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Global strong solution to a nonlinear Dirac type equation in one dimension
Nonlinear Analysis 80 (2013) 150–155
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Global strong solution to a nonlinear Dirac type equation in one dimension Yongqian Zhang School of Mathematical Sciences, Fudan University, Shanghai 200433, Key Laboratory of Mathematics for Nonlinear Sciences, China
article
abstract
info
Article history: Received 12 June 2012 Accepted 9 October 2012 Communicated by Enzo Mitidieri
This paper studies a class of nonlinear massless Dirac equations in one dimension, which include the equations for the massless Thirring model and the massless Gross–Neveu model. Under the assumptions of the initial data having small charge and being bounded, the global existence of the strong solution is established. The decay of the local charge is also proved. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Semilinear hyperbolic system Dirac equation Global solution Glimm functional Approximate conservation law
1. Introduction We consider the nonlinear massless Dirac equations
i(ut + ux ) = G1 (u, v), i(vt − vx ) = G2 (u, v),
(1.1)
with initial data
(u, v)|t =0 = (u0 (x), v0 (x))
(1.2)
where (x, t ) ∈ R , (u, v) ∈ C , G1 = ∂u W (u, v) + F1 (u, v) and G2 = ∂v W (u, v) + F2 (u, v) which satisfy the following: 2
2
(A1) W (u, v) : C2 → R1 is a polynomial in variables |u|2 and |v|2 ; (A2) Fk (k = 1, 2) are polynomials of cubic order in variables u, v and u, v , and moreover there exists a positive constant c such that
|uF1 (u, v)| + |v F2 (u, v)| ≤ c |u|2 |v|2 for (u, v) ∈ C2 . Many physical models verify (A1) and (A2)—for example:
• The massless Thirring model [1] W = α|u|2 |v|2 ,
α ∈ R1 ,
F1 = F2 = 0.
• The Federbusch model W = α(|u|2 + |v|2 )|u|2 |v|2 ,
α ∈ R1 ,
F1 = F2 = 0.
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.10.008
Y. Zhang / Nonlinear Analysis 80 (2013) 150–155
• The massless Gross–Neveu model [2] 2 (F1 , F2 ) = ∇(u,v) α uv + uv , α ∈ R1 ,
151
W = 0.
The nonlinear Dirac equation has been studied by many authors for the massless or massive models; see for examples [1–16], and the references therein. The global existence of a solution for the Cauchy problem has been established in [3,7,9,10] in Sobolev space H s for suitable s. Recently some studies have been devoted to the local low regularity solutions in a certain subspace of C ([0, T ]; H s (R1 )) for s > 0; see for instance [14–16] and references therein. In [13], using the explicit representation of the smooth solution by [10] (see also [14]), Huh got the strong solution in L2 (R1 ) of the nonlinear equations for the massless Thirring model and Federbusch model. The uniqueness in L∞ ([0, T ]; L2 (R1 ) ∩ L4 (R1 )) and the asymptotic behavior of the field were also established there. In [5], Candy used the Strichartz estimates to prove the global existence of an L2 -strong solution to the nonlinear Dirac equation for the massive Thirring model. For the nonlinear term being a polynomial of quintic or higher order, the small global solution has been constructed in [15] by using the end point Strichartz estimates. In contrast to the above, we consider in this paper the case where nonlinear terms include some cubic terms and may have more complex structure. Such kinds of equations include the massless Gross–Neveu model, for which, to our knowledge, the global existence of a strong solution in L2 is still an open problem [15]. And it was pointed out by Pelinovsky [15] that the a p p priori estimates of the Lp -norm (∥u(t )∥Lp + ∥v(t )∥Lp )1/p for the equations, like in the Gross–Neveu model, include nonlinear ∞ terms, which may lead to the blow-up of the L -norms and H 1 -norms. In this paper, we will overcome the difficulties caused by the nonlinear terms in the a priori estimates for p = 2 by introducing a Glimm type functional. To this end, for (1.1), we first have (2.1) for |u|2 and |v|2 . The assumptions (A1) and (A2) on G1 and G2 imply that the right hand sides in Eqs. (2.1) are bounded by O(1)|u|2 |v|2 , and therefore Eqs. (2.1) are similar to the approximate conservation law for the interaction between the elementary waves [17]; see also [18–20]. Then, following Bony’s approach [21], forany local H 1 -solution (u, v) ∞ with small initial charge we introduce a Glimm type functional L(t )+KQ (t ) to get the control on −∞ (|u(x, t )|2 +|v(x, t )|2 )dx
t ∞
|u(x, τ )|2 |v(x, τ )|2 dxdτ ; see Lemma 2.1. These bounds enable us to apply the characteristic method to Eqs. (2.1) to get the uniform L∞ bounds on |u|2 and |v|2 for extending the local H 1 -solution globally in time; see Lemmas 2.2 and 2.3 and
−∞
0
for the L∞ bounds. Such a functional was first given by Glimm [17] to establish the global existence of the BV solution for the system of conservation laws, and a similar technique has been used to study the general conservation laws, Boltzmann equations and wave maps in R1+1 ; see for instance [17–23] and the references therein. With these estimates on H 1 -solutions, our aim is to find the global solutions in the following sense. Definition 1.1. A function (u, v) is said to be a strong solution to (1.1)–(1.2) on the time interval [0, T ] if there exists a sequence of classical solutions (un , v n ) to (1.1) on R1 × [0, T ] such that (un , v n ) → (u, v) strongly in C ([0, T ]; L2 (R1 )) and (un , v n )|t =0 → (u0 , v0 ) strongly in L2 (R1 ). The main result is stated as follows. Theorem 1.1. Suppose that (u0 , v0 ) ∈ L2 (R1 )∩L∞ (R1 ) and that the norm ∥u0 ∥L2 (R1 ) +∥v0 ∥L2 (R1 ) is small. Then (1.1)–(1.2) admits a unique global strong solution (u, v) ∈ C ([0, ∞); L2 (R1 )) ∩ L∞ (R1 × [0, ∞)). Moreover, for any constants a < b, b
(|u(x, t )|2 + |v(x, t )|2 )dx → 0,
t → +∞.
