Global Low Regularity Solutions of Quasi-linear Wave Equations 7 Yi Zhou Zhen Lei 0 ∗ † 0 2 February 2, 2008 n a J 6 Keywords: quasi-linear wave equations, global low regularity solutions, unique- 2 ness, radial symmetry. ] P Abstract A . In this paper we prove the global existence and uniqueness of the low regu- h t larity solutions to the Cauchy problem of quasi-linear wave equations with radial a m symmetric initial data in three space dimensions. The results are based on the end-point Strichartz estimate together with the characteristic method. [ 1 v 1 Introduction 5 7 7 This paper studies the global existence and uniqueness of the low regularity solutions 1 0 to the Cauchy problem associated to the following quasi-linear wave equations in R1+3: 7 0 u a2(u)∆u = 0, t 0, x R3, (1.1) h/ tt − ≥ ∈ at where u = u(t,x) is a real-valued scalar function of (t,x), u = ∂2u, ∆ is the Laplace tt t m operator acting on space variables x, and a is a smooth real-valued function with v: a(0) > 0. Equation (1.1) is imposed on the following initial data i X u(0,x) = f(x),u (0,x) = g(x), x R3. (1.2) r t ∈ a Before going any further, we shall first recall a few historical results on the global classical solutions of system (1.1)-(1.2). In [12], Lindblad proved the global existence of ∗School of Mathematical Sciences, Fudan University, Shanghai 200433, P. R. China; Key Labora- tory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, P. R. China Email: [email protected]. †SchoolofMathematicalSciences,FudanUniversity,Shanghai200433,P.R.China;SchoolofMath- ematics and Statistics, Northeast Normal University, Changchun 130024, P. R. China; Current Ad- dress: AppliedandComput. Math, Caltech,Pasadena,CA91125;SchoolofMathematics andStatis- tics, Northeast NormalUniversity, Changchun 130024,P. R. China. Email: [email protected]; [email protected]. 1 classical solutions to system (1.1)-(1.2) for sufficiently small, radially symmetric initial data decaying rapidly as x or with compact support. Alinhac [1] improved the → ∞ global existence result of Lindblad [12] by removing the assumption that the initial data is radially symmetric. Recently, Lindblad [13] reproved the global existence result for more general quasi-linear wave equations by a different and simpler method. On the other hand, many authors studied the local well-posedness of low regular- ity solutions of quasi-linear wave equations [2, 3, 7, 8]. Recently, Smith and Tataru [16] established the local well-posedness of quasi-linear wave equations in three space dimensions with initial data in Hs(R3) Hs−1(R3) for s > 2. This result is sharp in × view of the counter example of Lindblad [14]. We point out here that in this paper Hs(R3) denotes the usual Sobolev space endowed with the norm kfkHs(R3) = (1+|ξ|2)2sfˆ L2(R3), where fˆ denotes the Fourier transform(cid:13)(cid:13)of f. For i(cid:13)(cid:13)nteger s 0, the above norm is ≥ equivalent to s f Hs(R3) = jf L2(R3). (1.3) k k k∇ k j=0 X Our main result in this paper is that the Cauchy problem (1.1)-(1.2) is globally