Mathematical Research Letters 5, 65 622 1998) COUNTEREXAMPLES TO LOCAL EXISTENCE FOR QUASILINEAR WAVE EQUATIONS Hans Lindblad 1. Introduction and themain argument In this paper, we study quasilinear wave equations and ask how regular the initial data must be to ensure that a local solution exists. We present counterexamples to local existence for typical model equations. We also give the proof in the quasilinear case as well. The proof for the semilinear case was given in Lindblad [7,9]. The semilinear case counter examples are sharp, in the sense that for initial data with slightly more regularity, a local solution exists. This was shown recently in Klainerman-Machedon [3-5], Ponce-Sideris [11] and Lindblad-Sogge [1] using space time estimates and generalizations of these estimates known as Strichartz estimates. For quasilinear equations, however, the optimal result is still unknown; there is a gap between the counterexamples we present here and a recent improvement on the local existence result by Tataru[15] and Bahouri- Chemin [1]. Consider the Cauchy problem for a quasilinear wave equation: 1.1) g ij u, u ) x i x j u = F u, u ), t, x) S T =[,T) R 3, i,j= u,x)=fx), u t,x)=gx), where x = t, g ij, and F are smooth functions of u and its first order derivatives. We also assume that {g ij } 3 i,j= is a symmetric matrix that is close to the coefficient matrix of = t 2 3 i=1 2 x ; i 1.2) j,k= g jk m jk 1/2, where { m =1, m jj = 1, j> m jk =, if j k, so that 1.1) is hyperbolic. Let Ḣγ denote the homogeneous Sobolev space with norm f Ḣγ = D x γ f L 2 where D x = x and set 1.3) ut, ) 2 γ = D x γ 1 u t t, x) 2 + D x γ ut, x) 2 ) dx. Received August 31, 1998. The author was supported in part by the National Science Foundation. 65
66 HANS LINDBLAD Since the norms 1.3) are more or less the only norms that are preserved for the linear wave equation u =, it is natural to look for existence in the spaces associated with these norms. We want to find the smallest possible γ such that 1.4) 1.5) f,g) Ḣγ R 3 ) Ḣγ 1 R 3 ), supp f supp g {x; x R}, implies that we have a local distributional solution of 1.1) for some T >, satisfying 1.6) u, t u) C [,T]; Ḣγ R 3 ) Ḣγ 1 R 3 ) ). We say that u is in H γ [,T]ifu, t u) L [,T]; Ḣγ R 3 ) Ḣγ 1 R 3 ), i.e., if the norm 1.3) is uniformly bounded on [,T]). Similarly we say that data f,g) is in H γ [] if f,g) Ḣγ R 3 ) Ḣγ 1 R 3 ), i.e., if the norm 1.3) is bounded initially when t = ). To avoid certain peculiarities concerning non-uniqueness we also require that u is a proper solution: Definition 1.1. We say that u is a proper solution of 1.1) if it is a distributional solution, and if u is the weak limit of a sequence of smooth solutions u ε to 1.1) with data φ ε f,φ ε g), where φ ε x) =φx/ε)ε n for some function φ satisfying φ C, φdx=1. Even if one has smooth data and hence a smooth solution there might still be another distributional solution which satisfies initial data in the space given by the norm 1.3). In fact, ut, x) = 2Ht x )/t satisfies u = u 3 in the sense of distribution theory. If γ<1/2 then ut, ) γ when t by homogeneity. Since ut, x) = is another solution with the same data it follows that we have non-uniqueness in the class 1.6) if γ<1/2. Definition 1.1 picks out the smooth solution if there is one. Our main theorem is the following: Theorem 1.2. Consider the Cauchy problem in R 1+3 ; 1.7) u = D l u ) D k l u, D = x 1 t ), u,x)=fx), u t,x)=gx), where l k l 2, l =, 1. Let γ = k. Then there exists data f,g) satisfying 1.4) 1.5), with f Ḣγ + g Ḣγ 1 arbitrarily small, such that 1.7) does not have any proper solution satisfying 1.6) in S T =[,T) R 3 for any T>. Remark 1.3. It also follows from the proof of the theorem that either we have non-existence of any distributional solution satisfying 1.6) or else we have nonuniqueness in a domain of dependence in this class see Definition 1.6 and Theorem 1.8). Also, stronger statements hold in the different cases see Lindblad[7-9].) I particular if k l = l = then there is no distributional solution in L 2 S T ).
