On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces

We proved the local well-posedness for the power-type nonlinear semi-relativistic or half-wave equation (NLHW) in Sobolev spaces. Our proofs mainly bases on the contraction mapping argument using Strichartz estimate. We also apply the technique of Christ-Colliander- Tao in \cite{ChristCollianderTao} to prove the ill-posedness for (NLHW) in some cases of the super-critical range.

The cases ν ∈ (1, 5] when d = 2 and ν ∈ (1, 3] when d ≥ 3 still remain open. It requires another technique rather than just Strichartz estimates. Note that conditions (1.3) and (1.4) allow the nonlinearity to have enough of regularity to apply the fractional derivative estimates (see Subsection 2.2). Finally, using the technique of Christ-Colliander-Tao given in [7], we are able to prove the ill-posedness for the (NLHW) in a certain range of the super-critical case. More precisely, we prove the ill-posedness in H γ with γ ∈ (−∞, −d/2] ∪ [0, γ c ) when γ c > 0, γ ∈ (−∞, −d/2] ∩ (−∞, γ c ) otherwise. (1.5) We expect that the ill-posedness still holds in the range γ ∈ (−d/2, max{0, γ c }) as for the classical nonlinear Schrödinger equation (see [7]). But it is not clear to us how to prove it at the moment. As mentioned above, the authors in [18] used the technique of [7] with the pseudo-Galilean transformation to prove the ill-posedness for the (NLFS) with negative exponent. Unfortunately, it seem to be difficult to control the error of the pseudo-Galilean transformation in high Sobolev norms. As one can check from the ill-posedness result of [18] (Theorem 1.5 there), this result holds only in the one dimensional case and does not hold in the two and three dimensional cases as claimed there. Moreover, the dependence of υ in the estimate (5.7) of [18] seems to be eliminated which may effect the proof of the ill-posedness given in [18]. We thus do not persuade the technique of [18] to prove the ill-posedness result for the (NLHW) in the range γ ∈ (−d/2, max{0, γ c }). We end this paragraph by noting that the techniques used in this note can be applied without any difficulty for the (NLFS). It somehow provides better results for those of [18]. Let us now recall known results about the local well-posedness and the ill-posedness for the (NLHW) in 1D. It is well-known that the (NLHW) is locally well-posed in H γ (R), with γ > 1/2 satisfying (1.3) if ν is not an odd integer, by using the energy method and the contraction mapping argument. When ν = 3, i.e. cubic nonlinearity, the (NLHW) is locally well-posed in H γ (R) with γ ≥ 1/2 (see e.g. [22], [24]). This result is optimal in the sense that the equation is ill-posed in H γ (R) provided γ < 1/2 (see e.g. [6]). The proof of this ill-posedness result is mainly based on the relation with the cubic Szegö equation, which can not be easily extended to general nonlinearity. To our knowledge, the local well-posedness for the generalized (NLHW) in H γ (R) with γ ≤ 1/2 seems to be an open question.
We denote the Littlewood-Paley projections by P 0 := χ 0 (D), P N := χ(N −1 D) with N = 2 k , k ∈ Z where χ 0 (D), χ(N −1 D) are the Fourier multipliers by χ 0 (ξ) and χ(N −1 ξ) respectively. Given γ ∈ R and 1 ≤ q ≤ ∞, the Sobolev and Besov spaces are defined by where S ′ is the space of tempered distributions. Now let S 0 be a subspace of the Schwartz space S consisting of functions φ satisfying D αφ (0) = 0 for all α ∈ N d where· is the Fourier transform on S and S ′ 0 is its topology dual space. One can see S ′ 0 as S ′ /P where P is the set of all polynomials on R d . The homogeneous Sobolev and Besov spaces are defined bẏ It is easy to see that the norms u B γ q and u Ḃ γ q do not depend on the choice of ϕ 0 , and S 0 is dense inḢ γ q ,Ḃ γ q . Under these settings, H γ q , B γ q ,Ḣ γ q andḂ γ q are Banach spaces with the norms u H γ q , u B γ q , u Ḣ γ q and u Ḃ γ q respectively (see e.g. [28]). In this note, we shall use H γ := H γ 2 , H γ :=Ḣ γ 2 . We note (see [3], [15]) that if 2 ≤ q < ∞, thenḂ γ q ⊂Ḣ γ q . The reverse inclusion holds for 1 < r ≤ 2. In particular,Ḃ γ 2 =Ḣ γ andḂ 0 2 =Ḣ 0 2 = L 2 . Moreover, if γ > 0, then H γ q = L q ∩Ḣ γ q and B γ q = L q ∩Ḃ γ q . In the sequel, a pair (p, q) is said to be admissible if We also denote for (p, q) ∈ [1, ∞] 2 , Our first result concerns with the local well-posedness for the (NLHW) in the sub-critical case. i. If T * < ∞, then u(t) H γ → ∞ as t → T * .
ii. u depends continuously on u 0 in the following sense. There exists 0 < T < T * such that if u 0,n → u 0 in H γ and if u n denotes the solution of the (NLHW) with initial data u 0,n , then 0 < T < T * (u 0,n ) for all n sufficiently large and u n is bounded in iii. Let β > γ be such that if ν is not an odd integer, ⌈β⌉ ≤ ν. If u 0 ∈ H β , then u ∈ C([0, T * ), H β ).
The continuous dependence can be improved to hold in C([0, T ], H γ ) if we assume that ν > 1 is an odd integer or ⌈γ⌉ ≤ ν − 1 otherwise (see Remark 1). We also have the following local well-posedness with small data scattering in the critical case.
and also, if ν is not an odd integer, (1.4). Then for all u 0 ∈ H γc , there exist T * ∈ (0, ∞] and a unique solution to the (NLHW) satisfying Moreover, if u 0 Ḣγc < ε for some ε > 0 small enough, then T * = ∞ and the solution is scattering in H γc , i.e. there exists u + 0 ∈ H γc such that Our final result is the following ill-posedness for the (NLHW). This note is organized as follows. In Section 2, after recalling Strichartz estimates for the linear half-wave equation and nonlinear fractional derivative estimates, we prove the local wellposedness given in Theorem 1.1 and Theorem 1.2. The proof of the ill-posedness will be given in Section 3.