well-posed in the Sobolev space H2(R3) H1(R3) provided that the initial data (f,g) are radially symmetric and have compac×t supports, and the H2(R3) H1(R3) norm of × the initial data (f,g) is small enough. Remark 1.1. Throughout this paper, we mean f + g by saying the H1(R3) H1(R3) k∇ k k k H2(R3) H1(R3) norm of a pair (f,g) and u(t, ) + u (t, ) by saying H1(R3) t H1(R3) the H2(×R3) H1(R3) norm of a function u(kt∇,x) at·tkime t. k · k × To state our result more precisely, we introduce the concept of strong solutions. Definition 1.2. We say u(t,x) is a strong solution to the Cauchy problem (1.1)-(1.2) onsome timeinterval[0,T ), if forasequence ofinitialdata(f ,g ) H3(R3) H2(R3) 0 n n which tends to (f,g) strongly in H2(R3) H1(R3) as n , ther∈e exists a×sequence × → ∞ of solutions u (t,x) in C [0,T );H3(R3) C1 [0,T );H2(R3) which solves the quasi- n 0 0 × linear wave equation (1.1) with the initial data (cid:0) (cid:1) (cid:0) (cid:1) u (0,x) = f (x), ∂ u (0,x) = g (x), n n t n n and tends to u weakly⋆ in L∞ [0,T],H2(R3) W1,∞ [0,T],H1(R3) for any T < T 0 × as n . → ∞ (cid:0) (cid:1) (cid:0) (cid:1) If T = + , then we say is u(t,x) a global strong solution to the Cauchy problem 0 ∞ (1.1)-(1.2). If(f,g)and(f ,g )areallradiallysymmetric functions, thenwe sayu(t,x) n n a radially symmetric strong solution to the Cauchy problem (1.1)-(1.2). Remark 1.3. The local existence of H3(R3) H2(R3) solutions for quasi-linear wave × equationsisguaranteedbytheclassicallocalexistence theoremprovidedthattheinitial data is in H3(R3) H2(R3). For example, see Hughes-Kato-Marsden [4]. × 2 Now we state our main result. Theorem 1.4. Assume that (H ): f and g are radially symmetric functions with compact supports: 1 supp (f,g) x R3, x 1 . (1.4) ⊆ ∈ | | ≤ (H2): (f,g) H2(R3) H(cid:8)1(R3)(cid:9)with(cid:8) (cid:9) ∈ × f + g ǫ (1.5) H1(R2) H1(R2) k∇ k k k ≤ for a positive constant ǫ. Then there exists a unique global strong solution u L∞ [0, );H2(R3) W1,∞ [0, );H1(R3) (1.6) ∈ loc ∞ ∩ loc ∞ to the quasi-linear wave eq(cid:0)uation (1.1) wi(cid:1)th initial(cid:0)data (1.2) prov(cid:1)ided that ǫ is small enough. Moreover, there exists a positive constant A and a small enough positive constant θ such that the global strong solution u(t,x) satisfies u(t, ) 2 + u (t, ) 2 (1.7) k∇ · kH1(R2) k t · kH1(R2) A f 2 + g 2 (1+t)θ ≤ k∇ kH1(R2) k kH1(R2) for all time t 0, where A de(cid:16)pends only on the functi(cid:17)on a and θ depends only on a ≥ and ǫ. There are a lot of results on the global well-posedness of low regularity solutions to semi-linear wave equations [5, 6, 10, 11, 15, 18], however, to the best of our knowledge, theorem 1.4 is the only result for global existence and uniqueness of low regularity solutions to quasi-linear wave equations. The rest of this paper is organized as follows. In section 2, we illustrate our main ideas of proving Theorem 1.4. More precisely, we will smooth the initial data (f,g) in (1.2) such that the local existence of solutions to the quasi-linear wave equations (1.1) with the regularized initial