COUNTEREXAMPLES FOR QUASILINEAR WAVE EQUATIONS 67 Remark 1.4. By a simple scaling argument one gets a counterexample to wellposedness, but it has lower regularity than our counterexamples: γ<k 1/2. Indeed, if u is a solution of 1.7) which blows up when t = T then u ε t, x) = ε k 2 ut/ε, x/ε) is a solution of the same equation with lifespan T ε = εt and u ε, ) γ = ε k 2+3/2 γ u, ) γ ifγ<k 1/2. By contrast, our counterexamples are designed to concentrate in one direction, close to a characteristic. Remark 1.5. In Klainerman-Machedon[3-5] it was proved that for semilinear wave equations satisfying the null condition one can in fact get local existence for data having the regularity predicted by the scaling argument. Although the nonlinearities in our theorem have a special form the same result should hold for any quadratic form in the derivatives of u that does not satisfy the null condition. Now, there is a unique way to write 1.7) in the form 1.1). In the semilinear case, k l 1, the equation is already in this form with g jk = m jk, where m jk is given by 1.2). In the quasilinear case, k l =2, g ij t, x) x i x j i,j= = x 2 2 x vt, x) i x x 1)2, where v = D l u. i= We now define the notion of a domain of dependence. Definition 1.6. Assume that Ω R + R 3 is an open set equipped with a Lorentzian metric g jk CΩ) such that inverse g jk satisfies 1.2). Then Ω is said to be a domain of dependence for the metric g ij if the closure of the causal past Λ t,x of each point t,x ) Ω is contained in Ω. Here Λ t,x is defined to be all points in Ω that can be joined to t,x ) by a Lipschitz continuous curve t, xt)) Ω, t t, satisfying 1.8) i,j= g ij x) dxi dt dx j dt, x t) =t, almost everywhere. Since a solution u to 1.1) gives rise to a unique metric g jk we say that Ω is a domain of dependence for the solution u if it is a domain of dependence for g jk. The importance of the concept of domain of dependence is Huygens principle which says that changing the initial data outside the intersection of the domain of dependence with the inital plane t = should not change the solution in the domain of dependence, or in other wor; information should not travel faster than the speed of light. This is true for smooth solutions. We also have uniqueness and continuous dependence of data for smooth solutions in a domain of dependence. Hence if there is a smooth solution in a domain of dependence it is unique within the class of proper solutions.
68 HANS LINDBLAD Lemma 1.7. Suppose u C Ω) is a solution to 1.7) in a domain of dependence Ω with initial data f,g) on Ω = {x;,x) Ω}. In the quasilinear case, k l =2, assume also that D l u L Ω) δ. Suppose that u ε C S T ), where S T =[,T) R 3, are solutions of 1.7) with data f ε,g ε ) where f ε,g ε ) f,g) in C Ω ). Then u ε u in C Ω S T ). Here C O) denotes the set of smooth function on O with the usual topology defined by the semi-norms ρ α,k f) = sup x K α fx), for K a compact subset of O. The proof of Lemma 1.7 is standard so we postpone it to an appendix. It is essential that Ω is a domain of dependence for Lemma 1.7 to be true; one nee exactly this in order to be able to use the energy method to get estimates. Theorem 1.8. There is an open set Ω R + R 3 and a solution u C Ω) of u = D l u ) D k l u, D = x 1 t ), l k l 2, l =, 1, such that Ω is a domain of dependence for u and writing 1.9) Ω t = {x;t, x) Ω}, we have that Ω is smooth, 1.1) β k, β 1 Ω t β ut, x) ) 2 dx { <, when t =,, when t>, where β =β,..., β 3 ) and β = β t β 3 x. In fact; 3 1.11) D k l ut, x) ) 2 dx =, t >. Ω t Furthermore, in the quasilinear case, k l =2, we can pick u with D l u L Ω) arbitrarily small. Proof of Theorem 1.2. By Theorem 1.8 we get a solution u in a domain of dependence Ω with initial data u,x) H k Ω ) and u t,x) H k 1 Ω ). We can extend these to f H k R 3 ) and g H k 1 R 3 ), see Stein [14]. If there exist a proper solution u of 1.7) in S T =[,T] R 3 with these data, it follows from Definition 1.1 and Lemma 1.7 that u is equal to u in S T Ω, contradicting 1.6). Let us now briefly describe how to construct the solution u and the domain of dependence Ω in Theorem 1.8. First we find a solution ut, x) =u 1 t, x 1 ), depending only on one space variable x 1 R, that develops a certain singularity for t> along a non timelike curve x 1 = µt), i.e., Ω 1 = {t, x 1 ); x 1 >µt)} is a domain of dependence). Then we will find a domain of dependence Ω Ω 1 R 2 for u 1 t, x 1 ) such that the solution is in H k Ω ) and such that the curve
