Lipschitz Metrics for a Class of Nonlinear Wave Equations

Transcription

1 Lipschitz Metrics for a Class of Nonlinear Wave Equations Alberto Bressan and Geng Chen * Department of Mathematics, Penn State University, University Park, Pa 1680, USA ** School of Mathematics Georgia Institute of Technology, Atlanta, Ga 3033, USA s: bressan@mathpsuedu, gchen73@mathgatechedu May 30, 015 Abstract The nonlinear wave equation u tt cucuu x x 0 determines a flow of conservative solutions taking values in the space H 1 R However, this flow is not continuous wrt the natural H 1 distance Aim of this paper is to construct a new metric which renders the flow uniformly Lipschitz continuous on bounded subsets of H 1 R For this purpose, H 1 is given the structure of a Finsler manifold, where the norm of tangent vectors is defined in terms of an optimal transportation problem For paths of piecewise smooth solutions, one can carefully estimate how the weighted length grows in time By the generic regularity result proved in [7], these piecewise regular paths are dense and can be used to construct a geodesic distance with the desired Lipschitz property 1 Introduction Aim of this paper is to understand the continuous dependence of solutions to the nonlinear wave equation u tt cu cuu x x 0 11 Roughly speaking, the analysis in [8, 17, ] shows that conservative solutions are unique, globally defined, and yield a flow on the space of couples u, u t H 1 R L R For each conservative solution, the total energy Et [u t + c u u x] dx 1 remains constant in time Precise results in this direction will be recalled in Section On the other hand, these solutions do not depend continuously on the initial data, wrt the distance in the normed space H 1 L 1

2 In the present paper we construct a new distance functional which renders Lipschitz continuous the flow generated by 11 We recall that, for solutions of the Hunter-Saxton or the Camassa- Holm equation, a similar task was achieved in [10, 13, 14, 0, 1] Developing ideas in [13], our distance will be determined by the minimum cost to transport an energy measure from one solution to the other While all previous papers dealt with first order equations, to define a suitable transportation distance between two solutions u, ũ of 11 one now faces three main difficulties: At any given time t, each solution determines two distinct measures These account for the energy µ t + of forward moving waves and the energy µ t of backward moving waves The distance between ut and ũt should be measured by the minimum cost for transporting µ t + to µ t + and µ t to µ t The above double transportation problem is considerably complicated by the fact that, while the total energy is conserved, some energy can be transferred from forward to backward moving waves, or viceversa These source terms must be accounted for, when designing an optimal double transportation plan As a wave front crosses waves of the opposite family, its speed can change As a consequence, the distance between two corresponding fronts in u and ũ may quickly increase, making the optimal transportation plan more costly To compensate for this effect, one needs to insert a weight function, accounting for the total energy of approaching waves In Section 3 we introduce a Finsler norm on tangent vectors, related to an energy transportation cost Given a smooth path γ : θ u θ, u θ t, one can then define its weighted length γ by integrating the norm of the tangent vector dγ/dθ Proposition 1, stated in Section 3 and proved in Section 4, contains the key estimate, describing how the norm of a tangent vector grows in time Assuming that, for θ [0, 1] and t [0, T ], all solutions u θ t, remain sufficiently regular so that the length of the path γ t : θ u θ t, u θ t t can still be computed, we obtain the bound γ t C T γ 0, for all t [0, T ] 13 Here the constant C T depends only on T and on a bound on the H 1 L norm of the initial data At this stage, it is natural to define the geodesic distance d { } u, u t, ũ, ũ t inf γ ; γ : [0, 1] H 1 L, γ0 u, u t, γ1 ũ, ũ t 14 By 13 we thus expect that, for any two solutions of 11 and any t [0, T ], this distance should satisfy d ut, u t t, ũt, ũ t t C T d u0, u t 0, ũ0, ũ t 0 15 This would imply that solutions depend Lipschitz continuously on the initial data, in the distance d To clinch this argument, one major difficulty must be overcome Indeed, smooth solutions may well develop singularities in finite time, [19] Given a path γ 0 of smooth initial data, there is no guarantee that at any time t [0, T ] the path γ t will be regular enough so that the

