R. We also ix an open interval I R, as well as unctions, g, h : I R. In particular, Γ := {((r), g(r)) r I} is the curve on which we impose the initial data, while h represents the initial data itsel. More speciically, our goal is to solve the ollowing Cauchy problem: (1) F (x, y, u, x u, y u) = 0, u((r), g(r)) = h(r). Our main result is as ollows: Theorem 1. Let F C 2 (R 2 R R 2 ), and let, g, h C 2 (I). Moreover, let r 0 I and p 0, q 0 R such that the ollowing admissibility conditions are satisied: (2) (3) (4) p 0 + g q 0 = h, F (, g, h, p 0, q 0 ) = 0, det p F (, g, h, p 0, q 0 ) q F (, g, h, p 0, q 0 ) Then, there exists a neighbourhood U R 2 o (, g) such that the Cauchy problem (1) has a unique solution u C 2 (U) that satisies the additional conditions (5) x u(, g) = p 0, y u(, g) = q 0. Remark. In the quasilinear case, F (x, y, z, p, q) = a(x, y, z)p + b(x, y, z)q c(x, y, z), one no longer needs to choose p 0, q 0 beorehand. To see this, note that: The noncharacteristic condition (4) is independent o p 0 and q 0, as it reduces to (6) det a(, g, h) b(, g, h) The remaining admissibility conditions (2), (3) are also unnecessary, as these now become p 0 + g q 0 = h, a(, g, h)p 0 + b(, g, h)q 0 = c(x, y, z), and (6) guarantees that the above yields exactly one possible pair (p 0, q 0 ). Remark. Furthermore, or the quasilinear case, we need only assume a, b, c C 1 and, g, h C 1. This yields a unique solution u C 1. Less regularity is required here than in the ully nonlinear case, since i one proves the quasilinear analogue o Theorem 1 directly (without reerring to the ully nonlinear case), then one can avoid altogether p, q, and second derivatives o u. 1

2 2 ARICK SHAO Proo o Theorem 1: Existence. The irst step or proving existence is to construct a set o similarly admissible data on Γ near (, g): Lemma 1. There exists an open interval J I containing r 0, and there exist unique w, v C 1 (J) such that (w, v) = (p 0, q 0 ), and such that the ollowing hold or every r J: (7) (8) (r) w(r) + g (r) v(r) = h (r), F ((r), g(r), h(r), w(r), v(r)) = 0, Furthermore, J, w, and v can be chosen such that or each r J, (9) det (r) g ] (r) p F ((r), g(r), h(r), w(r), v(r)) q F ((r), g(r), h(r), w(r), v(r)) Proo. Consider the map Φ : R r R 2 P,Q R2 given by Φ(r, P, Q) := ( (r) P + g (r) Q h (r), F ((r), g(r), h(r), P, Q)). Note that (2) and (3) imply that Φ(r 0, p 0, q 0 ) = 0. Moreover, since F,, g, h are C 2, then Φ is C 1. We now compute the derivative o Φ with respect to P and Q at (r 0, p 0, q 0 ): D P,Q Φ (r0,p 0,q 0 ) =. p F (, g, h, p 0, q 0 ) q F (, g, h, p 0, q 0 ) In particular, (4) implies D P,Q Φ (r0,p 0,q 0 ) is nonsingular. By the implicit unction theorem, there exists an open interval J I and a unique C 1 -unction Ψ = (w, v) : J R 2 such that Φ(r, Ψ(r)) = Φ(r, w(r), v(r)) = 0, r J. The deinition o Φ now implies that (7) and (8) hold or r J. Finally, by reducing J i necessary, then continuity implies that (9) also holds or r J. Now that we have w and v set, we can set up the characteristic equations. Set γ(r, s) := (x(r, s), y(r, s), z(r, s), p(r, s), q(r, s)), γ(r, s) := (x(r, s), y(r, s)), and recall that the characteristic equations are precisely the ollowing initial value problem: (10) s x(r, s) = p F ( γ(r, s)), s y(r, s) = q F ( γ(r, s)), s z(r, s) = p F ( γ(r, s)) p(r, s) + q F ( γ(r, s)) q(r, s), s p(r, s) = x F ( γ(r, s)) z F ( γ(r, s)) p(r, s), s q(r, s) = y F ( γ(r, s)) z F ( γ(r, s)) q(r, s), x(r, 0) = (r), y(r, 0) = g(r), z(r, 0) = h(r), p(r, 0) = w(r), q(r, 0) = v(r), Here, s represents the parameter along the characteristic curves, while r parametrises the characteristic curves corresponding to the point on Γ that they intersect. Since the right-hand sides o the equations in (10) are C 1 -unctions o γ(r, s), and since the initial data, g, h, w, r are at least C 1, then standard ODE theory yields: Lemma 2. There exists an open rectangle R R 2 about (r 0, 0), on which there exists a unique C 1 -unction γ : R R 2 R R 2 which solves the system (10). Since (7) and (8) represent consistency conditions or solutions, it is important to show that (7) and (8) are propagated along the characteristic curves: Lemma 3. The ollowing conditions hold or each (r, s) R: (11) (12) r x(r, s) p(r, s) + r y(r, s) q(r, s) = r z(r, s), F (x(r, s), y(r, s), z(r, s), p(r, s), q(r, s)) = 0.