COUNTEREXAMPLES FOR QUASILINEAR WAVE EQUATIONS 69 x 1 = µt), x 2 = x 3 =, lies on Ω, which ensures that the solution is not in H k Ω t ) for t>. This means that Ω contains the causal past, for the metric given by u 1, of the above curve for small t. The equation u = D l ud k l u, becomes 1.12) D + D u 1 + D u l 1 D k l u 1 =, where D ± = x 1 ± t. This equation can be solved by integrating along characteristics. By choosing particular initial data 1.13) D u 1,x 1 )=χ 3 k) x 1 ), D + u 1,x 1 )=, x 1 1.14) where χx 1 )= ε log s/4 α, we get a solution <α< 1 2,ε>, 1.15) u 1 C Ω 1 ), where Ω 1 = {t, x 1 ); µt) <x 1 < 2 t} R + R 1, for some function µt) with µ) =, such that Ω 1 is a domain of dependence and such that u 1 t, x 1 ) has a singularity along x 1 = µt). One sees this from the solution formulas obtained from integrating along characteristics see section 2). Essentially what is happening is that the initial data 1.13)-1.14) has a singularity when x 1 =. For the linear equation, u tt u x 1 x 1 =, the singularity would just have propagated along a characteristic, however the nonlinearity causes the solution to increase and this strengthens the singularity for t>. This is the same phenomena that causes blow-up for smooth initial data. The blow-up along characteristics occures at a time 1/χ. Because χ +) =, the blow-up happens directly close to x 1 =. Define Ω R + R 3 to be the largest domain of dependence for the metric obtained from the solution ut, x) =u 1 t, x 1 ) such that 1.16) Ω Ω 1 R 2, Ω = {x;,x) Ω} = B = {x; x 1,, ) < 1}. It follows from Definition 1.6 that the union and intersection of a finite number of domains of dependence is a domain of dependence so indeed a maximal domain exists.) The initial data 1.13)-1.14) was chosen so that 1.1) just is finite if t =, an easy calculation shows that this is equivalent to yχ y) 2 dy <. Let Ω t be as in 1.9) and 1.17) S t x 1 )={x 2,x 3 ) R 2 ;x 1,x 2,x 3 ) Ω t }, a t x 1 )= dx 2 dx 3. S t x 1 ) With this notation the integral in 1.11) becomes, if we integrate out the x 2 and x 3 variables 1.18) 2 t µt) a t x 1 ) D k l u 1 t, x 1 ) ) 2 dx 1.
61 HANS LINDBLAD The proof that this integral is infinite consists of estimating the two factors in the integrand from below, close to x 1 = µt). In the semilinear case the metric g jk is just m jk so Ω 1 is a domain of dependence if and only if µ t) 1 and it follows that Ω = Ω 1 R 2 Λ, where Λ = {t, x); x 1,, ) +t <1}. Hence for x 1 >µt); S t x 1 )={x 2,x 3 ); x 1 1) 2 + x 2 2 + x 3 2 < 1 t) 2 } so then a t x 1 )=π2 t x 1 )x 1 t). Also, the specific solution formulas are relatively simple. In particular if k l = l = 1, then it is easy to verify that 1.19) D u 1 t, x 1 )= χ x 1 t) 1+tχ x 1 t), satisfies 1.12)-1.13) when 1 + tχ x 1 t) >. Since χ +) = and χ > it follows that there is a function µt), with µ t) > 1 and µ) =, such that 1+tχ x 1 t) =, when x 1 = µt). Hence 1 + tχ x 1 t) Ct)x 1 µt)) so 1.2) 1 2 µt) a t x 1 ) D u 1 t, x 1 ) ) 2 dx 1 1 2 µt) x 1 t) dx 1 Ct) 2 x 1 µt)) 2 =. We refer the reader to Lindblad [9] for the complete proof in the semilinear case. In this paper we will concentrate on the quasilinear case. Here, the solution formulas are less explicit and given in terms of the characteristics for the equation 1.12). In the quasilinear case, estimating a t x 1 ) from below requires control of the geometry of the causal past for the metric whos inverse is given by 1.21) g ij t, x 1 ) x i x j = t 2 x 2 vt, i x1 ) x 1 t ) 2, where v = x 1 t ) l u 1. i=1 2. Thequasilinear case The quasilinear case of the equation in one space dimension 1.12) can be factored 2.1) 2.2) t + x 1)+v x 1 t ) ) x 1 t )u =, where v = x 1 t ) l u, u,x 1 )=, t u,x 1 )= χ 1 l) x 1 ). Here l =, 1. This equation can be solved by integrating along the characteristics. 2.1) is equivalent to 2.3) t + 1+v 1 v x 1 ) x 1 t )u =.