a
Here and in the sequel, by U ∈ C (I ; L2 (R1 )) for an interval I we mean that the function U : t ∈ I → U (·, t ) ∈ L2 (R1 ) is continuous with respect to t. The remaining part of the paper is organized as follows. In Section 2, on the basis of the existence of local H 1 -solutions, we will introduce a Glimm functional for H 1 -solutions and get the uniform L2 and L∞ bounds of the solution, which are independent of time and yield the bound of H 1 -norm for the solution. This enables us to extend the smooth solutions globally in time; see Proposition 2.1. In Section 3, we take the limit to get the global strong solution in L2 and prove the uniqueness of the strong solution and the decay of the local charge. 1 Remark 1.1. We can define the weak solution as follows. (u, v) ∈ C ([0, T ]; L2 (R1 )) ∩ L1 ((0, T ); L∞ loc (R )) is defined to be 1 the weak solution of (1.1)–(1.2) iff (u, v) solves Eqs. (1.1) in R × (0, T ) in the sense of the distributions with (u, v)|t =0 = 1 2 (u0 , v0 ).1 Then, we can get a global weak solution in the same way for (u0 , v0 ) ∈ L∞ loc (R ) with small L -norm.
2. The global classical solution We consider the case where (u0 , v0 ) ∈ Cc∞ in this section. (1.1) is a semilinear hyperbolic system. Local existence of the solutions in H 1 for the semilinear strictly hyperbolic system can be proved by standard methods using Duhamel’s principle and the fixed point argument; see for instance [4,7,20]. Now suppose that for (u0 , v0 ) ∈ Cc∞ , the initial value problem 1 This definition for the solution was suggested by the referees. The author would like to thank the referees for this.
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Y. Zhang / Nonlinear Analysis 80 (2013) 150–155
(1.1)–(1.2) has a smooth solution (u, v) ∈ C ([0, T∗ ); H 1 (R1 )) ∩ C 1 ([0, T∗ ); L2 (R1 )) for some T∗ > 0. Then the support of (u(x, t ), v(x, t )) is compact for each t ∈ [0, T∗ ). To extend the solution across the time t = T∗ we only need to show that sup
0≤t
∥u(t )∥H 1 + ∥v(t )∥H 1 ≤ C ′ exp(C ′′ T∗ )
where the constants C ′ and C ′′ are independent of T∗ . To this end, we multiply the first equation of (1.1) by u and the second equation by v ; then
2 (|u| )t + (|u|2 )x = 2ℜ(iF1 u), (|v|2 )t − (|v|2 )x = 2ℜ(iF2 v).
(2.1)
Here and in the sequel ℜZ denotes the real part of Z ∈ C. As in [21] (see also [22,23]), we define the following functional for the solution for any t ∈ [0, T∗ ): Q (t ) =
x
|u(x, t )|2 |v(y, t )|2 dxdy,
and L(t ) = D(t ) =
∞
(|u(x, t )|2 + |v(x, t )|2 )dx,
−∞ ∞
|u(x, t )|2 |v(x, t )|2 dx.
−∞
These functionals are given as in [21] (see also [22]). Q (t ) is called a Bony functional [22] and is similar to the Glimm interaction potential. The linear combination L(t ) + KQ (t ) for any constant K > 0 is called the Glimm type functional [17]. Direct computation shows that dQ (t ) dt
x
(|u(x, t )|2 )x |v(y, t )|2 dxdy +
|u(x, t )|2 (|v(y, t )|2 )y dxdy +2 |uF1 (u, v)(x, t )||v(y, t )|2 dxdy + 2 |u(x, t )|2 |v F2 (u, v)(y, t )|dxdy x
x
−∞
and dL(t ) dt
≤ 2cD(t ).
Therefore for any constant K > 0, dL(t ) dt
+K
dQ (t ) dt
≤ (−2 + 2L(t )c )K + 2c D(t ).
(2.2)
Lemma 2.1. There exist constants δ > 0 and K > 0 such that for the initial data satisfying L(0) + KQ (0) ≤ δ , the following holds: dL(t ) dt
+K
dQ (t ) dt
≤ −D(t )
(2.3)
for t ∈ [0, T∗ ). Therefore, L(t ) + KQ (t ) +
t
D(τ )dτ ≤ L(0) + KQ (0) 0
for t ∈ [0, T∗ ). Proof. First choose constants δ > 0 and K > 0 such that
−2 + 2δ c < −1/2,
−K /2 + 2c < −2.
If L(0) + KQ (0) ≤ δ , then (−2 + 2L(0)c )K + 2c ≤ −2. Let T0 = sup{T ; (−2 + 2L(t )c )K + 2c < −1, t ∈ [0, T )}.
(2.4)
Y. Zhang / Nonlinear Analysis 80 (2013) 150–155
153
Fig. 1. Domain Ω (x0 , t0 ).
