Global well-posedness and scattering for the quantum Zakharov system in

By Yung-Fu Fang and Kenji Nakanishi

Abstract

We study the Cauchy problem for the quantum Zakharov system in the class of square-integrable functions on the Euclidean space of general dimensions. The local well-posedness is proven for dimensions up to eight, together with global existence for dimensions up to five, as well as scattering for small initial data in dimensions greater than three.

1. Introduction

In this paper, we consider the global well-posedness and scattering of solutions in for the quantum Zakharov (QZ) system in general space dimensions . The system reads as

where and are the unknown quantities. See Reference 8, Reference 17, and Reference 18 for the physical meanings. All the coefficients are set to in this paper, since their sizes play no role in our analysis, nor do the signs of the quadratic terms on the right side. In other words, one can change their sizes, as well as the signs on the right side, in Equation 1.1 without changing the conclusions. It should be noted, however, that our estimates are not uniform with respect to the size.

When the quantum effect is absent, the system is reduced to the classical Zakharov system

The regular solutions of Equation 1.2 satisfy the conservation of mass

as well as the conservation of the Hamiltonian

Analogously, Equation 1.1 has the conservation of mass

as well as the conservation of the Hamiltonian

although we will not use the Hamiltonian structure in this paper, since our solutions will be merely in . Note that to have a positive definite quadratic part in the Hamiltonian, the signs of the two quadratic terms in Equation 1.1 should be the same, but we do not need it in this paper.

For simplicity, we transform Equation 1.1 into the first order equations in by the change of variable

where is the real part of . Thus the quantum Zakharov system Equation 1.1 becomes

where .

In this paper, we study the Cauchy problem for Equation 1.6 with initial data , locally and globally in time, just using the classical Strichartz estimates. In most of the preceding works, the initial data (and so the solutions) with more regularity were considered, except some recent results Reference 5Reference 7 in one dimension. It is needless to say that is the most important and basic function space in mathematics, and it is convenient to work with solutions with no derivative for various reasons, including numerical computations. The norm is physically important for the Zakharov (type) systems, as it measures the total electric energy of the plasma. For the mathematical analysis of PDE, it is important to solve the system in the invariant (conserved) function space, particularly for global analysis of the solutions. In comparison with the classical Zakharov system Equation 1.2, it is interesting to see how much the system (or its dynamics) is tamed by the quantum effect. It turns out that we can deal with the data in high space dimensions as well, unlike the classical case, where the well-posedness for is known Reference 2Reference 6Reference 13 only for , and the solution can blow up in Reference 9Reference 10. It is interesting because tremendous efforts have been devoted to constructing and analyzing rough solutions to nonlinear dispersive equations, where models arising in the mathematical physics tend to exhibit various mathematical challenges, making the corresponding analysis complicated even if the equations look very simple. This paper suggests that including some more physical effects in the equations can make our mathematical understanding much easier and better.

The main results of this paper are as follows.

Theorem 1.1 (Well-posedness).

For , the quantum Zakharov system Equation 1.6 is locally well-posed in . Moreover, if , then all the solutions are global in time, and there exist positive constants , , and , dependent only on , such that

A more precise form of the upper bound is given in the proof for each ; see Equation 5.4.

Theorem 1.2 (Small data scattering).

Let . Assume that is sufficiently small in -norm. Then the solution of Equation 1.6 is global in time, and there exist such that

In short, the solution scatters in as .

Remark 1.

It is worth noting that we do not need at all the dispersive nature of the (fourth order) wave equation, either for the small data scattering (Theorem 1.2) or for the local well-posedness in (Theorem 1.1). It means that in the left side of the second equation of Equation 1.6 could be replaced with any self-adjoint operator on for those results.

The outline of the paper is as follows. In Section 2, we prepare some notation and definitions. In Section 3, we recall the Strichartz estimates for the linear equations, which is the main tool for analysis in this paper. In Section 4, we prove the local well-posedness result, Theorem 1.1. In Section 5, we show the global well-posedness and polynomial bound of wave, Theorem 1.1. Finally in Section 6, we prove the scattering result, Theorem 1.2.

2. Notation and Duhamel formulae

Let us denote

Then we have . We also denote the Fourier transform of over by .

For , we denote by and the usual inhomogeneous and homogeneous Sobolev spaces equipped with the norms, respectively,

For and , denotes the inhomogeneous Besov space equipped with the norm

where and , with a fixed smooth and radial function satisfying for , 0 for . Similarly, the homogeneous Besov space is denoted by .

For any or any interval and any Banach space of functions on , the -valued norm in time is denoted by

for any , and similarly for . We often add the subscript in order to highlight the function space for .

The complex interpolation space and the real interpolation space are denoted respectively by and , for a couple of Banach spaces , , and .

For the fourth order Schrödinger equation of ,

we have the Duhamel formula

For the (square-root of) fourth order wave equation to ,

we have the Duhamel formula

3. Strichartz estimates

In this section, we recall the Strichartz estimates for the operators and , which follow from the standard arguments by the Fourier analysis. Following Reference 25, a pair is called Schrödinger admissible, for short -admissible, if

A pair is called biharmonic admissible, for short -admissible, if

Lemma 1 (Pausader Reference 25).

Let be a solution of Equation 2.3. For any -admissible pairs and , it satisfies

and for any -admissible pairs and and any ,

In both estimates, the implicit constants depend only on .

The proof is based on the work of Kenig-Ponce-Vega Reference 20 or the works of Ben-Artzi-Koch-Saut Reference 3, Pausader Reference 25, and Keel-Tao Reference 19, together with some modifications. We only sketch the proof.

Proof.

We consider the integral

where the phase function is given by . Then we have

The Hessian of the phase function can be expressed in the polar coordinate () as follows (for ):

where is the projection onto and is the orthogonal projection onto . Note that span is one dimensional and is dimensional. Hence we have

as well as for all multi-index such that . Then the decay estimate can be obtained by the stationary phase method as follows (cf. Reference 25, (3.6) or Reference 11, Theorem 2.2):

Via duality argument and argument (cf. Reference 19), we obtain the Strichartz estimate Equation 3.4 for -admissible exponents. Applying the Sobolev embedding to both sides of the estimate, we obtain Equation 3.3 for -admissible pairs unless or is . The last case is covered by the interpolation inequality as follows. If is -admissible, then and is -admissible. Hence there exists small such that the pairs of exponents defined by

are -admissible. Note that the pairs are near the pair . Then the Sobolev embeddings and the real interpolation, i.e.,

imply the interpolation inequality