COUNTEREXAMPLES FOR QUASILINEAR WAVE EQUATIONS 611 The characteristics are given by x 1 + t = constant respectively x 1 = φt, y) where 2.4) d φt, y)) φt, y) =1+vt, dt 1 vt, φt, y)), φ,y)=y. 2.3) says that x 1 t )u is constant along the characteristics 2.4) so 2.5) x 1 t )v ) t, φt, y)) = χ y), v,y)=, if l =, vt, φt, y)) = χy), if l =1. Let R + =, ), R + =[, ) and let C l C k Ω ) = C l Ω ) C k Ω ), where C k Ω ) = C k Ω).Wehave Proposition 2.1. Let <ε<1 and suppose that χ C C R + ) and 2.6) ε <χ<, χ <, χ >, χ)=, χ +) =, χ ) =. Then there are functions φ C C R + R + ) C 1 R + R + \{, )} ), ν C CR + ) and v C C Ω 1), where 2.7) Ω 1 = {t, x 1 ); x 1 >µt) =φt, νt)), t }, such that 2.4) 2.5) hold when y>νt), t, and 2.8) ν)=, ν t), φ y t, y) >, y > νt), φ y t, νt))=, vt, x 1 ) <ε, x 1 t )vt, x 1 ) <, when t> and t, x 1 ) Ω 1. If l =then νt) =and 2.9) <φ y t, y) <e 2tχ y) /1 ε), when y>. If l =1, then ν t) > when t>, and there is a function M CR + ) such that 2.1) <φ y t, y) <Mt) y νt) ), when y>νt), t >. Remark. Note that Ω 1 is a domain of dependence since ν t). Furthermore φt, νt)) = so the function vt, x 1 ) defined by solving x 1 = φt, y) for y as a function of t, x 1 ) in 2.5) has a singularity along the curve x 1 = µt) =φt, νt)), which lies in Ω 1.
612 HANS LINDBLAD The calculations simplify if we introduce new variables 2.11) s = x 1 + t, q = x 1 t, Us, q) =us q)/2, s + q)/2). Then 2.1) becomes, 2.12) 2.13) s + V s, q) q ) q Us, q) =, Vs, q) = qus, l q)2 l, Uy, y) =, U q y, y) =χ 1 l) y)2 l. The characteristics for 2.12) are given by s = constant respectively q = hs, y) where 2.14) d hs, y) =V s, hs, y)), hy, y) =y. By 2.12), du q s, hs, y))/ =so 2.15) V q s, hs, y)) = χ y), Vy, y) =, if l =, V s, hs, y)) = χy), if l =1. Proof of Proposition 2.1. We will first prove Proposition 2.1 in the case l =. Differentiating 2.14) with respect to y gives d h ys, y) =V q s, hs, y))h y s, y), h s y, y)+h y y, y) =1, Since h s y, y) = by 2.14) and 2.13) it follows that which has the solution d h ys, y) =χ y)h y s, y), h y y, y) =1, 2.16) h y s, y) =e s y)χ y). Since h y s, y) >, when y>, we can solve the equation q = hs, y) for y as a function of s, q) so 2.15) defines a function Us, q) =V s, q), in Ω 1 = {s, q); q>hs, )}, which satisfies 2.12)-2.13). Furthermore by 2.15) V s, hs, y)) = s y V q s, hs, z))h y s, z)dz s y χ z) dz,