We want to show that T0 = T∗ . Otherwise, suppose that T0 < T∗ ; then with (2.2) we have (2.4) for t ∈ [0, T0 ). Therefore, by the continuity of L(t ) and Q (t ) for 0 ≤ t < T∗ , we have the following: L(t ) + KQ (t ) ≤ δ,
(−2 + 2L(t )c )K + 2c ≤ −2,
t ∈ [0, T0 ],
which leads to a contradiction to the definition of T0 on using again the continuity of L(t ) for 0 ≤ t < T∗ . Thus the proof is complete. Next we shall use the above estimates to get the L∞ bound of the solution for 0 < t < T∗ . Define
Ω (x0 , t0 ) = {(x, t ) : 0 < t < t0 , x0 − (t0 − t ) < x < x0 + (t0 − t )}, and
ΓR (x0 , x0 ) = {(x, t ) : 0 < t < t0 , x = x0 + (t0 − t )}, ΓL (x0 , x0 ) = {(x, t ) : 0 < t < t0 , x = x0 − (t0 − t )}; see Fig. 1. Lemma 2.2. For any t0 ∈ (0, T∗ ), the following hold:
√
2
|u| ≤
2
1
2
x 0 + t0 − t
|u| |v| dxdt + √
2c
2
Ω (x0 ,t0 )
ΓR (x0 ,t0 )
|u0 (x)|2 dx
x 0 − t0 + t
and
√
2
|v| ≤
2
1
2
x 0 + t0 − t
|u| |v| dxdt + √
2c
ΓL (x0 ,t0 )
2
Ω (x0 ,t0 )
|v0 (x)|2 dx.
x 0 − t0 + t
Proof. Integrating (2.1) over Ω (x0 , t0 ), then by Green formula we have
|u|2 =
2
ΓR (x0 ,t0 )
Ω (x0 ,t0 )
2ℜ(iF1 u)dxdt +
x 0 + t0 − t
|u0 (x)|2 dx
x 0 − t0 + t
|u|2 |v|2 dxdt +
≤ 2c Ω (x0 ,t0 )
x 0 + t0 − t
|u0 (x)|2 dx
x 0 − t0 + t
and
|v|2 =
2
ΓL (x0 ,t0 )
Ω (x0 ,t0 )
2ℜ(iF2 v)dxdt +
≤ 2c
x 0 + t0 − t
|v0 (x)|2 dx
x 0 − t0 + t
|u|2 |v|2 dxdt +
Ω (x0 ,t0 )
The proof is complete.
x 0 + t0 − t
|v0 (x)|2 dx.
x 0 − t0 + t
Lemma 2.3. Let δ and K be the constants given in Lemma 2.1. Then for any constants a and b with a < b, the following hold: sup
0≤t
|u(x, t )|2 + |v(x, t )|2 ≤ C1
sup
a−T∗
|u0 (x)|2 + |v0 (x)|2 ,
and b
|u(x, t )|2 + |v(x, t )|2 dx ≤ C1 a
√
where C1 = exp((2 2c + 2
√
b
|u0 (x − t )|2 + |v0 (x + t )|2 dx, a
2c )δ).
154
Y. Zhang / Nonlinear Analysis 80 (2013) 150–155
Proof. Using the characteristic method, by (2.1) and (A2) we have d dt
|u(x0 + (t − t0 ), t )|2 = 2ℜ(iF1 u) ≤ 2c |u(x0 + (t − t0 ), t )|2 |v(x0 + (t − t0 ), t )|2
and d dt
|v(x0 − (t − t0 ), t )|2 = 2ℜ(iF2 v) ≤ 2c |u(x0 − (t − t0 ), t )|2 |v(x0 − (t − t0 ), t )|2 ,
for any (x0 , t0 ) with a − (T∗ − t0 ) < x0 < b + (T∗ − t0 ). Then,
t0
|v(x0 + (τ − t0 ), τ )| dτ |u(x0 , t0 )| ≤ |u(x0 − t0 , 0)| exp 2c 0 2 2 |v| = |u(x0 − t0 , 0)| exp 2c 2
2
2
ΓL (x0 ,t0 )
≤ |u0 (x0 − t0 )|2 exp I1 (x0 , t0 ) and
t0 |v(x0 , t0 )|2 ≤ |v(x0 + t0 , 0)|2 exp 2c |u(x0 − (τ − t0 ), τ )|2 dτ 0 = |v(x0 + t0 , 0)|2 exp 2c |u|2 ΓR (x0 ,t0 )
≤ |v(x0 + t0 , 0)| exp I2 (x0 , t0 ) , where the estimates for Γ (x ,t ) |u|2 and Γ (x ,t ) |v|2 in Lemma 2.2 are used, and R 0 0 L 0 0 √ 2 √ x0 +t0 I1 (x0 , t0 ) = 2 2c |u|2 |v|2 dxdt + 2c |u0 |2 dx, 2
Ω (x0 ,t0 )
√
I2 (x0 , t0 ) = 2 2c
2
x 0 − t0 2
√
2
|u| |v| dxdt +
x0 +t0
2c
Ω (x0 ,t0 )
|v0 |2 dx.
x 0 − t0
Therefore these lead to the desired result by Lemma 2.1. The proof is complete.
