Local Well-posedness for 2-D Schrodinger Equation on Irrational Tori and Bounds on Sobolev Norms

In this paper we consider the cubic Schrodinger equation in two space dimensions on irrational tori. Our main result is an improvement of the Strichartz estimates on irrational tori. Using this estimate we obtain a local well-posedness result in H^s for s>131/416. We also obtain improved growth bounds for higher order Sobolev norms.


Introduction
The equation we consider in this paper is the cubic, Schrödinger equation on irrational tori, namely, where α = ±1. T is the time of existence of the solutions and T 2 θ is the irrational tori, R 2 /θ 1 Z × θ 2 Z for θ 1 , θ 2 > 0 and θ 1 /θ 2 irrational. The equation is called focusing for α = −1 and defocusing for α = 1.
The equation (1) posed T 2 , has been studied widely for its importance in the theory of differential equations. The equation is energy subcritical, see [13], and for defocusing equation, for any initial data in H 1 there is global well-posedness and global bounds on the Sobolev norm of the solution, see [1]. In addition there have been many results on the well-posedness of (1) for both focusing and defocusing case for rough initial data (in H s for s < 1) on two dimensional torus, see [3], [5], [6], [7], [9] and also on more general compact manifolds, see [4], [14].
One of the main tools in proving local well-posedness is the Strichartz estimates, i.e. estimates of the form (2) e it∆ f L 4 , for some s 0 ≥ 0 and f ∈ H s 0 2 (T 2 θ ). Our main focus in this paper will be on the improvement of this estimate on irrational tori. As one can see for θ 1 = θ 2 = 1 we get the usual (flat) torus.
Although the domain resembles the flat torus, the tools used to prove (2) are fundamentally different. The reason behind this difference is that the symbol of the Laplacian on flat torus at any (m, n)-level is m 2 + n 2 whereas the symbol of it on a irrational torus is of the form (θ 1 m) 2 + (θ 2 n) 2 . Thus the method of counting lattice points on a circle to get (2) cannot be applied here. In 3-d, Bourgain [3], used bounds on the l p -norms on the number of lattice points on the ellipsoid and Jarnick's estimate [11] to get (2) with s 0 = 1 3 . A slight modification of his method in 2-d gives us a 1 4 -derivative loss in (2). But this result was already proven for not only on irrational tori but also on any two dimensional compact manifold, see [4]. This remedy was overcome in Catoire and Wang's paper [6] using Jarnick's estimate, see [11], in two dimensions without passing to the l p -norms of the number of lattice points on ellipsoids. They obtained (2) with s 0 2 = 1 6 . The first part of this paper will be consisted of our main result, improving (2) to s 0 2 = 131 832 using a counting argument of Huxley, [10]. In the second part of the paper, using the theory of Bourgain spaces, we prove local well-posedness for initial data in H s , s > s 0 and also polynomial bounds on the growth of the Sobolev norms of the solution for the defocusing case. On 2-d flat tori, we should note that the local well-posedness theory gives the exponential bound [2]. Also in [2], Bourgain improved this exponential bound with the polynomial bound u(t) H s C t 2(s−1)+ using the following polynomial estimate: If there exists a constant r ∈ (0, 1) and δ > 0 such that for any time t 0 we have, then we get It suffices to prove this result for t being an integer multiple of δ and the rest follows from induction. This result was later improved by Staffilani, see [12] to u(t) H s C t (s−1)+ .

Notations
Throughout the paper, , which will be used for Sobolev spaces too. We will use (.) + to denote (.) ǫ for all ǫ > 0 with implicit constants depending on ǫ and will use the usual Japanese bracket notation, x = (1 + x 2 ) 1/2 .
The linear propogator of the Schrödinger equation will be denoted as e it∆ , where it is defined on the Fourier side as ( The Bourgain spaces X s,b will be defined as the closure of compactly supported smooth functions under the norm , and the restricted norm will be given as For any operator D and positive number N , χ being the characteristic function, χ D N u is Also, in this paper we will use s 0 = 131 416 .

Preliminaries
When we say the equation (1) is locally well-posed in H s , we mean that there exist a time T LW P = T LW P ( u 0 H s ) such that the solution exists and is unique in X s,b T LW P ⊂ C([0, T LW P ), H s ) and depends continuously on the initial data. We say that the the equation is globally well-posed when T LW P can be taken arbitrarily large.
By Duhamel Principle, we know that the smooth solutions of (1) satisfy the integral equation Thus, our main concern is to find the fixed point to the integral operator To do that, we will use Banach Fixed Point Theorem on the set of functions and get a contraction for sufficiently small T .
We also note that the equation has mass and energy conservations, namely, and, Thus, for the defocusing equation, i.e. α = 1, we have global bounds on the H 1 -norm of the solution. This also says, for defocusing equation we have H 1 global well-posedness.