COUNTEREXAMPLES FOR QUASILINEAR WAVE EQUATIONS 613 since χ z) < so 2.17) V s, hs, y)) χy) χs) <ε<1, and V q s, hs, y)) <. We now want to change back to the original variables t, x 1 ). The characteristics q = hs, y) can be expressed as x 1 = φt, y), if we introduce t as a new parameter: s = st, y) and set φt, y) =hs, y)+t. In fact, t =s q)/2 =s hs, y))/2 so dt/ =1 V s, hs, y))/2 >. Since hφ + t, y) =φ t we have h y s, y) 2.18) φ y t, y) = >, when y>, 1 V s, hs, y)) so, u 1 t, x 1 )=Vt + x 1,x 1 t) is a solution to 2.1) 2.2) in the set Ω 1 = {t, x 1 ); x 1 >µt) =φt, )}. Furthermore, s y 2t = hs, y) y = s y h s z,y) dz = s y V z,hs, y)) dz, which in view of 2.18) and 2.17) proves 2.9). Let us now deal with the case l = 1, which in fact is much simpler. By 2.4) and 2.5) d φt, y) =1+χy) dt 1 χy), φ,y)=y, so φt, y) =y + 1+χy) 1 χy) t, and χ y) φ y t, y) =1+ 1 χy)) 2 t. Since χ +) = and χ ) = it follows that there is a function νt) >, when t>, such that φ y t, νt))=, when t>. and ν+) =. We have φ yy t, y) = 1 χy))χ y)+2χ y) 2 1 χy)) 3 t>, so <φ y t, y) < y νt)) sup φ yy t, z), y νt), z>νt) which proves 2.1). Furthermore = d dt φ yt, νt)) = χ νt)) 1 χνt)) + φ yyt, νt))ν t), so ν t) > when t>. This concludes the proof of Proposition 2.1. We now have to show that the integral 1.18) is infinity for t>.
614 HANS LINDBLAD Lemma 2.2. Let φ, ν, v and Ω 1 be as in Proposition 2.1 and suppose that u C 2 Ω 1 ) satisfy 2.1) 2.2) in Ω 1. Then 2.19) 2 t µt) a t x 1 ) x 1 t ) 2 ut, x 1 ) ) 2 dx 1 = ν t) νt) ) 2 1 a t φt, y)) χ 2 l) dy y) ) φ y t, y) 2. 1 vt, φt, y)) Proof. Using the coordinates 2.11) 2.14), we obtain U qq s, hs, y) ) = 1 h y s, y) which, together with 2.18), proves that d dy U ) 1 q s, hs, y) = h y s, y) d dy U qy, y) = χ2 l) y) h y s, y), 2.2) x 1 t ) 2 u ) t, φt, y)) = χ 2 l) y) φ y t, y)1 vt, φt, y))). 2.19) follows if we use 2.2) and change variables x 1 = φt, y), dx 1 = φ y t, y)dy. Since φ y t, νt)) = and v <ε<1 it will follow that the integral 2.19) is infinity for t>. In view of 1.16) 1.17) we have 2.21) a φ,y)) = a y) = dx 2 dx 3 = π2y y 2 ), x 2 2 +x 3 2 1 1 y) 2 and we will prove in Proposition 2.4 below there is a constant c> such that 2.22) a t φt, y)) cy if y 1/4, t<1/16, Assuming 2.22) for a while, the proof of that the integral 2.19) is infinity now follows from: Lemma 2.3. There is χ satisfyingthe assumtions in Proposition 2.1 such that νt) 2 2 2.23) y χ y)) dy <, ν t) ) 2 dy 2.24) y χ 2 l) y) =, t >. φ y t, y) Proof. Let us first deal with the case l =. It is then easy to choose χ so the integral 2.23) is finite but 2.24) is infinity. We claim that y χy) = C ε log w/1 + w 2/α ) α/2) α 1 dw, <α< 2,