Now, to get the H 1 bound of the solution, we differentiate Eqs. (1.1) with respect to x; then i(uxt + uxx ) = ∇u,v,u,v G1 (u, v) · (ux , vx , ux , v x ), i(vxt − vxx ) = ∇u,v,u,v G2 (u, v) · (ux , vx , ux , v x ),
which lead to the following: sup
0≤t
∥u(t )∥H 1 + ∥v(t )∥H 1 ≤ ∥u0 ∥H 1 + ∥v0 ∥H 1 exp(C ′ T∗ ) < ∞,
where the constant C ′ depends only on the L∞ bounds of (u, v) given in Lemma 2.3. Therefore, the local solution could be extended globally. Since Cc∞ (R1 ) is dense in H 1 (R1 ), with the above estimates, we get the following. Proposition 2.1. Suppose that (u0 , v0 ) ∈ H 1 (R1 ) and that ∥u0 ∥L2 (R1 ) +∥v0 ∥L2 (R1 ) is small enough. Then (1.1)–(1.2) has a unique global classical solution (u, v) ∈ C ([0, +∞); H 1 (R1 )) ∩ C 1 ([0, +∞); L2 (R1 )) ∩ L∞ (R1 × [0, ∞)). Moreover, there exists some positive constant C2 , depending only on ∥u0 ∥2L2 + ∥v0 ∥2L2 , such that sup t >0,x∈R1
|u(x, t )|2 + |v(x, t )|2 ≤ C2 sup |u0 (x)|2 + |v0 (x)|2 ,
b
|u(x, t )|2 + |v(x, t )|2 dx ≤ C2 a
for any a < b.
x∈R1
a
b
|u0 (x − t )|2 + |v0 (x + t )|2 dx,
Y. Zhang / Nonlinear Analysis 80 (2013) 150–155
155
3. Proof of the main result Let (uj,0 , vj,0 ) ∈ Cc∞ (R1 ) be a sequence of smooth functions such that (uj,0 , vj,0 ) → (u0 , v0 ) strongly in L2 and such that supj ∥(uj,0 , vj,0 )∥L∞ < ∞. Denote by (uj , vj ) the H 1 -solutions of (1.1) with initial data (uj,0 , vj,0 ). Then by Proposition 2.1 we derive the following: sup (∥uj (t )∥L∞ + ∥vj (t )∥L∞ ) < ∞.
(3.1)
t > 0 ,j
Now for any k, j,
(uk − uj )t + (uk − uj )x = iG1 (uj , vj ) − iG1 (uk , vk ), (vk − vj )t − (vk − vj )x = iG2 (uj , vj ) − iG2 (uk , vk ),
which, together with (3.1), gives the following by the energy estimate method:
∥(uk − uj )(t )∥2L2 + ∥(vk − vj )(t )∥2L2 ≤ exp(O(1)t ) ∥(uk − uj )(0)∥2L2 + ∥(vk − vj )(0)∥2L2 , where the bounds of O(1) depend only on the L∞ bounds of the solutions (uj , vj ) (j = 1, 2, . . .) for t ≥ 0. This yields the 2 1 convergence of the sequence {(uj , vj )}∞ j=0 to a function (u, v) in C ([0, T ); L (R )) for any T > 0. Then (u, v) is a global strong solution to (1.1)–(1.2). Moreover, by using Proposition 2.1 and by repeating the argument of taking the limit, we have b
|u(x, t )|2 + |v(x, t )|2 dx ≤ C1
a
b
|u0 (x − t )|2 + |v0 (x + t )|2 dx a
b +t
|u0 (y)| dy + 2
= C1 a+t
b−t
|v0 (y)| dy . 2
a −t
Then the decay of the local charge follows. In the same way, we can prove the following for any strong solutions (u′ , v ′ ) and (u′′ , v ′′ ):
∥(u′ − u′′ )(t )∥2L2 + ∥(v ′ − v ′′ )(t )∥2L2 ≤ exp(O(1)t ) ∥(u′ − u′′ )(0)∥2L2 + ∥(v ′ − v ′′ )(0)∥2L2 for t ≥ 0, where the bounds of O(1) depend only on the L∞ bounds of the solutions. This leads to the uniqueness of the strong solution. The proof of Theorem 1.1 is complete. Acknowledgments This work was partially supported by NSFC Project 11031001 and 11121101, by the 111 Project B08018 and by the Ministry of Education of China. The author would like to thank the referees for helpful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
W.E. Thirring, A soluble relativistic field theory, Ann. Phys. 3 (1958) 91–112. D.J. Gross, A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10 (1974) 3235–3253. A. Bachelot, Global existence of large amplitude solutions for nonlinear massless Dirac equation, Portucaliae Math. 46 (Fasc. Supl.) (1989) 455–473. A. Bachelot, Global Cauchy problem for semilinear hyperbolic systems with nonlocal interactions. Applications to Dirac equations, J. Math. Pures Appl. 86 (2006) 201–236. T. Candy, Global existence for an L2 critical nonlinear Dirac equation in one dimension, Adv. Differential Equations 16 (7–8) (2011) 643–666. M. Chungunova, D. Pelinovsky, Block-diagonalization of the symmetric first-order coupled-mode system, SIAM J. Appl. Dyn. Syst. 5 (2006) 66–83. V. Delgado, Global solution of the Cauchy problem for the (classical) coupled Maxwell–Dirac and other nonlinear Dirac equations, Proc. Amer. Math. Soc. 69 (2) (1978) 289–296. J.P. Dias, M. Figueira, Time decay for the solutions of a nonlinear Dirac equation in one space dimension, Ric. Mat. 35 (1986) 309–316. J.P. Dias, M. Figueira, Sur l’existence d’une solution globale pour une equation de Dirac non linéaire avec masse nulle, C. R. Acad. Sci. Paris, Sér. I 305 (1987) 469–472. J.P. Dias, M. Figueira, Remarque sur le problème de Cauchy pour une equation de Dirac non linéaire avec masse nulle, Portucaliae Math. 45 (4) (1988) 327–335. M. Escobedo, L. Vega, A semilinear Dirac equation in H s (R3 ) for s > 1, SIAM J. Math. Anal. 28 (2) (1997) 338–362. M.J. Esteban, M. Lewin, E. Séré, Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc. 45 (4) (2008) 535–598. H. Huh, Global strong solution to the Thirring model in critical space, J. Math. Anal. Appl. 381 (2011) 513–520. S. Machihara, T. Omoso, The explicit solutions to the nonlinear Dirac equation and Dirac–Klein–Gordon equation, Ric. Mat. 56 (1) (2007) 19–30. D. Pelinovsky, Survey on global existence in the nonlinear Dirac equations in one dimension, 26 November 2010. arXiv:1011.5925vl [math-ph]. S. Selberg, A. Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension, Differential Integral Equations 23 (3–4) (2010) 265–278. J. Glimm, Solution in the large for nonlinear systems of conservation laws, Comm. Pure Appl. Math. 18 (1965) 695–715. A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, Oxford University Press Inc., New York, 2000. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2010. L. Tartar, From Hyperbolic Systems to Kinetic Theory, Springer, 2008. J.M. Bony, Solution globales bornées pour les modèles discrets de l’équation de Boltzmann, en dimension 1 d’espace, Journées Équations Aux Derivées Partielles (Saint Jean de Monts 1987) XVI (1987) p1–p10. S.-Y. Ha, A.E. Tzavaras, Lyapunov functionals and L1 -stability for discrete velocity Boltzmann equations, Comm. Math. Phys. 239 (2003) 65–92. Y. Zhou, Uniqueness of weak solutions in 1 + 1 dimensional wave maps, Math. Z. 232 (1999) 707–719.
Contents lists available at SciVerse ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Global strong solution to a nonlinear Dirac type equation in one dimension Yongqian Zhang School of Mathematical Sciences, Fudan University, Shanghai 200433, Key Laboratory of Mathematics for Nonlinear Sciences, China
article
abstract
info
Article history: Received 12 June 2012 Accepted 9 October 2012 Communicated by Enzo Mitidieri
This paper studies a class of nonlinear massless Dirac equations in one dimension, which include the equations for the massless Thirring model and the massless Gross–Neveu model. Under the assumptions of the initial data having small charge and being bounded, the global existence of the strong solution is established. The decay of the local charge is also proved. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Semilinear hyperbolic system Dirac equation Global solution Glimm functional Approximate conservation law
1. Introduction We consider the nonlinear massless Dirac equations
i(ut + ux ) = G1 (u, v), i(vt − vx ) = G2 (u, v),
(1.1)
with initial data
(u, v)|t =0 = (u0 (x), v0 (x))
(1.2)
where (x, t ) ∈ R , (u, v) ∈ C , G1 = ∂u W (u, v) + F1 (u, v) and G2 = ∂v W (u, v) + F2 (u, v) which satisfy the following: 2
2
(A1) W (u, v) : C2 → R1 is a polynomial in variables |u|2 and |v|2 ; (A2) Fk (k = 1, 2) are polynomials of cubic order in variables u, v and u, v , and moreover there exists a positive constant c such that
|uF1 (u, v)| + |v F2 (u, v)| ≤ c |u|2 |v|2 for (u, v) ∈ C2 . Many physical models verify (A1) and (A2)—for example:
• The massless Thirring model [1] W = α|u|2 |v|2 ,
α ∈ R1 ,
F1 = F2 = 0.
• The Federbusch model W = α(|u|2 + |v|2 )|u|2 |v|2 ,
α ∈ R1 ,
F1 = F2 = 0.
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.10.008
Y. Zhang / Nonlinear Analysis 80 (2013) 150–155
• The massless Gross–Neveu model [2] 2 (F1 , F2 ) = ∇(u,v) α uv + uv , α ∈ R1 ,
151
W = 0.
The nonlinear Dirac equation has been studied by many authors for the massless or massive models; see for examples [1–16], and the references therein. The global existence of a solution for the Cauchy problem has been established in [3,7,9,10] in Sobolev space H s for suitable s. Recently some studies have been devoted to the local low regularity solutions in a certain subspace of C ([0, T ]; H s (R1 )) for s > 0; see for instance [14–16] and references therein. In [13], using the explicit representation of the smooth solution by [10] (see also [14]), Huh got the strong solution in L2 (R1 ) of the nonlinear equations for the massless Thirring model and Federbusch model. The uniqueness in L∞ ([0, T ]; L2 (R1 ) ∩ L4 (R1 )) and the asymptotic behavior of the field were also established there. In [5], Candy used the Strichartz estimates to prove the global existence of an L2 -strong solution to the nonlinear Dirac equation for the massive Thirring model. For the nonlinear term being a polynomial of quintic or higher order, the small global solution has been constructed in [15] by using the end point Strichartz estimates. In contrast to the above, we consider in this paper the case where nonlinear terms include some cubic terms and may have more complex structure. Such kinds of equations include the massless Gross–Neveu model, for which, to our knowledge, the global existence of a strong solution in L2 is still an open problem [15]. And it was pointed out by Pelinovsky [15] that the a p p priori estimates of the Lp -norm (∥u(t )∥Lp + ∥v(t )∥Lp )1/p