A sharp condition for the well-posedness of the linear KdV-type equation

To make the illposedness argument more transparent the argument is rewritten to reduce the equation to the constant dispersion case. Minor errors are corrected. Accepted to the Proceedings of the AMS.


Introduction
This paper is concerned with the study of the equation where a j are real-valued functions. This is the most general linear form of the KdV, one of the most studied dispersive equations, and used as an important model in understanding behavior of linear and non-linear waves. Such an equation with non-constant dispersive coefficient a 3 describes nonisotropic dispersion and its study is of use for the quasi-linear analogues of (1).
Another motivation, for the study of the well-posedness of (1) is understanding the relative strength of dispersive and non-dispersive effects present in the equation. In particular, from the geometrical optics expansion for the equation, c.f. the classical book of Whitham [11], the dispersive coefficient a 3 guides the propagation of the wave packets, while the term a 2 ∂ 2 x can lead to the growth of the amplitudes of the wave packets of (1). In light of these heuristics, it is natural to expect that well-posedness requires non-degeneracy of a 3 , which prevents the collapse of the wave packets, namely 0 < ε ≤ |a 3 | ≤ 1 ε for some ε, and a condition on a 2 to ensure dispersion dominates anti-diffusion effects. Craig-Goodman [4] proved well-posedness in the Sobolev spaces H s for a 2 ≡ a 1 ≡ 0 under the non-degeneracy of coefficient a 3 and ill-posedness for some degenerate cases of a 3 . In a follow up paper, Craig-Kappeler-Strauss [3] proved well-posedness with non-degenerate dispersion and −a 2 ≥ 0, as well as extensions to the quasi-linear analogues. These results were extended in [1] to allow for the "anti-diffusion" in a 2 , as long as x 1 2 + |a 2 | ≤ C, under some additional assumptions on other coefficients, and to systems of equations.
In the current paper, the condition on the diffusion coefficient a 2 is extended to a sharp one for the well-posedness in H s , where well-posedness means existence of C 0 [0,T ] H s distributional solutions of (1), that are unique and depend continuously on data in the C 0 [0,T ] H s topology. Namely a condition on the diffusion coefficient a 2 along the flow is obtained, that separates well-posedness from illposedness (in the sense of violating continuous dependence) of (1) with non-degenerate dispersion. This is qualitatively similar to the necessity of a Mizohato condition |sup x,t|ω|=1 t 0 ℑb(x+sω)·ωds| < ∞ for the well-posedness Schrödinger equation ∂ t u + i△u + b(x)∇u = 0 in [9], see also [5], [6], [8] and references therein for more refined results on the variable coefficient Schrödinger equation. The well-posedness is proved by the "gauged energy method" and the condition on the gauge captures the a 2 condition. Ill-posedness is proved by an explicit geometrical optics construction.
The rest of the paper is organized as follows. In the section 2 the main results of the paper are stated. Well-posedness is proved in the section 3, and ill-posedness in section 4. Some results of this paper were obtained during my Ph.D. studies at the University of Chicago, under the supervision of Carlos Kenig. I would like to thank Carlos Kenig and Cristian Rios for helpful discussions. Finally, I would like to thank the anonymous referee for helpful comments.
The following functional space notation is used. Let The following assumptions are made for the coefficients of (1) x . Note, that by (A1) and (A2), a 3 has a constant sign.
For the well-posedness arguments, positive constants will depend on C N for some N and will not be made explicit.
Moreover, for any δ > 1 2 , the solution additionally satisfies u ∈ L 2 [0,T ] H s+1 x −2δ dx and there is ã C =C(s, δ) Estimate (2) implies continuous dependence for (1), while estimate (3) is a manifestation of a local smoothing effect of (1).   However, the equivalence breaks down if absolute values are removed from a 3 in (A3). Thus a 3 > 0 can be assumed without loss of generality, as long as (A3N ) is replaced with While preparing this paper for publication, I have learned of a preprint by Ambrose-Wright [2] that treats an analogue of (1) in the periodic case. Their argument for the well-posedness is also based on the "gauged energy method", however in the case of R the smoothness of the coefficients does not imply integrability that is often needed. Additionally, this paper also proves that (1) possesses a local smoothing effect, which is not present in the periodic case. The ill-posedness result in [2] is done by a spectral method, which only works in the time independent case of (1).