COUNTEREXAMPLES FOR QUASILINEAR WAVE EQUATIONS 615 will do. Note that yχ y) χ y) 1/α 1 =Cε) 1/α /1 + y 2/α ). By 2.9), 2.24) is bounded from below by ν t) yχ y) 2 e 2tχ y) dy = b ε yχ y)e 2tz dz b ε c ε e 2tz z 1/α 1 dz =, b ε >, where we have made the change of variables z = χ y), dz = χ y)dy. From the same calculation when t = it follows that the integral 2.23) is finite. If l = 1 then by 2.1), 2.24) is bounded from below by a constant times ν t) νt) y χ y) ) 2 dy Mt) 2 y νt)) =. It only remains to prove 2.22); Proposition 2.4. Let v, φ and Ω 1 be as in Proposition 2.1 and assume in addition that v ε =1/8. Let Ω {t, x); t, x 1 ) Ω 1, t < 1/16} be the largest domain of dependence for the metric whose inverse is given by 2.25) g ij t, x) x i x j i,j= = 2 t x 2 vt, i x1 ) t x 1) 2, i=1 such that Ω {} R 3 = {x;x 1 1) 2 + x 2 2 + x 3 2 < 1}. Then 2.26) Ω = {t, x); t, x 1 ) Ω 1 and x 2 2 + x 3 2 <Dt, x 1 )}, where 2.27) Dt, φt, y)) 1 4 y, when y 1 4, t 1 16. Proof of Proposition 2.4. It follows from Definition 1.6 that t,x ) Ωifand only if t,x 1 ) Ω 1 and all Lipschitz continuous curves from t,x ) that satisfy 1.8) intersect the hyperplane t = in the ball {x;x 1 1) 2 + x 2 2 + x 3 2 < 1}. Hence Proposition 2.4 follows from Lemma 2.6 below. Now recall Definition 1.6 of domain of dependence. An easy calculation shows that if g ij is given by 2.25) then the inverse satisfies 2.28) i,j= g ij t, x) dxi dx j ) dx 2 ) dx i 2 ) dx = +vt, x 1 2 ) + dx1. i=1
616 HANS LINDBLAD Lemma 2.5. Let ts),x 1 s),x 2 s),x 3 s)) be a Lipschitz continuous curve parameterized so that ts)+x 1 s) =s and let g ij be given by 2.28). Then 2.29) i,j= g ij t, x) dxi dx j, x = t, is equivalent to 2.3) where dqs) Rs)+vts),x 1 s)), dx 2.31) qs) =x 1 2 s) ) 2 dx 3 s) ) 2. s) ts), and Rs) = + Proof. The proof just follows from factoring 2.28): dx dx1 + v dx + dx1 )) dx + dx1 ) ) dx 2 2 ) dx 3 2 = dq + v R. Lemma 2.6. Assume that V s, q) C Ω 1 ) and V q. Suppose that hs, y) satisfies 2.32) dhs, y) = V s, hs, y)), hy, y) =y, and consider a curve as in Lemma 2.5 satisfying 2.3) with qb) =hb, y) for some b>y> and let rs) = x 2 s) 2 + x 3 s) 2. Then 2.33) qa) ha, y) and ra) rb) 2 qa) ha, y) b a, a b. Suppose in addition that ra) 2 =2a a 2 when qa) =a and V 1/8. Then 2.34) rb) y/2, when t =b hb, y))/2 1/8 and <y<1/2. Proof. Subtracting 2.32) from 2.3) gives d qs) hs, y) ) Rs)+V s, qs)) V s, hs, y)).