data is guaranteed by Remark 1.3. Then we state the main a priori estimates (2.15)-(2.18) for the local classical solutions involving only the H2(R3) H1(R3) norm of the initial data. The global existence and the uniform bound × (1.7) for strong solutions to the Cauchy problem (1.1)-(1.2) is a direct consequence of (2.18) (see Remark 2.2). We discuss how much the characteristics of the quasi-linear wave equations (1.1) deviate from that of the linear wave equation in section 3, and some other useful properties related to characteristics are also proved. In section 4, we use the characteristic method to prove the a priori estimates presented in section 2 by assuming that the weighted end-point Strichartz estimate (2.18) is true. To prove (2.18), we invoke Klainerman’s vector fields and the generalized energy method to explore decay properties of energy away from the characteristics in section 5. Then we estimate Hardy-Littlewood Maximal functions to get (2.18) in section 6. Section 7 studies the localexistence of general strong solutions to the Cauchy problem (1.1)-(1.2) inSobolevspaceH2(R3) H1(R3)andtheirstabilityinSobolevspaceH1(R3) L2(R3), × × which implies the uniqueness of the global strong solution and completes the proof of theorem 1.4. 3 2 A priori estimates This section illustrates our main observations and ideas in the proof of Theorem 1.4, and also lays some groundwork for the latter sections. First of all, for any given radially symmetric function ρ( x ) C∞(R3), ρ 0, ρ(x)dx = 1, ρ 0 for x 2, | | ∈ 0 ≥ ≡ | | ≥ R3 Z we define the mollification ( f)(x) of functions f Lp(R3), 1 p , by n J ∈ ≤ ≤ ∞ ( f)(x) = n3 ρ n(x y) f(y)dy, n = 1,2,3, (2.1) n J − ··· R3 Z (cid:0) (cid:1) For any rotation matrix Q, it is easy to see that ( f)(Qx) = n3 ρ n(Qx y) f(y)dy n J − R3 Z (cid:0) (cid:1) = n3 ρ n(Qx Qy) f(Qy)dy − R3 Z (cid:0) (cid:1) = n3 ρ n(x y) f(Qy)dy. − R3 Z (cid:0) (cid:1) Thus, for any radially symmetric function f(x) = f( x ), the mollified function is also | | radially symmetric: ( f)(x) = ( f)( x ). n n J J | | Our first step of proving Theorem 1.4 is to regularize the initial data (f,g) H2(R3) H1(R3) so that the mollified functions (f ,g ) = f, g H3(R3) ∈ n n n n H2(R3) ×by using the mollification operator defined in (2.1). JBy tJhe loc∈al existenc×e (cid:0) (cid:1) theorem (see Remark 1.3), there admits a unique solution u (t,x) in H3(R3) H2(R3) n × tothequasi-linear wave equation(1.1)withtheinitialdata(f ,g ). Thus, toshow that n n there exists a unique global strong solution to the Cauchy problem (1.1)-(1.2), we only need to achieve the a priori estimate for u in H2(R3) H1(R3) uniformly with respect n to n, which only involves the H2(R3) H1(R3) norm×(but not the H3(R3) H2(R3) × × norm) of the initial data (f ,g ). Note that (f ,g ) is also radially symmetric. In what n n n n follows, we will drop the subscript n and intend to achieve a priori H2(R3) H1(R3) × estimate for classical solutions u(t,x) to the quasi-linear wave equations (1.1) only using the H2(R3) H1(R3) norm of the initial