SELF-SIMILAR SOLUTIONS TO NONLINEAR DIRAC EQUATIONS AND AN APPLICATION TO NONUNIQUENESS

. Self-similar solutions to nonlinear Dirac systems (1) and (2) are constructed. As an application, we obtain nonuniqueness of strong solution in super-critical space C ([0 ,T ]; H s ( R )) ( s < 0) to the system (1) which is L 2 ( R ) scaling critical equations. Therefore the well-posedness theory breaks down in Sobolev spaces of negative order.


1.
Introduction. We are interested in the initial value problem for the following nonlinear Dirac equations and with the initial data U j (x, 0) = u j (x). Here U j : R 1+1 → C for j = 1, 2 andŪ is a complex conjugate of U . The systems (1) and (2) have the charge conservation 54 HYUNGJIN HUH time T > 0 and solution U j ∈ C([0, T ], H s (R)) (s ≥ 0). Furthermore, in [2], it is proven that a local solution in L 2 can be extended to a global one by excluding concentration of the L 2 norm at a point. In this study we construct self-similar solutions to (1), (2) and establish the ill-posedness of the initial value problem to the nonlinear Dirac equations (1) in super-critical space C([0, T ]; H s (R)) (s < 0) by showing nonuniqueness of solutions. Therefore the well-posedness theory of (1) breaks down in Sobolev spaces of negative order. We could not prove nonuniqueness of (2) because the cubic terms |U 1 | 2 U 2 , |U 2 | 2 U 1 are not integrable. We refer to Remark 1 in section 3 for details.
Nonuniqueness of solution for Burgers' equation has been shown in [5] by constructing a nontrivial solution which converges to zero in the H s (s <?1/2) topology as t → 0 + . Nonuniqueness of solutions of the one-dimensional nonlinear Schrödinger equation has been studied in [3] where the author showed that there exist nonzero weak solutions varying continuously in the H s (s < 0), with vanishing initial data. Making use of self-similar solutions inside the null cone {(x, t) ∈ R 2 : |x| ≤ |t|}, we will build particular particular nontrivial solutions which converge to zero in the H s (s < 0) norm as t → 0 + . To construct proper self-similar solutions, special algebraic structure of nonlinear terms should be considered crucially. Our main result is as follows.
Theorem 1.1. There exist strong solutions to (1) satisfying not identically vanishing, with initial data u j (x) ≡ 0.
For the precise meaning of strong solution, we refer to section 3. We will reduce (1) to a simpler equation (3) and construct self-similar solutions inside the null cone. The self-similar solutions are constructed in two separate regions and glued continuously. To construct a self-similar solution of (2), we will consider the special algebraic structure of nonlinearity which was used in [6]. In section 2, the self-similar solutions of (1) and (2) are constructed. In section 3, we introduce the definition of strong solutions and prove Theorem 1.1. We use the standard Sobolev spaces H s (R) with the norm f H s = (1 − ∆) s/2 f L 2 .
2. Self similar solutions. In this section, we construct self similar solutions of equations (1) and (2). We reduce PDEs (1) and (2) to ODEs of the self-similar variable y = x/t. Then the ODEs are solved, considering the special algebraic structure of nonlinearity, inside the null cone.
2.1. Self similar solution of (1). We consider the change of variables U 1 = U , U 2 = iV and suppose that U , V are real valued functions. Then the system (1) reduces to Then any solution to (3) becomes a solution of (1). We construct self-similar solutions of the form

NONUNIQUENESS OF SOLUTION TO NONLINEAR DIRAC EQUATIONS 55
Then the system (3) reduces to where we use the variable y = x/t. We can check that d Assuming that the constant of integration is zero, we derive From now on, we consider the case −1 < y < 1. Making use of (5), we have from (4) We start with the choice of +, in a region −1 < y ≤ 0, which leads to where a > 0 is a constant of integration. With the choice of −, in a region 0 ≤ y < 1, we have Through the similar process to u, we have for |x| ≥ t.
Note that U and V are continuous across the axis {(x, t)| x = 0 and t > 0}. 56 HYUNGJIN HUH 2.2. Self similar solution of (2). Let us try to find solutions to (2) of the form where V j are complex-valued functions. Then the Thirring equations (2) reduce to where we use the variable y = x/t. For −1 < y < 1, the system (8) can be rewritten as .
Then we have (1 + y) Observing |e i y 0 f (s)ds | = 1 for a real-valued function f which was used in [6], we deduce Then the system (9) reads as Now we define, in the region t ≥ 0, and for |x| ≥ t.

Proof of Theorem 1.1. Let us introduce the definition of strong solution to
(1), (2) which can be written in the following form where F 1 , F 2 are cubic polynomials.
Definition 3.1. Consider the Cauchy problem (10) with initial data (u 1 (x), u 2 (x)) ∈ (H s (R)) 2 . It is said that U = (U 1 , U 2 ) is a strong solution to the Cauchy problem on the time interval [0, T ] provided that satisfies the equations (10) in the following sense. For any φ ∈ C ∞ 0 (R × (−T, T )), we have where F j (U 1 , U 2 ) ∈ L 1 loc (R × (−T, T )). Now we prove Theorem 1.1. We will show that the functions U and V in section 2.1 are nontrivial strong solutions of (1) with the trivial initial data U (x, 0) ≡ 0 ≡ V (x, 0). First of all, we know that U (·, t), V (·, t) ∈ L p (R) (1 ≤ p < 4) for each t > 0. Here we just show the case of V : dy. Let Then we can check that 0 < g(y) ≤ 1 2 for a ≥ 2. Actually we have g(−1) = g(1) = 1 2 . Then it is easy to check V (·, t) ∈ L p (R) (1 ≤ p < 4) for each t > 0. Moreover, taking into account L p (R) → H ) ≤ C lim t→0 + ( U L p (R) + V L p (R) ) = 0, which shows that the nontrivial solutions converge to zero in the H s (s < 0) norm as t → 0 + . We note that nonlinear terms V 2 U , U 2 V , V 3 , U 3 are integrable. Here we just show the case of V 2 U . Making use of (6) and (7), we have dy.
Remark 1. Let us consider functions U j in section 2.2. Put V 1 (0) = 1 = V 2 (0) for simplicity. We can check U j (·, t) ∈ L p (R) (1 ≤ p < 2) for each t > 0. In fact, we have where we use change of variable y = x/t. Taking into account L p (R) → H 1 2 − 1 p (R) for 1 < p < 2, we can verify (11) and lim t→0 + U j H ≤ C lim t→0 + U j L p (R) = 0, for 1 < p < 2. Therefore U j converges to zero in the H s (s < 0) norm as t → 0 + . Here we should note that the nonlinear terms |U 1 | 2 U 2 , |U 2 | 2 U 1 are not integrable. We have, for instance, Therefore we have a problem in understanding U j as a solution to (2) in the sense of distribution.
The main difference between the self-similar solutions for equations (1) and (2) is that, when one writes the nature of h is essentially different: for system (1), it behaves like (t − x) 1/4 as x → t; in system (2), it is a constant. This is the essential point that makes the nonlinearity integrable for (1) but not for (2).