COUNTEREXAMPLES FOR QUASILINEAR WAVE EQUATIONS 617 Since Rs) > and qb) =hb, y) it follows by continuity that qs) hs, y) > when s<b. Hence, after integration using that V q, b a Rs) qa) ha, y), since rb) ra) b b Rs) b a Rs). a a This proves 2.33). If we pick a so that qa) =a then a y and qa) ha, y) =a y + a y V s, hs, y) 9a y)/8 9a/8 9ra) 2 /8, since ra) 2 a when a 1. 2.34) follows from 2.33) since b a b y 16t/7, b y =2t + hb, y) hy, y) =2t + b y V s, hs, y)) 2t +b y)/8. t hb,y)=qb) x 1 +t =s x 1 -t = hs,y) x 1 -t = qs) qa) -ha,y) 2 y a=qa) t+hb,y) x 1 Figure 3.1
618 HANS LINDBLAD r ra) 2 =2a-a 2 ra) rb) y a=qa) t+hb,y) x 1 Figure 3.2 3. Appendix I: Proof of Lemma 1.7 First we note that Defintion 1.6 has two equivalent formulations: Lemma 3.1. If Ω is a domain of dependence and t,x ) Ω then there is an ε> such that the closure of the cone Λ ε t x is contained in Ω. Here Λε t x Λ t x is the cone cosistingof all points that can be joined to t,x ) by a smooth curve t, xt)) Ω, t t, satisfying 3.1) g ij x) dxi dt dx j dt ε, x t) =t. Proof. Suppose that x n t) is a sequence of curves satisfying 3.1) with ε =1/n. Then there is a constant C such that dx n /dt C for all n. Hence we can pick a subsequence, which we also denote by x n, such that x n t) xt) uniformly, where xt) is Lipschitz continuous satisfying satisfying dx/dt C. We claim that xt) satisfies 3.1) with ε =. This follows since g ij x n t)) g ij xt)) uniformely and the set of vectors satisfying g ij xt))y i Y j εy 2, with Y > is a convex cone. Lemma 3.2. Let Ω is a domain of dependence and t,x ) Ω. Then there is a smooth function φ such that with H = {t, x); t < φx)} we have that
COUNTEREXAMPLES FOR QUASILINEAR WAVE EQUATIONS 619 t,x ) H Ω and H is space like, 3.2) i.e., g ij t, x)n i x)n j x) > if t = φx), Nx) =1, x φx)). Proof. Pick t >t such that t,x ) Ω and Λ ε t x as in Lemma 3.1. Note that Λ ε t x = {t, x); t φ εx)} where φ ε x) = sup{t;t, x) Λ ε t x } is Lipschitz continuous. In fact if t, x) Λ ε t x then the cone consisting of all line segments from t, x) with length less than δ and slope satisfying g ij t, x)y i Y j εy 2 /2 lies in Λ ε t x. It follows that φ ε is Lipschitz continuous and satisfies 3.2) with in the right hand side replaced by ε/4, if ε is sufficiently small. The same then is true for φ ε x) = min{φ ε x),t }. By regularizing φ = φ ε ψ δ, where ψ δ x) =ψx/δ)/δ 3, ψ C, ψdx = 1, we get a smooth function satisfying 3.2) which is arbitrarily close to φ ε for which 3.2) hold and φx )=t if δ is sufficiently small. To prove Lemma 1.7, we must show that in any compact subset K Ω [,T) R 3 u ε ten the solution u in Theorem 1.8. Since every compact compact subset of Ω is contained in the interior of a union of sets H as in Lemma 3.2, it suffices to prove Lemma 1.7 for these. Let E N w, t) = sup α wt, ) 2 L 2 H s ) )1/2, s t I N w, t) = sup α N s t β N sup β wt, x), x H s where H s = {x;s, x) H}. Since u is smooth in H it follows that we have the boun 3.3) E M u, t) K M, I M u, t) K M, for any M. By assumption we also have I l u, t) δ. We need the following version of Sobolev type lemma: Lemma 3.3. Suppose that t, x) H and t 1. Then 3.4) wt, x) C sup α ws, ) L 2 H s ), s t α 3 where C depen on H but is independent of t, x). Proof. First we will prove that 3.5) wt, x) sup t α α w,y) + C t sup y x t/2 α 1 s t α =2 α ws, ) L2 y x t s)/2, ).