Main Results
Theorem 4.1. The 2-d cubic Schrödinger equation (1) is locally well-posed for initial data And we will also prove, x) be the solution to the defocusing cubic Schrödinger equation (1). Then for any time t, we have,

4.1.
Proof of The Results.

Proof of Strichartz estimates.
To be able to prove (2), we will use a counting argument by Huxley, [10]: Proof.
where, to pass to the last inequality we used: Proof. For any finite sum over n, Here, to pass to the second line we used Plancherel's equality. In the third line we used that the Fourier transform of φ is a Schwartz function and decays faster than any polynomial, and thus we can choose an α > 1. Also to pass to the last line we used Young's inequality.
and hence, using l N 2 , we obtain, Therefore, the result.
The proof of this result is standard, see [8]. Hence, to be able to use the Banach Fixed Point Theorem, we have to control the right hand side of the inequality in the appropriate X s,b space. And since our nonlinearity is cubic, that means have to show a trilinear estimate: such that, Hence, it is clear that once we prove Proposition 4.7, Theorem 4.1, i.e. the local wellposedness will follow.
We will prove Proposition 4.7 using the duality argument Hence, to prove Proposition 4.7, we will bound the integral on the right hand side of this equality by

Proof. (Proof of Proposition 4.7)
We first need a fundamental bilinear Strichartz estimate: Proof. Let P I be the partition of Z 2 into boxes I of size N 1 . We can decompose u 2 as u 2 = I u 2 ), which follows from the convolution property, we have, Using this bilinear Strichartz estimate we can also prove: With an abuse of notation we will callũ i = u i . For n ∈ Z 2 , Q(n) being the symbol of Laplacian we have, by the definition of linear propagator, where v i (τ, x) = 1 (2π) 2 n∈Z 2 u i (τ − Q(n), n)e ix.n . Thus, and for each i we use, Also, using embedding X 0,(1/4)+ ⊂ L 4 t L 2 x , which is obtained by interpolation between X 0,b ⊂ L ∞ t L 2 x for b > 1/2 and X 0,0 = L 2 t L 2 x , we see that, Now we can prove a crude interpolation between (3) and (4) and get: Proof. We have, and we also have, Note that (5) ≤ N 1 u 1 X 0,b u 2 X 0,(1/4)+ . Then for fixed u 1 interpolating between this result and (6), we get, for somes ∈ [s 0 , s) and b ′ < 1/2. Also note that (5) ≤ N 1 u 1 X 0,(1/4)+ u 2 X 0,b ′ . Thus, for fixed u 2 , interpolating between this result and (7) we obtain, Thus we get that, if u i ∈ X 0,b , i ∈ {1, 2, 3, 4} are functions s.t. their space Fourier We are almost ready to finish the proof of the proposition. All we need now is to guarantee the existence of b, b ′ which satisfy the conditions of Proposition 4.6. But for that we need better estimates on the restrictions of functions on the eigenspaces of the Laplacian. Let ℘ k be the projection onto the e k , the eigenspace of Laplacian corresponding to the eigenvalue µ k . Also for each e k we see that µ k e k = −∆e k which implies that hence, e k 's are supported on µ k = (θ 1 m 1 ) 2 + (θ 2 m 2 ) 2 = Q(m 1 , m 2 ) i.e. they are supported on µ s k = |Q(m 1 , m 2 )| 2s . This gives that Since ℘ k u L 2 x ≤ u L 2 x and that e k 's form an orthonormal basis, we can define Sobolev space H s , with the norm, u 2 H s = k µ k s ℘ k u 2 L 2 which we will be using later in the paper.

Now if we consider the integrals of the form
and we will use this observation in our estimates. Now we can show the existence of 1/4 < b ′ < 1/2 < b s.t. for every s > s 0 , which will finish the proof of local well-posedness.
As we mentioned before, we will bound, | u 1 u 2 u 3 u 4 dxdt|. To do so, it is enough to bound this integral for where N i is a dyadic integer. Let without loss of generality that N 1 ≤ N 2 ≤ N 3 and let s ′ ∈ (s 0 , s) then for the range N 4 ≤ 8N 3 , Hence, for the range of the frequencies, write N 4 = 2 n N 3 for n ≤ 3 and then we have and summing in N 1 , N 2 , N 4 and n ≤ 3 we get where we used the H s -orthogonality of the operators χ The proof of the theorem will mainly follow Bourgain's arguments in [2], i.e. we will use Lemma 1.1. For that, first we need to observe that for s > 1, in the proof of Proposition 4.7 if we take u 1 = u 2 = u 3 = u and s ′ = 1−, redoing the calculations we get This says, we can choose the local well-posedness interval depending only on u(0) H 1 .
Thus we can find T 0 > 0 such that, for any time τ ≥ 0 the solution exists for t ∈ [τ, τ + T 0 ]. Now we need to find r ∈ (0, 1) such that for any t ∈ [τ, τ + T 0 ], Since L 2 x -norm of the solution is conserved, it is enough to show this estimate inḢ s x . Without loss of generality we can take τ = 0. Since for some s − 1 ≥ σ > 0, writing H s−σ as the interpolation space between H 1 and H s we will obtain, we get, Thus we get the desired bound using u X s,b T 0 u(0) H s in the local well-posedness interval.
The term I is harder to deal with since the highest order derivatives acts on u. The main problem here is that, because of the term (D s u) 2 in the integrand we expect to have a bound of the form II u 2 which is not useful. To remedy that problem, we will try to get for some σ > 0 to be determined. In the following estimates we will mainly follow Zhong's arguments in [14].
Since the summand is summable in L's and N 's, we get the result for σ = 2(b − 1/4)−.
Now we are left with the last case,