data (f,g). × We turn to focus on the standard energy estimates for the quasi-linear wave equa- tions (1.1). For any integer s 1, define the standard energy by ≥ s−1 1 E u(t) = ∂bu (t, ) 2 +a2(u) ∂bu(t, ) 2 dx, (2.2) s t 2 | · | |∇ · | R3 (cid:2) (cid:3) Xb=0 Z n o where ∂ = (∂ , ). By the standard energy estimates and Nirenberg inequality, it is t ∇ rather easy to get (see [16] and the references therein) d dtEs (u(t) ≤ C0k∂u(t,·)kL∞(R3)Es u(t) , (cid:2) (cid:3) (cid:2) (cid:3) 4 provided that u is bounded and a2(u) is strictly greater than 0, where C is a positive 0 constant depending only on a and u L∞(R3). Consequently, it follows that k k T Es u(T) Es u(0) exp C0 ∂u(t, ) L∞(R3)dt . ≤ k · k Z0 n o (cid:2) (cid:3) (cid:2) (cid:3) If the inequality T 1 k∂u(t,·)kL∞(R3)dt ≤ K ln(1+T) (2.3) Z0 is valid for all time T 0 and a uniform (big) constant K > 0, then there holds the ≥ following a priori energy estimate C0 E2 u(T) E2 u(0) (1+T)K , (2.4) ≤ which only involves the H2(R(cid:2)3) H(cid:3)1(R3)(cid:2)norm(cid:3)of the initial data (f,g). The energy × E u(T) admits a dynamic growth, but will be still bounded for all time T 0. By 2 ≥ the counterexample given by Lindblad [14], the inequality (2.3) may not be true for the (cid:2) (cid:3) quasi-linear wave equation (1.1) with general initial data (f,g) in H2(R3) H1(R3). × We will prove in this paper that the inequality (2.3) is true (see Remark 2.2) for all timet 0 if theinitialdata(f,g)satisfies theassumptions (H ) (H ) inTheorem 1.4. 1 2 ≥ − The global existence and the uniform bound (1.7) for strong solutions to the Cauchy problem of quasi-linear wave equation (1.1)-(1.2) are then direct consequences of (2.4) (by letting θ = C0), as what we have mentioned at the beginning of this section. K To present our main ideas of proving the a priori estimate (2.3), we begin with defining the characteristics of the quasi-linear wave equations (1.1). Let r (τ;β) denote afamily ofminus characteristics passing throughthepoint (t,r) − with the intersection (0,β), β 0, with σ axis (see Figure 1): ≥ dr−(τ;β) = a(u) τ,r (τ;β) , dτ − − (2.5)  (cid:0) (cid:1) r (0;β) = β.  − For convenience, sometimes we also denote the above minus characteristics as t (σ;β) − (see Figure 1): dt−(σ;β) = 1 , dσ −a(u) t−(σ;β),σ (2.6)  t−(β;β) = 0. (cid:0) (cid:1) Similarly, let r+(τ;α) denote a family of plus characteristics passing through the point (t,r) with the intersection (α,0), α 0, with τ axis (see Figure 1): ≥ dr+(τ;α) = a(u) τ,r (τ;α) , dτ + (2.7)  (cid:0) (cid:1) r (α;α) = 0.  +  5 τ (t, r) r(τ; α) + r(τ; β) α(t, r) t+(σ; α) − t(σ; β) − 1 β(t, r) σ Figure 1: which is also written as t (σ;α) (see Figure 1): + dt+(σ;α) = 1 , dσ a(u) t+(σ;α),σ (2.8)  t (0;α) = α,(cid:0) (cid:1) +  When starting from (0,γ), γ > 0, a plus characteristic r (τ;γ) is defined by + dr+(τ;γ) = a(u) τ,r (τ;γ) , dτ + (2.9) r (0;γ) = γ, (cid:0) (cid:1)  + and sometimes we also write it as t (σ;γ): + dt+(σ;γ) = 1 , dσ a(u) t+(σ;γ),σ (2.10)  t (γ;γ) = 0. (cid:0) (cid:1) +  Note that in the above definitions (2.5)-(2.10), the parameters α, β and γ depend on the point (t,r), and the point (t,r) also depends on the parameters (α,β) or (γ,β). To show their dependence, we also write α = α(t,r), β = β(t,r), γ = γ(t,r) and 6 t = t(α,β), r = r(α,β) or t = t(γ,β), r = r(γ,β). It is obviously that t r;α(t,r) = t, r t;α(t,r) = r, + + t (cid:0)r;γ(t,r)(cid:1)= t, r (cid:0)t;γ(t,r)(cid:1)= r, + +    t (cid:0)r;β(t,r)(cid:1) = t, r (cid:0)t;β(t,r)(cid:1) = r,  − −  (2.11)   α (cid:0)τ,r+(τ;α(cid:1)) = α t+((cid:0)σ;α),σ =(cid:1) α, γ(cid:0)τ,r (τ;γ)(cid:1)= γ (cid:0)t (σ;γ),σ (cid:1)= γ,  + +     β(cid:0)τ,r−(τ;β)(cid:1) = β(cid:0)t−(σ;β),σ(cid:1) = β.    These notations seem rath(cid:0)er tedious a(cid:1)nd co(cid:0)mplicated.(cid:1)However, the reader will see the advantages of the above convention. Next, we define the cone t = (t,r) r +1 , D { | ≤ 4 } and assume that (without loss of generality) a(0) = 1. (2.12) As in [12], we introduce the following weighted differential operators: 1 L = ∂ a(u)∂ . (2.13) ± t r a(u) ± p We will prove the following crucpial Theorem: Theorem 2.1. Suppose that the assumptions (H ) (H ) in Theorem 1.4 are satisfied 1 2 − and u(t,r) is the classical solution to the Cauchy problem of the quasi-linear wave equation (1.1)-(1.2). Suppose furthermore that K ( 1) and ǫ ( 1) satisfy ≫ ≪ K4ǫ 1. (2.14) ≤ Then there exists positive constants C > 0, j = 1,2, ,6, such that for all time j ··· t 0, there holds ≥ A priori estimates for u: u(t,r) C1ǫ , 3 | | ≤ (1+t)5 (2.15)   u(t,r) C2Kǫ(1+t)−1+K2 . | | ≤   7 A priori estimates for L v with v = ru: ± 1 L v(t,r) C3ǫ(1+t)K , | + | ≤ 1+β(t,r)   L v(t,r) C4ǫ C ǫ for t (r;1) t < t (r;0), | − | ≤ 1+γ(t,r) 1−K1 ≤ 4 + ≤ + (2.16)    L v(t,r) (cid:0) C4ǫ(cid:1) for t t (r;0). | − | ≤ 1−K1 ≥ +  1+α(t,r)     A priori estimates f(cid:0)or ∂u h(cid:1)olds away from the cone : D ∂u(t,r) C5Kǫ , for r > max t +1,r (t;0) , | | ≤ 1−K2 {4 + } (1+t) 1+γ(t,r)  (2.17)   ∂u(t,r) (cid:0) C5Kǫ (cid:1) , for t +1 < r r (t;0). | | ≤ (1+t) 1+α(t,r) 1−K2 4 ≤ +   Weighted a priori esti(cid:0)mate fo(cid:1)r ∂u holds inside the cone : D t (1+τ)1−K4 sup ∂u(τ,σ) 2dτ C2K4ǫ2. (2.18) Z0 σ≤τ4+1| | ≤ 6 (cid:2) (cid:3) Remark 2.2. With the aid of (2.14), estimates (2.17) and (2.18) indeed imply (2.3) by the following calculations: t ∂u(τ, ) L∞dτ | · | Z0 t t sup ∂u(τ,σ) dτ + sup ∂u(τ,σ) dτ ≤ Z0 σ>τ4+1| | Z0 σ≤τ4+1| | (cid:2) (cid:3) (cid:2) (cid:3) C5Kǫln(1+t)+C 1 (1+t)−1+K8 12 ≤ − t 1 (1+τ)1−K4 (cid:2)sup ∂u(τ,σ) 2(cid:3)dτ 2 ×(cid:16)Z0 σ≤τ4+1| | (cid:17) (cid:2) (cid:3) CKǫln(1+t)+CK2ǫ 1 (1+t)−1+K8 21 ≤ − CK2ǫln(1+t). (cid:2) (cid:3) ≤ We mention that in this paper, C will be used to denote a generic positive constant independent of any function appeared throughout this paper except for a, and its meaning may vary from line to line. The proof of (2.15) will be completed by using (2.16) at the beginning of section 4. (2.16) will be proved by using (2.18) and the characteristic method also in section 4. The properties of characteristics will be discussed in section 3. Then we use (2.15) and (2.16) to prove (2.17) at the end of section 4. The strategy of proving the series of 8 estimates (2.15)-(2.18) is that by assuming their validity first, we show these estimates are still valid when the constants C are replaced by 1C for j = 1,2, ,6. j 2 j ··· The proof of (2.18) relies on the estimation of Hardy-Littlewood Maximal func- tions and involves Klainerman’s vector fields and generalized energy method. To best illustrate our ideas, let us review the end-point Strichartz estimate by Klainerman and Machedon [9]. Consider the following linear problem φ ∆φ = 0, t 0,x R3, tt − ≥ ∈ (2.19) φ(0,x) = φ (r),φ (0,x) = 0, x R3.  0 t ∈ Extend the initialdata φ (r) and the solution φ(t,r) of system (2.19) into even 0 functions with respect to r, and then let φ = rφ (r), r R1,. (2.20) 0 0 ∈ d’Alembert formula gives e φ(t,r) φ (r +t)+φ (r t) 0 0 = − 2r φe (t+r) φe (t r) 0 0 = − − 2r e1 t+r e = φ′(λ)dλ. 2r 0 Zt−r Differentiating the above equality with respeect to t yields 1 t+r φ (t,r) = φ′′(λ)dλ. (2.21) t 2r 0 Zt−r We recall some properties for Hardy-Littleweood Maximal functions. The Hardy- Littlewood maximal operator in Rn is defined on L1 (Rn) by loc 1 Mf(x) = sup f(y) dy, f L1 (Rn), Q | | ∈ loc Q | | ZQ where the supremum is taken over all cubes containing x. The following property is well-known (see [17]): Proposition 2.3. The Hardy-Littlewood maximal operator is of strong (p,p) type for 1 < p : ≤ ∞ Mf Lp(Rn), Mf Lp(Rn) C f Lp(Rn) for f Lp(Rn),1 < p , ∈ k k ≤ k k ∈ ≤ ∞ where C > 0 is a uniform constant. 9 To simplify our notations, we will denote the Lp and Sobolev norms of φ(t,r) with respect to r by φ(t, ) and φ(t, ) , compared with the notations for the Sobolev Lp Hs norms of a funcktion·wkith respkect to· kx R3 which are denoted by φ(t, ) and Lp(R3) ∈ k · k f(t, ) Hs(R3) (see (1.3)). k · k Using (2.21), Hardy inequality and Proposition 2.3, we have (see Klainerman and Machedon [9]) T 1 φ (t, ) 2 dt 2 (2.22) k t · kL∞ (cid:16)Z0 (cid:17) T 1 M φ′′(λ) (t) 2dt 2 ≤ 0 (cid:16)Z0 (cid:17) (cid:12)∞ (cid:2) (cid:3) (cid:12) 1 C (cid:12) φe′(λ) 2 +(cid:12)λφ′′(λ) 2 dλ 2 ≤ | 0 | | 0 | (cid:16)Z0 (cid:17) (cid:2) φ′(x) (cid:3) ≤ Ckφ′0′kL2(R3) +C 0x L2(R3) | | C φ′′ . (cid:13) (cid:13) ≤ k 0kL2(R3) (cid:13) (cid:13) One of our key observations is that there holds weighted end-point Strichartz estimate inside the cone : D T 1 (1+t)δ φ (t,r) 2dt 2 (2.23) k t kL∞(r≤4t+1) (cid:16)Z0 (cid:17) (cid:2)T (cid:3) 1 M (1+λ)δφ′′(λ) (t) 2dt 2 ≤ 0 (cid:16)Z0 (cid:17) (cid:12)∞ (cid:2) (cid:3) (cid:12) 1 C (cid:12) φ′(λ) 2 +e λφ′′(λ)(cid:12)2 dλ 2 ≤ | 0 | | 0 | (cid:16)Z0 (cid:17) C φ′′ (cid:2) (cid:3) ≤ k 0kL2(R3) provided that φ has a compact support, where δ is a positive constant. Using equality 0 (2.21) and estimate (2.23), we have T φt(t,r) L∞dt (2.24) k k Z0 T T φ (t,r) dt+ φ (t,r) dt ≤ k t kL∞(r≤4t+1) k t kL∞(r>4t+1) Z0 Z0 1 C 1 (1+t)1−2δ 2 T 1 − (1+t)δ φ (t,r) 2dt 2 ≤ 2δ 1 k t kL∞(r≤4t+1) (cid:2) − (cid:3) (cid:16)Z0 (cid:17) + C φ′′ ln(1+t) (cid:2) (cid:3) k 0kL1 ≤ C(δ)kφ′0′kL2(R3)ln(1+t) for 1 < δ < 1. 2 10