Well-posedness
The main ingredient in the proof of the Theorem 1 is stated as the following Proposition, which is an a priori L 2 estimate for a slightly more general version of (1), that comes from commuting derivatives.
with L from (1). The following assumptions are made on A 0 ∈ C 0 [0,T ] S 0 , the Pseudo-Differential operator of standard symbol class of order 0 (Cf. Chapter VI of [10]): (A4): The S 0 semi-norms of A 0 are bounded for t ∈ [0, T ] and their size depends on constants C N from (A1)-(A3).
Proposition 7. Suppose that the coefficients a j of (1) satisfy (A1)-(A3) and A 0 satisfies (A4). Then there exists a constant C and for any δ > 1 2 there is a constantC, such that for any u ∈ Remark 8. If A 0 ≡ 0, then N = 0 in (A2) can be chosen for the Proposition 7.
The proof of the Proposition 7 is done by a change of variables (gauge) followed by the application of the energy estimates. The proof is broken into several preliminary results.
A gauge is a smooth invertible function, which for the purposes of the argument needs to have 3 bounded derivatives: (5) gives: with comparability constants dependent only on the constant in the Definition 9. Therefore, to prove Proposition 7 it suffices to prove (6) and (7) for v satisfying (8).
Proof. It suffices to show one sided inequalities in (9) as φ −1 satisfies the same estimates as φ. The first comparability follows from For the second, a similar computation and Cauchy-Schwartz implies It is clear from (9) that (6) is equivalent for u and v, whereas an estimate (6) and (7) hold for v.
The energy method involves multiplying (8) by The following Lemma summarizes the energy estimates for L or L φ : is an L 2 x pairing. Proof of Lemma 11. The computation is immediate by computing the adjoint L * of L using the Calculus of PDO. Alternatively, as L is a differential operator, the same computation can be also done by a repeated integration by parts. Indeed, the operator ∂ k x is skew-adjoint for odd k, which implies that principal parts of odd order terms are eliminated by integration by parts. For example Using these identities and more integration by parts establishes A similar analysis for Re(a 2 ∂ 2 x v, v) completes the proof.
Applying Lemma 11 to L φ , shows that the only term of order higher than 0 is Thus, if this term were negative, an a priori estimate would be obtained for v. This motivates the choice of a gauge φ that should satisfy A choice of equality in this equation can be made and this choice is enough for the estimate (6), but by exploiting the inequality the local smoothing estimate (7) is proved. The exact choice of a gauge is summarized in the following Lemma where c δ = 0 or 1. Then φ is a gauge in the sense of the Definition 9, and is independent of δ if c δ = 0.
Proof. The ODE for φ is solved explicitly as A computation for ∂ t φ and ∂ j x φ for j = 1, 2 and 3 and using (A1)-(A3) finishes the proof. Proof of Proposition 7. By the Lemma 10 it suffices to prove the Proposition for v satisfying (8).
Applying the Lemma 11 for L φ implies that whereb 0 is obtained from the Lemma 11 applied to L φ . With φ chosen from the Lemma 12, this implies By (A4), A 0 : L 2 → L 2 is bounded. Moreover, by the Definition 9 and (A2), φ ∈ L ∞ andb 0 ∈ L ∞ . Hence For c δ = 0 an application of Grownwall Lemma implies (6) for v.
Whereas moving ∂ x v term to the left hand side for c δ = 1 and integrating in time gives Using (6) completes the proof of (7).
Proposition 7 can be strengthened to an H s estimate.
Proposition 13. Let L be as in (1), whose coefficients a j satisfy (A1)-(A3). Then for any s ∈ R there exist constants C(s) andC(s, δ) for any δ > 1 2 , such that for any u ∈ C 1 [0,T ] H s ∩ C 0 [0,T ] H s+3 the following estimates hold where L * is the adjoint of L. Moreover Corollary 14. By the Theorem 23.1.2 on page 387 in [7], the proof of the Theorem 1 reduces to the Proportion 13.
The Proposition 13 is reduced to the Proposition 7. Observe, that where J s is a Pseudo Differential Operator with symbol ξ s . Therefore to prove (10) where A s ∈ S 0 , whose semi-norms depend on the coefficient bounds (A2) for N = N (s) and hence satisfies (A4).
Proof. From the first term in the Calculus of PDO [J s L]J −s ∈ S 2 . A further expansion of [J s , a 3 ∂ 3 x ] gives: where the substitution ξ 2 = ξ 2 − 1 was used and the terms of order s were absorbed into the remainder. Performing a similar computation for the remaining terms in [J s L] and composition with J −s verifies (11).
Remark 16. A simple computation shows that the adjoint L * of the operator L from (1) is Both L * and L(T − t) satisfy (A1)-(A3). This completes the proof of Theorem 1 by the Corollary 14.