3 tangent vectors dγ t /dθ are meaningfully defined see Fig 1 We remark that a similar issue was encountered in the analysis of hyperbolic conservation laws [6] For a path of piecewise smooth solutions with finitely many shocks, a weighted norm on a suitable family of tangent vectors was introduced in [5] However, a lengthy effort was later required [9, 1], in order to construct paths of approximate solutions which retained enough regularity, so that their length could still be estimated in terms of these tangent vectors ~ ut ~ ut ~ ut? u0 ~ v θ 0 θ u 0 u0 ut v θ t u θt ut u0 ~ u0 ut Figure 1: Left: due to singularity formation, a smooth path of initial data γ 0 : θ u θ 0 may lose regularity at a later time T In this case, the weighted length γ T can no longer be computed by integrating the norm of a tangent vector Right: by a small perturbation of the initial data, one obtains a path of solutions θ u θ which remain piecewise smooth, for all except finitely many values of θ [0, 1] In the present context, we can take advantage of the generic regularity results recently proved in [7] These can be summarized as follows i For an open dense set of initial data u0,, u t 0, u 0, u 1 C 3 R H 1 R C R L R 16 the corresponding solution u ut, x of 11 is piecewise smooth in the t-x plane, with singularities occurring along a finite set of smooth curves ii Every path of initial data θ γ 0 θ u θ 0, uθ 1 can be approximated by a second path θ γ 0 θ ũ θ 0, ũθ 1 such that, for all but finitely many values of θ [0, 1], the corresponding solution ũ θ remains piecewise smooth on the domain [0, T ] R Using this dense set of piecewise regular paths, we can thus define a geodesic distance on the space H 1 L, with the desired Lipschitz property Our main results are contained in Proposition 1, which establishes the basic estimate 3 on the size of tangent vectors Theorem 5, providing the bound 63 on how the length of a path of solutions can grow in time Theorem 7, showing that, by 76, the flow generated by the wave equation 11 is Lipschitz continuous wrt the geodesic distance d We remark that, for hyperbolic conservation laws, the distance constructed in [5, 9, 1] is equivalent to the L 1 distance On the contrary, our new metric is not equivalent to the norm 3

4 distance on H 1 L The completion of H 1 L wrt the geodesic distance includes a family of measures This should not come as a surprise, since it was already observed in [17, ] that conservative solutions can occasionally be measure-valued In Section 7 we compare the geodesic distance 14 with more familiar distances found in the literature In one direction, we show that d u 0, u 1, ũ 0, ũ 1 C u 0 ũ 0 H 1 + u 0 ũ 0 W 11 + u 1 ũ 1 L + u 1 ũ 1 L 1, for some constant C On the other hand, let µ and µ be the positive measures having densities respectively u t + c uu x and ũ t + c ũũ x 17 wrt Lebesgue measure Then the geodesic distance d dominates the Wasserstein distance between the two measures Namely { } sup f dµ fd µ ; f C 1 1 d u, u t, ũ, ũ t 18 All of the present analysis is concerned with conservative solutions to 11 For dissipative solutions, studied in [15, 19, 5, 6], the continuous dependence for general initial data in H 1 L remains an open question For scalar conservation laws, an entirely different approach to continuous dependence, relying on an L formulation, was developed in [, 3, 4] Conservative solutions to the nonlinear wave equation In this section we review the main results in [7, 8, 17] on the Cauchy problem for the quasilinear second order wave equation u tt cu cuu x x 0, 1 with initial data u0, x u 0 x, u t 0, x u 1 x Here c : R R + is a smooth, uniformly positive function, such that cu c 0 > 0 3 Consider the variables { R ut + cuu x, S u t cuu x, so that u t R + S By 1, the variables R, S satisfy, u x R S c R t cr x c R S, S t + cs x c S R

5 Multiplying the first equation in 6 by R and the second one by S, one obtains balance laws for R and S, namely R t cr x c c R S RS, S t + cs x c c R S RS As a consequence, for smooth solutions the following quantity is conserved: 7 e u t + c u x R + S 8 We think of R / and S / as the energy of backward and forward moving waves, respectively These are not separately conserved Indeed, by 7 energy is transferred from forward to backward waves, and viceversa The main results on the existence of solutions to the Cauchy problem can be summarized as follows Theorem 1 Let c : R R be a smooth function satisfying 3 Assume that the initial data u 0 in is absolutely continuous, and that u 0 x L, u 1 L Then the Cauchy problem 1- admits a weak solution u ut, x, defined for all t, x R R In the t-x plane, the function u is locally Hölder continuous with exponent 1/ This solution t ut, is continuously differentiable as a map with values in L p loc, for all 1 p < Moreover, it is Lipschitz continuous wrt the L distance, ie ut, us, L L t s 9 for all t, s R The equation 1 is satisfied in distributional sense, ie [ φ t u t cuφ x cu u x] dxdt 0 10 for all test functions φ Cc 1 The maps t u t t, and t u x t, are continuous with values in L p loc R, for every p [1, [ Theorem In the same setting as Theorem 1, a unique solution u ut, x exists which is conservative in the following sense There exists two families of positive Radon measures on the real line: {µ t } and {µ t +}, depending continuously on t in the weak topology of measures, with the following properties i At every time t one has µ t R + µ t +R E 0 [ u 1x + cu 0 xu 0,x x ] dx 11 ii For each t, the absolutely continuous parts of µ t and µ t + wrt the Lebesgue measure have densities respectively given by R u t + cuu x, S u t cuu x 1 iii For almost every t R, the singular parts of µ t and µ t + are concentrated on the set where c u 0 5