3 THE CAUCHY PROBLEM VIA THE METHOD OF CHARACTERISTICS 3 Proo. First, by the chain rule and then by (10), s F ( γ(r, s))] = x F ( γ(r, s)) s x(r, s) + y F ( γ(r, s)) s y(r, s) + z F ( γ(r, s)) s z(r, s) = 0. + p F ( γ(r, s)) s p(r, s) + q F ( γ(r, s)) s q(r, s) Since (8) is simply that F ( γ(r, 0)) = 0, we can conclude (12). Next, or (11), we deine A(r, s) := r x(r, s) p(r, s) + r y(r, s) q(r, s) r z(r, s). A direct computation using (10) and (12) yields s A(r, s) = r p F ( γ(r, s)) p(r, s) + r x(r, s) x F ( γ(r, s)) z F ( γ(r, s)) p(r, s)] + r q F ( γ(r, s)) q(r, s) + r y(r, s) y F ( γ(r, s)) z F ( γ(r, s)) q(r, s)] r p F ( γ(r, s)) p(r, s) + q F ( γ(r, s)) q(r, s)] = r x(r, s) x F ( γ(r, s)) + z F ( γ(r, s)) r x(r, s) p(r, s) r y(r, s) y F ( γ(r, s)) + z F ( γ(r, s)) r y(r, s) q(r, s) p F ( γ(r, s)) r p(r, s) q F ( γ(r, s)) r q(r, s) = r F ( γ(r, s))] + z F ( γ(r, s)) A(r, s) = z F ( γ(r, s)) A(r, s). Since A(r, 0) = 0 by (7), then either Gronwall s inequality or solving the above directly yields A(r, s) = 0 or all (r, s) R, which is the remaining relation (11). In particular, Lemma 2 implies that the projected characteristic map γ deines a C 1 change o variables rom (r, s) to (x, y) = (x(r, s), y(r, s)). Next, we show that γ can be locally inverted: Lemma 4. There exists an open rectangle R R about (r 0, 0) on which γ R is one-to-one and Dγ R is everywhere nonsingular. Furthermore, this local inverse φ o γ R is C 1. Proo. A direct computation using (10) shows that at (r 0, 0), we have Dγ (r0,0) = p F (, g, h, p 0, q 0 ) q F (, g, h, p 0, q 0 ) Since (4) implies the above is nonsingular, the inverse unction theorem yields the desired φ. We can now construct our solution u by u = z φ C 1 (U), U = γ(r ), which by the invertibility in Lemma 4 is equivalent to (13) u(x(r, s), y(r, s)) = z(r, s), (r, s) R. (14) By the chain rule, we compute rom (13) that r z(r, s) = r x(r, s) x u(x(r, s), y(r, s)) + r y(r, s) y u(x(r, s), y(r, s)), s z(r, s) = s x(r, s) x u(x(r, s), y(r, s)) + s y(r, s) y u(x(r, s), y(r, s)). Moreover, rom (11) and (10), we have (15) r z(r, s) = r x(r, s) p(r, s) + r y(r, s) q(r, s), s z(r, s) = s x(r, s) p(r, s) + s y(r, s) q(r, s). Since Dγ is invertible on R, one then concludes that (16) x u(x(r, s), y(r, s)) = p(r, s), y u(x(r, s), y(r, s)) = q(r, s).