62 HANS LINDBLAD Let hτ,y) =wt τ,x τy). following identity Then integrating by parts twice gives us the t τh ττ τ,y) dτ = th τ t, y) ht, y)+h,y). Since wt, x) =h, ) = h,y) we get t wt, x) dy = h ττ τ,y) τdydτ + ht, y) thτ t, y) ) dy. y 1 2 y 1 2 y 1 2 Here t y 1 2 t ) 1 h ττ τ,y) τdydτ C h ττ τ,y) 2 τ 3 2 dτ dy y 1 2 τ t ) 1 C h ττ τ,z/τ) 2 2 dτ dz y τ 2 τ C t ) 1 sup α wt τ,x z) 2 2 dz. τ t α =2 z τ 2 Since ht, y) =w,x ty) and h τ t, y) = w t,x ty) y w,x ty), 3.5) follows. Since the conce {s, y); y x t s)/2} Hit follows that the second term in the right hand side of 3.5) is bounded by the right hand side of 3.4). In order to bound the first term in the right hand of 3.5) we use Sobolev Lemma for H. In the semilinear case we have α u ε u) =F α, where F α = α D l u ε D k l u ε D l ud k l u ), and l k l 1. Note that F α is a sum of terms of the form β 1 u ε u) β 2 u ε u), β 1 u) β 2 u ε u), β 1 u ε u) β 2 u), with β 1 α /2 + 1 and β 2 α +1. Let mt) =E N u ε u, t), and nt) =I N 3 u ε u, t), for N 8, say. In view of 3.3) and Lemma 3.3, nt) Cmt), we have 3.6) F α t, ) L 2 H t ) Cmt)+C)mt) Cmt)+C)mt). Recall the energy inequality; 3.7) w t, ) L 2 H t ) w, ) L 2 H ) + t F s, ) L 2 H s ),
COUNTEREXAMPLES FOR QUASILINEAR WAVE EQUATIONS 621 if w = F and H is spacelike. We also have the simple inequality 3.8) wt, ) L 2 H t ) w, ) L 2 H t ) + t These two inequalities together with 3.6) implies that mt) C m) + t t ws, ) L 2 H t ). ) ms) +K)ms). It follows that mt) C m), if m) is sufficiently small and t 1. assumption m), when ε. This proves that u ε u in K. Let g = g ij i j. In the quasilinear case we have By gd l u ε )u ε u) = gd l u ε ) gd l u)) u. If nt) is sufficiently small it follows by inspection of 3.2) that H is spacelike also with respect to the metric g jk D l u ε ). Assuming that this is true we will show that nt) remains small if its sufficiently small when t =. Differentiating the equation gives gd l u ε ) α u ε u) =F α, where F α satisfies the same estimate as in the semilinear case. We now have to use the energy inequality for variable coefficients g w = F that are sufficiently close to see Hörmander [2]), w t, ) L 2 H t ) K w, ) L 2 H ) + t ) t ) F s, ) L 2 H s ) exp K g s), where g s) = sup i g jk s, x), x H j,k,i s provided that H is spacelike and g = j,k gjk m jk 1/2. Here m =1, m jj = 1, j > and m jk =ifj k. Using this inequality we can now proceed as in the semilinear case. References [1] H. Bahouri and J.-Y. Chemin, Equations d ondes quasilinears et estimations de Strichartz, preprint 1998. [2] L. Hörmander, Lectures on Nonlinear hyperbolic differential equations, Mathmatiques & Applications [Mathematics & Applications], 26, Springer-Verlag, Berlin, 1997. [3] S. Klainerman and M. Machedon, Space-time estimates for null-forms and the local existence theorem, Comm. Pure and Appl. Math. 46 1993), 1221 1268.
622 HANS LINDBLAD [4] S. Klainerman and M. Machedon, Finite energy solutions of the Yang-Mills equations in R 3+1, Ann. of Math. 142 1995), 39 119. [5] S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Duke Math. J. 81 1996), 99 133. [6] H. Lindblad, Global solutions for nonlinear wave equations with small initial data, Comm. Pure Appl. Math. 45 1992), 163 196. [7] H. Lindblad, A sharp counterexample to local existence of low regularity solutions to nonlinear wave equations, Duke Math. J. 72 1993), 53 539. [8] H. Lindblad, Counterexamples to local existence for nonlinear wave equations. Journées Equations aux Dérivées Partielles, Saint-Jean-de-Monts, Ecole Polytechnique, Palaiseau 1994). [9] H. Lindblad, Counterexamples to local existence for semilinear wave equations, Amer. J. Math. 118 1996), 1 16. [1] H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 13 1995), 357 426. [11] G. Ponce and T. Sideris, Local regularity of nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations 18 1993), 169 177. [12] J. Rauch, Explosion for some semilinear wave equations, J. Differential Equations 74 1988), 29-33. [13] J. Shatah and A. Shadi Tahvildar-Zadeh, On the Cauchy Problem for equivariant wave maps, Comm. Pure Appl. Math. 47 1994), 719 754. [14] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 3, Princeton University Press, Princeton, N.J., 197. [15] D. Tataru, Strichartz estimates for operators with non-smooth coefficients and the nonlinear wave equation, preprint 1998. University of California San Diego Department of Mathematics, 95 Gilman Drive, La Jolla, CA 9237 E-mail address: lindblad@math.ucsd.edu