Local well-posedness
In this section, we will give the proofs of Theorem 1.1 and Theorem 1.2. Our proofs are based on the standard contraction mapping argument using Strichartz estimates and nonlinear fractional derivatives.

Linear estimates
In this subsection, we recall Strichartz estimates for the linear half-wave equation. [21]). Let d ≥ 2, γ ∈ R and u be a (weak) solution to the linear half-wave equation, namely for some data u 0 , F . Then for all (p, q) and (a, b) admissible pairs, where γ p,q and γ a ′ ,b ′ are as in (1.6). In particular, Here (a, a ′ ) and (b, b ′ ) are conjugate pairs.
The proof of this result is based on the scaling technique. We refer the reader to [2, Section 8.3] for more details.
Corollary 1. Let d ≥ 2 and γ ∈ R. If u is a (weak) solution to the linear half-wave equation for some data u 0 , F , then for all (p, q) admissible satisfying q < ∞, (2.10) Proof. We firstly remark that (2.9) together with the Littlewood-Paley theorem yield for any (p, q) admissible satisfying q < ∞, The estimate (2.10) then follows by using the fact that γ p,q > 0 for all (p, q) is admissible satisfying q < ∞.

Nonlinear estimates
In this subsection, we recall some nonlinear fractional derivative estimates related to our purpose. Let us start with the following fractional Leibniz rule (or Kato-Ponce inequality).
Proposition 1. Let γ ≥ 0, 1 < r < ∞ and 1 < p 1 , p 2 , q 1 , q 2 ≤ ∞ satisfying Then there exists C = C(d, γ, r, p 1 , q 1 , p 2 , q 2 ) > 0 such that for all u, v ∈ S , We refer to [17] for the proof of above inequalities and more general results. We also have the following fractional chain rule.
A similar estimate holds withḢ γ r ,Ḣ γ p -norms are replaced by H γ r , H γ p -norms respectively. A next result will give a good control on the nonlinear term which allows us to use the contraction mapping argument.
The above lemma follows the same spirit as in [18,Lemma 3.5] (see also [10]) using the argument of [9, Lemma 3.1].
Proof. We only give a sketch of the proof in the case d ≥ 4, the cases d = 2, 3 are treated similarly. By interpolation, we can assume that ν − 1 = m/n > 2, m, n ∈ N with gcd(m, n) = 1. We proceed as in [18] and set By Bernstein's inequality, we have This implies that for θ ∈ (0, 1) which will be chosen later, We next use Estimating the n highest frequencies by (2.18) and the rest by (2.19), we get For an arbitrary δ > 0, we set Using the fact that c N (t) ≤c N (t) andc Nj (t) (N 1 /N j ) δc N1 (t) for j = 2, ..., m and similar estimates for primes, we see that We can rewrite the above quantity in the right hand side as . By choosing θ = 1/(ν − 2) ∈ (0, 1) and δ > 0 so that .
Here condition ν > 3 ensures that m − 2n > 0. Summing in N m , then in N m−1 ,..., then in N 2 , we have The Hölder inequality with the fact that (ν − 3)n ≥ 1 implies and similarly for c ′ (t) ℓ q (2 Z ) . The Minkowski inequality then implies Since it converges to u in the ditribution sense, so the limit is u(t). Thus . The proof is complete.