for the equations, like in the Gross–Neveu model, include nonlinear ∞ terms, which may lead to the blow-up of the L -norms and H 1 -norms. In this paper, we will overcome the difficulties caused by the nonlinear terms in the a priori estimates for p = 2 by introducing a Glimm type functional. To this end, for (1.1), we first have (2.1) for |u|2 and |v|2 . The assumptions (A1) and (A2) on G1 and G2 imply that the right hand sides in Eqs. (2.1) are bounded by O(1)|u|2 |v|2 , and therefore Eqs. (2.1) are similar to the approximate conservation law for the interaction between the elementary waves [17]; see also [18–20]. Then, following Bony’s approach [21], forany local H 1 -solution (u, v) ∞ with small initial charge we introduce a Glimm type functional L(t )+KQ (t ) to get the control on −∞ (|u(x, t )|2 +|v(x, t )|2 )dx
t ∞
|u(x, τ )|2 |v(x, τ )|2 dxdτ ; see Lemma 2.1. These bounds enable us to apply the characteristic method to Eqs. (2.1) to get the uniform L∞ bounds on |u|2 and |v|2 for extending the local H 1 -solution globally in time; see Lemmas 2.2 and 2.3 and
−∞
0
for the L∞ bounds. Such a functional was first given by Glimm [17] to establish the global existence of the BV solution for the system of conservation laws, and a similar technique has been used to study the general conservation laws, Boltzmann equations and wave maps in R1+1 ; see for instance [17–23] and the references therein. With these estimates on H 1 -solutions, our aim is to find the global solutions in the following sense. Definition 1.1. A function (u, v) is said to be a strong solution to (1.1)–(1.2) on the time interval [0, T ] if there exists a sequence of classical solutions (un , v n ) to (1.1) on R1 × [0, T ] such that (un , v n ) → (u, v) strongly in C ([0, T ]; L2 (R1 )) and (un , v n )|t =0 → (u0 , v0 ) strongly in L2 (R1 ). The main result is stated as follows. Theorem 1.1. Suppose that (u0 , v0 ) ∈ L2 (R1 )∩L∞ (R1 ) and that the norm ∥u0 ∥L2 (R1 ) +∥v0 ∥L2 (R1 ) is small. Then (1.1)–(1.2) admits a unique global strong solution (u, v) ∈ C ([0, ∞); L2 (R1 )) ∩ L∞ (R1 × [0, ∞)). Moreover, for any constants a < b, b
(|u(x, t )|2 + |v(x, t )|2 )dx → 0,
t → +∞.
a
Here and in the sequel, by U ∈ C (I ; L2 (R1 )) for an interval I we mean that the function U : t ∈ I → U (·, t ) ∈ L2 (R1 ) is continuous with respect to t. The remaining part of the paper is organized as follows. In Section 2, on the basis of the existence of local H 1 -solutions, we will introduce a Glimm functional for H 1 -solutions and get the uniform L2 and L∞ bounds of the solution, which are independent of time and yield the bound of H 1 -norm for the solution. This enables us to extend the smooth solutions globally in time; see Proposition 2.1. In Section 3, we take the limit to get the global strong solution in L2 and prove the uniqueness of the strong solution and the decay of the local charge. 1 Remark 1.1. We can define the weak solution as follows. (u, v) ∈ C ([0, T ]; L2 (R1 )) ∩ L1 ((0, T ); L∞ loc (R )) is defined to be 1 the weak solution of (1.1)–(1.2) iff (u, v) solves Eqs. (1.1) in R × (0, T ) in the sense of the distributions with (u, v)|t =0 = 1 2 (u0 , v0 ).1 Then, we can get a global weak solution in the same way for (u0 , v0 ) ∈ L∞ loc (R ) with small L -norm.
2. The global classical solution We consider the case where (u0 , v0 ) ∈ Cc∞ in this section. (1.1) is a semilinear hyperbolic system. Local existence of the solutions in H 1 for the semilinear strictly hyperbolic system can be proved by standard methods using Duhamel’s principle and the fixed point argument; see for instance [4,7,20]. Now suppose that for (u0 , v0 ) ∈ Cc∞ , the initial value problem 1 This definition for the solution was suggested by the referees. The author would like to thank the referees for this.
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Y. Zhang / Nonlinear Analysis 80 (2013) 150–155
(1.1)–(1.2) has a smooth solution (u, v) ∈ C ([0, T∗ ); H 1 (R1 )) ∩ C 1 ([0, T∗ ); L2 (R1 )) for some T∗ > 0. Then the support of (u(x, t ), v(x, t )) is compact for each t ∈ [0, T∗ ). To extend the solution across the time t = T∗ we only need to show that sup
0≤t
∥u(t )∥H 1 + ∥v(t )∥H 1 ≤ C ′ exp(C ′′ T∗ )
where the constants C ′ and C ′′ are independent of T∗ . To this end, we multiply the first equation of (1.1) by u and the second equation by v ; then
2 (|u| )t + (|u|2 )x = 2ℜ(iF1 u), (|v|2 )t − (|v|2 )x = 2ℜ(iF2 v).
(2.1)
Here and in the sequel ℜZ denotes the real part of Z ∈ C. As in [21] (see also [22,23]), we define the following functional for the solution for any t ∈ [0, T∗ ): Q (t ) =
x
|u(x, t )|2 |v(y, t )|2 dxdy,
and L(t ) = D(t ) =
∞
(|u(x, t )|2 + |v(x, t )|2 )dx,
−∞ ∞
|u(x, t )|2 |v(x, t )|2 dx.