4 4 ARICK SHAO Note that (16) can be restated as x u = p φ, y u = q φ. Since the right-hand sides above are C 1, it ollows that u C 2 (U). Finally, we show that u indeed solves (1). The initial condition holds, since by (13), u((r), g(r)) = u(γ(r, 0)) = z(r, 0) = h(r). In addition, u solves the PDE, since by (12), (13), and (16), Since setting (r, s) = (r 0, 0) in (16) yields F (γ(r, s), u(γ(r, s)), u(γ(r, s))) = F ( γ(r, s)) = 0. x u(, g) = p 0, y u(, g) = q 0, this completes the proo o existence in Theorem 1. Proo o Theorem 1: Uniqueness. Suppose ū is another C 2 -solution to (1) on U. Since F (x, y, ū(x, y), ū(x, y)) = 0, (x, y) U, taking partial derivatives o the above yields the ollowing relations: (17) 0 = x F (x, y, ū(x, y), ū(x, y)) + z F (x, y, ū(x, y), ū(x, y)) x ū(x, y) + p F (x, y, ū(x, y), ū(x, y)) xxū(x, 2 y) + q F (x, y, ū(x, y), ū(x, y)) xyū(x, 2 y), 0 = y F (x, y, ū(x, y), ū(x, y)) + z F (x, y, ū(x, y), ū(x, y)) y ū(x, y) + p F (x, y, ū(x, y), ū(x, y)) yxū(x, 2 y) + q F (x, y, ū(x, y), ū(x, y)) yyū(x, 2 y). Next, we deine the unctions λ = ( x, ȳ) : R R 2 via the initial value problem (18) s x(r, s) = p F (λ(r, s), ū(λ(r, s)), ū(λ(r, s))), x(r, 0) = (r) = x(r, 0), s ȳ(r, s) = q F (λ(r, s), ū(λ(r, s)), ū(λ(r, s))), ȳ(r, 0) = g(r) = y(r, 0). Indeed, standard ODE theory indicates that x and ȳ exist, at least locally near R {s = 0}, and are C 1 (since the right-hand sides o (18) are C 1 -unctions o λ). Given λ, we next deine (19) z(r, s) = ū(λ(r, s)), p(r, s) = x ū(λ(r, s)), q(r, s) = y ū(λ(r, s)). and, or convenience, the shorthands Lemma 5. The ollowing relations hold: λ(r, s) := ( x(r, s), ȳ(r, s), z(r, s), p(r, s), q(r, s)). (20) z(r, 0) = h(r) = z(r, 0), p(r, 0) = w(r) = p(r, 0), q(r, 0) = v(r) = q(r, 0). Proo. The irst relation in (20) is an immediate consequence o the assumption that ū solves (1). For the remaining relations, we note rom (1) that on the initial data curve Γ, F ( λ(r, 0)) = F ((r), g(r), ū((r), g(r)), ū((r), g(r))) = 0, (r) p(r, 0) + g (r) q(r, 0) = d ( (r),g (r))ū((r), g(r)) = h (r). Since we have assumed p(r 0, 0) = x ū(, g) = p 0, q(r 0, 0) = y ū(, g) = q 0, the relations or p and q in (20) ollow rom the uniqueness o w, v in Lemma 1.

5 THE CAUCHY PROBLEM VIA THE METHOD OF CHARACTERISTICS 5 Lemma 6. The ollowing identities hold or any (r, s) R : (21) s z(r, s) = p F ( λ(r, s)) p(r, s) + q F ( λ(r, s)) q(r, s), s p(r, s) = x F ( λ(r, s)) z F ( λ(r, s)) p(r, s), s q(r, s) = y F ( λ(r, s)) z F ( λ(r, s)) q(r, s). Proo. The irst relation in (21) ollows immediately rom the chain rule and (18): s z(r, s) = s x(r, s) x ū(λ(r, s)) + s ȳ(r, s) y ū(λ(r, s)) Applying similar computations to p, we see that = p F ( λ(r, s)) p(r, s) + q F ( λ(r, s)) q(r, s). s p(r, s) = s x(r, s) xx ū(λ(r, s)) + s ȳ(r, s) yx ū(λ(r, s)) = p F ( λ(r, s)) xx ū(λ(r, s)) + q F ( λ(r, s)) yx ū(λ(r, s)). Recalling the irst equation in (17) results in the irst relation in (21). The remaining relation or q in (21) can be derived using analogous methods. Finally, combining (18), (20), and (21), we see that λ solves the same initial value problem (10) as γ. Thus, by standard uniqueness results or ODEs, we see that λ = γ. It then ollows that ū(x, y) = z(φ(x, y)) = z(φ(x, y)) = u(x, y), (x, y) U, which completes the proo o uniqueness.