Proof of Theorem 1.1
We now give the proof of Theorem 1.1 by using the standard fixed point argument in a suitable Banach space. Thanks to (1.2), we are able to choose p > max(ν − 1, 4) when d = 2 and p > max(ν − 1, 2) when d ≥ 3 such that γ > d/2 − 1/p and then choose q ∈ [2, ∞) such that Step 1. Existence. Let us consider where I = [0, T ] and M, T > 0 to be chosen later. By the Duhamel formula, it suffices to prove that the functional is a contraction on (X, d). The Strichartz estimate (2.10) yields where F (u) = |u| ν−1 u and similarly for F (v). By our assumptions on ν, Corollary 2 gives and The Sobolev embedding with the fact that γ − γ p,q > d/q implies L p (I, H γ−γp,q q ) ⊂ L p (I, L ∞ ). Thus, we get This shows that for all u, v ∈ X, there exists C > 0 independent of u 0 ∈ H γ and T such that Therefore, if we set M = 2C u 0 H γ and choose T > 0 small enough so that CT 1− ν−1 p M ν−1 ≤ 1 2 , then X is stable by Φ and Φ is a contraction on X. By the fixed point theorem, there exists a unique u ∈ X so that Φ(u) = u.
Step 2. Uniqueness. Consider u, v ∈ C(I, H γ ) ∩ L p (I, L ∞ ) two solutions of the (NLHW). Since the uniqueness is a local property (see [5,Chapter 4]), it suffices to show u = v for T is small. We have from (2.22) that Since u L p (I,L ∞ ) is small if T is small and similarly for v, we see that if T > 0 small enough, Step 3. Item i. Since the time of existence constructed in Step 1 only depends on H γ -norm of the initial data. The blowup alternative follows by standard argument (see e.g. [5,Chapter 4]).
Step 4. Item ii. Let u 0,n → u 0 in H γ and C, T = T (u 0 ) be as in Step 1. Set M = 4C u 0 H γ . It follows that 2C u 0,n H γ ≤ M for sufficiently large n. Thus the solution u n constructed in Step 1 belongs to X with T = T (u 0 ) for n large enough. We have from Strichartz estimate (2.10) and (2.21) that ). We also have from (2.22) and the choice of T that This yields that u n → u in L ∞ (I, ) for any admissible pair (a, b) with b < ∞. The convergence in C(I, H γ−ǫ ) follows from the boundedness in L ∞ (I, H γ ), the convergence in L ∞ (I, L 2 ) and that Step 5. Item iii. If u 0 ∈ H β for some β > γ satisfying ⌈β⌉ ≤ ν if ν > 1 is not an odd integer, then Step 1 shows the existence of H β solution defined on some maximal interval [0, T ). Since H β solution is also a H γ solution, thus T ≤ T * . Suppose that T < T * . Then the unitary property of e itΛ and Lemma imply that for all 0 ≤ t < T . The Gronwall's inequality then gives for all 0 ≤ t < T . Using the fact that u ∈ L ν−1 loc ([0, T * ), L ∞ ), we see that lim sup u(t) H β < ∞ as t → T which is a contradiction to the blowup alternative in H β .
Remark 1. If we assume that ν > 1 is an odd integer or ⌈γ⌉ ≤ ν − 1 otherwise, then the continuous dependence holds in C(I, H γ ). To see this, we consider X as above equipped with the following metric Using Item (ii) of Corollary 2, we have . The Sobolev embedding then implies for all u, v ∈ X, Therefore, the continuity in C(I, H γ ) follows as in Step 4.