−∞
These functionals are given as in [21] (see also [22]). Q (t ) is called a Bony functional [22] and is similar to the Glimm interaction potential. The linear combination L(t ) + KQ (t ) for any constant K > 0 is called the Glimm type functional [17]. Direct computation shows that dQ (t ) dt
x
(|u(x, t )|2 )x |v(y, t )|2 dxdy +
|u(x, t )|2 (|v(y, t )|2 )y dxdy +2 |uF1 (u, v)(x, t )||v(y, t )|2 dxdy + 2 |u(x, t )|2 |v F2 (u, v)(y, t )|dxdy x
x
−∞
and dL(t ) dt
≤ 2cD(t ).
Therefore for any constant K > 0, dL(t ) dt
+K
dQ (t ) dt
≤ (−2 + 2L(t )c )K + 2c D(t ).
(2.2)
Lemma 2.1. There exist constants δ > 0 and K > 0 such that for the initial data satisfying L(0) + KQ (0) ≤ δ , the following holds: dL(t ) dt
+K
dQ (t ) dt
≤ −D(t )
(2.3)
for t ∈ [0, T∗ ). Therefore, L(t ) + KQ (t ) +
t
D(τ )dτ ≤ L(0) + KQ (0) 0
for t ∈ [0, T∗ ). Proof. First choose constants δ > 0 and K > 0 such that
−2 + 2δ c < −1/2,
−K /2 + 2c < −2.
If L(0) + KQ (0) ≤ δ , then (−2 + 2L(0)c )K + 2c ≤ −2. Let T0 = sup{T ; (−2 + 2L(t )c )K + 2c < −1, t ∈ [0, T )}.
(2.4)
Y. Zhang / Nonlinear Analysis 80 (2013) 150–155
153
Fig. 1. Domain Ω (x0 , t0 ).
We want to show that T0 = T∗ . Otherwise, suppose that T0 < T∗ ; then with (2.2) we have (2.4) for t ∈ [0, T0 ). Therefore, by the continuity of L(t ) and Q (t ) for 0 ≤ t < T∗ , we have the following: L(t ) + KQ (t ) ≤ δ,
(−2 + 2L(t )c )K + 2c ≤ −2,
t ∈ [0, T0 ],
which leads to a contradiction to the definition of T0 on using again the continuity of L(t ) for 0 ≤ t < T∗ . Thus the proof is complete. Next we shall use the above estimates to get the L∞ bound of the solution for 0 < t < T∗ . Define
Ω (x0 , t0 ) = {(x, t ) : 0 < t < t0 , x0 − (t0 − t ) < x < x0 + (t0 − t )}, and
ΓR (x0 , x0 ) = {(x, t ) : 0 < t < t0 , x = x0 + (t0 − t )}, ΓL (x0 , x0 ) = {(x, t ) : 0 < t < t0 , x = x0 − (t0 − t )}; see Fig. 1. Lemma 2.2. For any t0 ∈ (0, T∗ ), the following hold:
√
2
|u| ≤
2
1
2
x 0 + t0 − t
|u| |v| dxdt + √
2c
2
Ω (x0 ,t0 )
ΓR (x0 ,t0 )
|u0 (x)|2 dx
x 0 − t0 + t
and
√
2
|v| ≤
2
1
2
x 0 + t0 − t
|u| |v| dxdt + √
2c
ΓL (x0 ,t0 )
2
Ω (x0 ,t0 )
|v0 (x)|2 dx.
x 0 − t0 + t
Proof. Integrating (2.1) over Ω (x0 , t0 ), then by Green formula we have
|u|2 =
2
ΓR (x0 ,t0 )
Ω (x0 ,t0 )
2ℜ(iF1 u)dxdt +
x 0 + t0 − t
|u0 (x)|2 dx
x 0 − t0 + t
|u|2 |v|2 dxdt +
≤ 2c Ω (x0 ,t0 )
x 0 + t0 − t
|u0 (x)|2 dx
x 0 − t0 + t
and
|v|2 =
2
ΓL (x0 ,t0 )
Ω (x0 ,t0 )
2ℜ(iF2 v)dxdt +
≤ 2c
x 0 + t0 − t
|v0 (x)|2 dx
x 0 − t0 + t
|u|2 |v|2 dxdt +
Ω (x0 ,t0 )
The proof is complete.
x 0 + t0 − t
|v0 (x)|2 dx.
x 0 − t0 + t
Lemma 2.3. Let δ and K be the constants given in Lemma 2.1. Then for any constants a and b with a < b, the following hold: sup
0≤t
|u(x, t )|2 + |v(x, t )|2 ≤ C1
sup
a−T∗
|u0 (x)|2 + |v0 (x)|2 ,
and b
|u(x, t )|2 + |v(x, t )|2 dx ≤ C1 a
√
where C1 = exp((2 2c + 2
√
b
|u0 (x − t )|2 + |v0 (x + t )|2 dx, a
2c )δ).