Ill-posedness
Ill-posedness is proved by justifying the formal geometrical optics argument, cf [4], for a special choice of initial data. It is instructive to first consider the case of constant dispersion a 3 ≡ 1: Then the condition (A3N ′ ) is equivalent to The general case of (1) is later reduced to illposedness for (12). For this reduction it is desirable to relax the condition (A2) to smooth, but not necessarily bounded coefficients: x and c 1 , c 0 ∈ C 0 t,x . From now on, the notation C = C(α) means that there exists a constant C ≥ 1 that depends continuously α and may depend on the norms of coefficients c j evaluated on some compact set, whose size also depends on α. Constants required to be small are reciprocal of the large constants.
Using (14), (13) and the estimate above implies 0 with comparability constant chosen to be 2. The estimates (15) and (16) are the main ingredient for the proof of the following theorem.
Theorem 18. Suppose Then there exists a sequence t n → 0, and sequences x n 0 and ξ n , η n such that v n ∈ C 1 t L 2 x ∩ C 0 t H 3 x from (14) and g n from (12) satisfy Proof. By (17), there exist x 0 ∈ R and N > 0, such that e 1 3 x 0 Let t n = N 3ξ 2 with ξ = ξ(x 0 , N ) to be chosen below. From now on only t, such that 0 ≤ t ≤ t n will be considered. For this range of t, the small parameter η = η(x 0 − 3ξ 2 t) > 0 can be chosen to depend only on (x 0 , N ). As the choice of x 0 and η completely determines ψ, ψ is independent of ξ.
This estimate requires a comparison of (16) and (20). To this end, by (16) and the Fundamental Theorem of Calculus A computation shows, that for smooth functions b j andb j . Using this computation and a 3 (x, t)( ∂y ∂x ) 3 ≡ 1 substitute (25) into (1) to get where the coefficients c j satisfy (A2') and, in particular The relationship between f and g is identical to (25), thus (26) implies Therefore, (1) can be reduced to (12), which was analyzed in the Theorem 18.
Lemma 21. Suppose (A1), (A2) and (A3N ′ ) hold. Let s ∈ R. Then there exists a sequence u n ∈ C 1 H s ∩ C 0 t H s+3 and t n → 0, such that Note, that (17) for c 2 (y, t) defined by (28) is equivalent to (A3N'). Therefore, Theorem 18 applies to (27). Define u n by applying (25) to v n from the Theorem (18), which can be written explicitly as Let f n = ∂ t u n + Lu n and g n defined by (28). Then (18), (26) and (29) imply up to a subsequence u n (t n ) L 2 x ≥ n( u n (0) L 2 x + tn 0 (∂ t + L)u n (t) L 2 x dt) > 0 (32) Likewise, (19) holds for u n instead of v n . This completes the proof of (30) for s = 0 by taking w n := u n .
For the general s ∈ R, commute J s with L as in Lemma 15: J s (∂ t + L) = (∂ t +L)J s , wherẽ L = L + [J s L]J −s . By Lemma 15L = P + A s (x, t, ∂ x ), where A s ∈ S 0 and the differential operator P satisfies (A1), (A2) and (A3N'). Defineũ n via (31) with L replaced by P . I.e.ũ n differs from u n by a factor of ( a3(x0,t) a3(x,t) ) s . Further define w n (x, t) = J −sũ n (x, t) Hence w n (t) H s = u n (t) L 2 . Applying (32) toũ n implies w n (t n ) H s ≥ n( w n (0) H s x + tn 0 (∂ t + P )ũ n (t) L 2 x dt) > 0 By (19) for n ≥ A s L 2 →L 2 tn 0 A sũn L 2 ≤ w n (0) H s x As J s f = (∂ t + P )ũ n + A sũn , combining the last two estimates implies that w n satisfies (30) by passing to a subsequence.

Proof of illposedness.
Corollary 22. Lemma 21 implies that (2) fails or, more generally, for any T > 0 there is no non-decreasing function C(T ′ ) : [0, T ] → R, such that holds for all u ∈ C 0 [0,T ] H s solving (1). Proof. Assuming (33), for the sake of contradiction, and using (30), implies that C(t n ) ≥ n for all n ∈ N. As t n → 0 and C(t) is non-decreasing in t, C(t) ≥ n for all t > 0 and n ∈ N. This is a contradiction. Proof. Suppose, for the sake of contradiction, that (4) holds for some [0, T ] and some continuous function C(t 0 , t) for 0 ≤ t 0 ≤ t ≤ T . Define a non-decreasing function C(T ′ ) = sup 0≤t0≤t≤T ′ C(t 0 , t). Then by the Duhamel principle every solution of (1) satisfies u(t) = S(t, 0)u 0 +