Proof of Theorem 1.2
We now turn to the proof of the local well-posedness and small data scattering in critical case by following the same argument as in [10].
Step 1. Existence. We only treat for d ≥ 4, the ones for d = 2, d = 3 are completely similar. Let us consider where I = [0, T ] and T, M, N > 0 will be chosen later. One can check (see e.g. [4] or [5]) that (X, d) is a complete metric space. Using the Duhamel formula the Strichartz estimate (2.9) yields A similar estimate holds for u hom L ∞ (I,Ḣ γc ) . We see that u hom L 2 (I,Ḃ ≤ ε for some ε > 0 small enough which will be chosen later, provided that either u 0 Ḣγc is small or it is satisfied some T > 0 small enough by the dominated convergence theorem. Therefore, we can take T = ∞ in the first case and T be this finite time in the second. On the other hand, using again (2.9), we have A same estimate holds for u inh L ∞ (I,Ḣ γc ) . Corollary 2 and Lemma 2.3 give Similarly, we have This implies for all u, v ∈ X, there exists C > 0 independent of u 0 ∈ H γc such that Φ(u) Now by setting N = 2ε and M = 2C u 0 Ḣγc and choosing ε > 0 small enough such that CN 2 M ν−3 ≤ min{1/2, ε/M }, we see that X is stable by Φ and Φ is a contraction on X. By the fixed point theorem, there exists a unique solution u ∈ X to the (NLHW). Note that when u 0 Ḣγc is small enough, we can take T = ∞.
Step 2. Uniqueness. The uniqueness in C ∞ (I, H γc ) ∩ L 2 (I, B γc−γ 2,2 ⋆ 2 ⋆ ) follows as in Step 2 of the proof of Theorem 1.1 using (2.25). Here u can be small as T is small.
Step 3. Scattering. The global existence when u 0 Ḣγc is small is given in Step 1. It remains to show the scattering property. Thanks to (2.24), we see that as t 1 , t 2 → +∞. We have from (2.25) that which also tends to zero as t 1 , t 2 → +∞. This implies that the limit exists in H γc . Moreover, we have The unitary property of e itΛ in L 2 , (2.26) and (2.27) imply that u(t) − e itΛ u + 0 H γc → 0 when t → +∞. This completes the proof of Theorem 1.2.

Ill-posedness
In this section, we will give the proof of Theorem 1.3. We follow closely the argument of [7] using small dispersion analysis and decoherence arguments.

Small dispersion analysis
Now let us consider for 0 < δ ≪ 1 the following equation Note that (3.28) can be transformed back to the (NLHW) by using Lemma 3.1. Let k > d/2 be an integer. If ν is not an odd integer, then we assume also the additional regularity condition ν ≥ k + 1. Let φ 0 be a Schwartz function. Then there exists C, c > 0 such that if 0 < δ ≤ c sufficiently small, then there exists a unique solution
Proof. We refer the reader to [7,Lemma 2.1] where the small dispersion analysis is invented to prove the ill-posedness for the classical nonlinear Schrödinger equation. The same proof can be applied to the nonlinear half-wave equation without any difficulty. By using the energy method, we end up with the following estimate Thus, if |t| ≤ c| log δ| c for suitably small 0 < δ ≤ c, then exp(C(1 + |t|) C ) ≤ δ −1/2 and (3.29) follows.
Remark 2. By the same argument as in [7], we can get the following better estimate for all |t| ≤ c| log δ| c , where H k,k is the weighted Sobolev space Now let λ > 0 and set It is easy to see that u (δ,λ) is a solution of the (NLHW).
Here we use the fact that This completes the proof of (3.32).

Proof of Theorem 1.3
We are now able to prove Theorem 1.3. We only consider the case t ≥ 0, the one for t < 0 is similar. Let ǫ ∈ (0, 1] be fixed and set Note that we are considering here γ < γ c . This implies that 0 < λ ≤ δ ≪ 1, and Lemma 3.2 gives u (δ,λ) (0) H γ ≤ Cǫ.
We now split the proof to several cases.
Therefore, for any ε > 0, there exists a solution of the (NLHW) satisfying for some t ∈ (0, ε). Thus for any t > 0, the solution map S ∋ u(0) → u(t) for the Cauchy problem (NLHW) fails to be continuous at 0 in the H γ -topology.
This shows that the solution map fails to be uniformly continuous on L 2 .