On the Cauchy Problem for Von Neumann-Landau Wave Equation

In present paper we prove the local well-posedness for Von Neumann-Landau wave equation by the T. Kato’s method.


Introduction
For the stationary Von Neumann-Landau wave equation, Chen investigated the Dirichlet problems [1], where the generalized solution is studied by Function-analytic method.The present paper is related to the Cauchy problem: the Von Neumann-Landau wave equation where 2 2 1 , , u t x y is an unknown complex valued function on 1 2n +  and f is a nonlinear complex valued function.
If the plus "+" is replaced by the minus "−" on right hand in Equation (1), then the resulted equation is the Schrödinger equation.For the Schrödinger equation, the well-posedness problem is investigated for various nonlinear terms f .In terms of the nonlinear terms f , the problem (1) can be divided into the subcritical case and the critical case for 1 H solutions.We are concerned with the subcritical case and obtain a local well-posedness result by the T. Kato's method.
The paper is organized as follows.Section 2 contains the list of assumptions on the interaction term f and the main result is presented.Section 3 is concerned with the Strichartz estimates.Finally, in Section 4, the main result is proved.

Statement of the Main Result
In this section we list the assumptions on the interaction term f and state the main result.Firstly, we recall that the definition of admissible pair [2].
Definition 2.1.Fix We say that a pair ( ) , q r of exponents is admissible if and ( ) Remark 2.1.The pairs ( ) The two pairs are called the endpoint cases.
Secondly, let ( ) for all , for all measurable function u and a.e.
for all , t I ∈ where : .
x y The main result is the following theorem:  : : there exists a unique strong 1 H -solution u to the Equation where 2, r α = + and ( ) , q r is an admissible pair.
the unique solution u is defined on a maximal interval ( ) (iv) There is the blowup alternative: If max , T < ∞ then ( ) ( )

Strichartz Estimates
In this subsection, we recall that the Strichartz estimates.Let ( ) .
It is easy to verify that the L is a self-adjoint unbounded operator on ( ) .
n H  Then, by Stone theorem we see that itL e is an unitary group on ( ) e itL can be expressed explicitly by Fourier transform.
, e e , d d .4π The following result is the fundamental estimate for e .
where p′ is the dual exponent of , p defined by the formula 1 1 1.

p p + = ′
Proof.For the proof please see [3] or [4].□ The following estimates, known as Strichartz estimates, are key points in the method introduced by T. Kato [5].

Lemma 2. Let ( )
, q r and ( ) , q r   be any admissible exponents.Then, we have the homogeneous Strichartz estimate and the inhomogeneous Strichartz estimate for any interval J and real number 0 .t Proof.For the proof please see [3] or [4] in the non-endpoint case.On the other hand, the proof in the endpoint case follows from the theorem 1.2 in [6] and the lemma 1 in the present paper.□

The Proof of Theorem
Proof.Let ( ) one easily verifies that for any , .
Using (17), we deduce from Hölder's inequality that And it follows from Remark 1.3.1 (vii) in [2] that We now proceed in four steps.
Step 1. Proof of (i).Fix , 0, A T > to be chosen later, and let 2 r α = + , q be such that ( ) , q r is an ad- missible pair, and set ( ) , equipped with the distance , , , = .
We claim that ( ) be a Cauchy sequence.Clearly, , , , lim inf ; , :  is measurable, and we deduce easily that , .
Using the embedding ( ) ( ) and Hölder's inequality in time, we deduce from the above estimates that ( ) and , .
It follows from ( 22) and Strichartz estimates (lemma 2) that , , , , , Also, we deduce from (23) that Finally, note that .q q′ > We now proceed as follows.For any ( ) be the unique positive number so that It then follows from ( 26) and ( 28) that for any 0 ( ) and by ( 27) we obtain ( , .
In particular,  is a strict contraction on .E By Banach's fixed-point theorem,  has a unique fixed , . By the definition 2.2, we con-clude that u is a strong 1 For uniqueness, assume that , u v are two strong .
For simplicity, we set .
Note that ( ) where the constant , , and the constant 2 C is independent of J by above inequalities.Note that , q q ′ < we conclude that 0 w = by the lemma 4.2.2 in [2].So .u v = Step 2. Proof of (ii).Suppose that ( ) in R B as .k → ∞ By the part (i), we denote k u and u by the unique solution of (1) corresponding to the initial value ( ) 0 k u and u , respectively.We will show that and the estimate (29) which implies that (27) holds for .
Next, we need to estimate 0 0 e e d .
A similar identity holds for .
k u We use the assumption ( ) Note that 1 f and 2 f are also 1 , C so that .
There, if we prove that also a Cauchy sequence in

1 H
-solution of (1) on [ ] ) and Strichartz estimates (16), then we obtain ( ) Note that the choosing of the time T in (28), it follows from (27) with (30) that

(
z ∈  and some constant 3 .C Therefore, arguing as in Step 1, we obtain the estimate