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Y. Zhang / Nonlinear Analysis 80 (2013) 150–155
Proof. Using the characteristic method, by (2.1) and (A2) we have d dt
|u(x0 + (t − t0 ), t )|2 = 2ℜ(iF1 u) ≤ 2c |u(x0 + (t − t0 ), t )|2 |v(x0 + (t − t0 ), t )|2
and d dt
|v(x0 − (t − t0 ), t )|2 = 2ℜ(iF2 v) ≤ 2c |u(x0 − (t − t0 ), t )|2 |v(x0 − (t − t0 ), t )|2 ,
for any (x0 , t0 ) with a − (T∗ − t0 ) < x0 < b + (T∗ − t0 ). Then,
t0
|v(x0 + (τ − t0 ), τ )| dτ |u(x0 , t0 )| ≤ |u(x0 − t0 , 0)| exp 2c 0 2 2 |v| = |u(x0 − t0 , 0)| exp 2c 2
2
2
ΓL (x0 ,t0 )
≤ |u0 (x0 − t0 )|2 exp I1 (x0 , t0 ) and
t0 |v(x0 , t0 )|2 ≤ |v(x0 + t0 , 0)|2 exp 2c |u(x0 − (τ − t0 ), τ )|2 dτ 0 = |v(x0 + t0 , 0)|2 exp 2c |u|2 ΓR (x0 ,t0 )
≤ |v(x0 + t0 , 0)| exp I2 (x0 , t0 ) , where the estimates for Γ (x ,t ) |u|2 and Γ (x ,t ) |v|2 in Lemma 2.2 are used, and R 0 0 L 0 0 √ 2 √ x0 +t0 I1 (x0 , t0 ) = 2 2c |u|2 |v|2 dxdt + 2c |u0 |2 dx, 2
Ω (x0 ,t0 )
√
I2 (x0 , t0 ) = 2 2c
2
x 0 − t0 2
√
2
|u| |v| dxdt +
x0 +t0
2c
Ω (x0 ,t0 )
|v0 |2 dx.
x 0 − t0
Therefore these lead to the desired result by Lemma 2.1. The proof is complete.
Now, to get the H 1 bound of the solution, we differentiate Eqs. (1.1) with respect to x; then i(uxt + uxx ) = ∇u,v,u,v G1 (u, v) · (ux , vx , ux , v x ), i(vxt − vxx ) = ∇u,v,u,v G2 (u, v) · (ux , vx , ux , v x ),
which lead to the following: sup
0≤t
∥u(t )∥H 1 + ∥v(t )∥H 1 ≤ ∥u0 ∥H 1 + ∥v0 ∥H 1 exp(C ′ T∗ ) < ∞,
where the constant C ′ depends only on the L∞ bounds of (u, v) given in Lemma 2.3. Therefore, the local solution could be extended globally. Since Cc∞ (R1 ) is dense in H 1 (R1 ), with the above estimates, we get the following. Proposition 2.1. Suppose that (u0 , v0 ) ∈ H 1 (R1 ) and that ∥u0 ∥L2 (R1 ) +∥v0 ∥L2 (R1 ) is small enough. Then (1.1)–(1.2) has a unique global classical solution (u, v) ∈ C ([0, +∞); H 1 (R1 )) ∩ C 1 ([0, +∞); L2 (R1 )) ∩ L∞ (R1 × [0, ∞)). Moreover, there exists some positive constant C2 , depending only on ∥u0 ∥2L2 + ∥v0 ∥2L2 , such that sup t >0,x∈R1
|u(x, t )|2 + |v(x, t )|2 ≤ C2 sup |u0 (x)|2 + |v0 (x)|2 ,
b
|u(x, t )|2 + |v(x, t )|2 dx ≤ C2 a
for any a < b.
x∈R1
a
b
|u0 (x − t )|2 + |v0 (x + t )|2 dx,
Y. Zhang / Nonlinear Analysis 80 (2013) 150–155
155
3. Proof of the main result Let (uj,0 , vj,0 ) ∈ Cc∞ (R1 ) be a sequence of smooth functions such that (uj,0 , vj,0 ) → (u0 , v0 ) strongly in L2 and such that supj ∥(uj,0 , vj,0 )∥L∞ < ∞. Denote by (uj , vj ) the H 1 -solutions of (1.1) with initial data (uj,0 , vj,0 ). Then by Proposition 2.1 we derive the following: sup (∥uj (t )∥L∞ + ∥vj (t )∥L∞ ) < ∞.
(3.1)
t > 0 ,j
Now for any k, j,
(uk − uj )t + (uk − uj )x = iG1 (uj , vj ) − iG1 (uk , vk ), (vk − vj )t − (vk − vj )x = iG2 (uj , vj ) − iG2 (uk , vk ),
which, together with (3.1), gives the following by the energy estimate method:
∥(uk − uj )(t )∥2L2 + ∥(vk − vj )(t )∥2L2 ≤ exp(O(1)t ) ∥(uk − uj )(0)∥2L2 + ∥(vk − vj )(0)∥2L2 , where the bounds of O(1) depend only on the L∞ bounds of the solutions (uj , vj ) (j = 1, 2, . . .) for t ≥ 0. This yields the 2 1 convergence of the sequence {(uj , vj )}∞ j=0 to a function (u, v) in C ([0, T ); L (R )) for any T > 0. Then (u, v) is a global strong solution to (1.1)–(1.2). Moreover, by using Proposition 2.1 and by repeating the argument of taking the limit, we have b
|u(x, t )|2 + |v(x, t )|2 dx ≤ C1
a
b
|u0 (x − t )|2 + |v0 (x + t )|2 dx a
b +t
|u0 (y)| dy + 2
= C1 a+t
b−t
|v0 (y)| dy . 2
a −t
Then the decay of the local charge follows. In the same way, we can prove the following for any strong solutions (u′ , v ′ ) and (u′′ , v ′′ ):
∥(u′ − u′′ )(t )∥2L2 + ∥(v ′ − v ′′ )(t )∥2L2 ≤ exp(O(1)t ) ∥(u′ − u′′ )(0)∥2L2 + ∥(v ′ − v ′′ )(0)∥2L2 for t ≥ 0, where the bounds of O(1) depend only on the L∞ bounds of the solutions. This leads to the uniqueness of the strong solution. The proof of Theorem 1.1 is complete. Acknowledgments This work was partially supported by NSFC Project 11031001 and 11121101, by the 111 Project B08018 and by the Ministry of Education of China. The author would like to thank the referees for helpful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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