6 iv The measures µ t and µ t + provide measure-valued solutions respectively to the balance laws ξ t cξ x c c R S RS, 13 η t + cη x c c R S RS The existence part of the above theorems was proved in [17] The uniqueness of conservative solutions has been recently established in [8] Remark 1 By 13 the total energy, represented by the positive measure µ t µ t + + µ t, is conserved in time Occasionally, some of this energy is concentrated on a set of measure zero At a time τ when this happens, µ τ has a non-trivial singular part and hence its absolutely continuous part satisfies [ u t τ, x + c uτ, x ] u xτ, x dx < E 0 The condition iii puts some restrictions on the set of such times τ In particular, if c u 0 for all u, then this set has measure zero Remark For any t 0, the conservation of the total energy implies u t t L E 0 u 1 + c u 0 u 0,x dx 14 Hence 9 holds with Lipschitz constant L E 0 Moreover, one has the bounds ut, L u 0 L + t E 0, u x t, L E0 c 0 15 This yields an a priori bound on ut, H 1, and hence on ut, L, depending only on time and on the total energy E 0 In turn, since the wave speed c is smooth, we obtain an a priori bound on cu and c u 3 First order variations For simplicity, in this section we consider solutions of 1 with bounded support precisely, we shall assume that all our solutions satisfy More ut, x 0 for x / [0, L 0 ], t [0, T ] 31 Because of finite propagation speed, this is hardly a restriction Let u, R, S provide a smooth solution to 1, 4, and consider a family of perturbed solutions of the form u ε u + εv + oε, { R ε R + εr + oε, S ε S + εs + oε 3 6

7 From 5 it follows u ε x u ε t Rε S ε cu ε Rε + S ε R + S + ε r + s + oε, 33 R S cu + ε r s cu ε R S c u c u v + oε 34 Under the assumption 31, given r, s, the perturbation v is uniquely determined by v x R Sc u c u v + r s, vt, cu Furthermore, we have v t r + s 36 A direct computation shows that the first order perturbations v, s, r satisfy the linear equations v tt c v xx cc u x v x + c u x + cc u x + cc u xx v 37 c r t cur x c R x v + c R S v + c Rr Ss, c c s t + cus x c S x v + c S R v + c Ss Rr c 38 By the assumptions 3 on the wave speed cu, all functions c /, c /, c /, are smooth functions of u We shall introduce a weighted norm on tangent vectors r, s, which takes into account the total energy of waves which are approaching a given wave located at x This is described by the weights W x x 1 + S y dy, W + x x R y dy 39 In addition, consider the function at c R S S R t, x dx 310 c As proved in [8], the function τ τ + 0 c c R S RS t, x dx dt is Hölder continuous and absolutely continuous on bounded time intervals, and has sub-linear growth In particular see in the proof of Lemma 1 in [8], one has T 0 at dt C T, 311 7

8 for some constant C T depending only on T and on the total energy E 0 By 7 it follows x Wt cwx cs c + c S R R S dy c 0 S + at, 31 + W t + + cw x + cr c + x c R S S R dy c 0 R + at On the space of tangent vectors v, r, s we introduce a Finsler norm, having the form v, r, s inf r, w, s, z, 313 u,r,s u,r,s r, s,w,z where the infimum is taken over the set of vertical displacements r, s and shifts w, z which satisfy r r wr x + c w zs, 8c 314 s s zs x + c w zr 8c This norm is defined as r, w, s, z u,r,s κ 1 { w 1 + R W + z 1 + S W +} dx +κ { r W + s W +} dx v Rw +κ 3 + c Sz } {1 + R W S W + dx c { wx } +κ 4 + c W w zs + z x + c W w zr + dx { Rwx } +κ 5 + c W w zsr + Sz x + c W w zrs + dx { +κ 6 R r + R w x + c R Sw z W + S s + S z x + c S Rw z W }dx + κ 1 I 1 + κ I + κ 3 I 3 + κ 4 I 4 + κ 5 I 5 + κ 6 I 6, for suitable constants κ 1,, κ 6 to be determined later 315 To motivate 313, consider a profile R and a perturbation R ε, as shown in figure In first approximation, R ε R + εr Notice that we could also obtain the profile R ε starting from the graph of R, performing a horizontal shift in the amount εw and then a vertical shift in the amount ε r, provided that r r wr x 316 8

9 As a first guess, one could thus define a norm r by optimizing the choice of r, w, subject to 316 However, a detailed analysis has shown that this approach does not work Indeed, it does not take into account the fact that, when backward and forward moving waves cross each other, by 6 their sizes R, S are modified Compared with 316, the additional term in the first equation of 314 accounts for this interaction Notice that w z is the relative shift of backward wrt forward waves R ε R εr ε ~ r x x+ εw Figure : A perturbation of the R-component of the solution to the variational wave equation We now explain the meaning of each integral on the right hand side of 315 The integral of w 1 + R can be interpreted as the cost for transporting the base measure with density 1 + R from the point x to the point x + εwx Similarly, the integral of z 1 + S accounts for the cost of transporting the measure with density 1 + S from x to x + εzx Here, as in all other terms, we insert the weights W ± coming from the interaction potential I accounts for the vertical shifts in the graphs of R, S We interpret the integrand as the change in arctan R times the density 1 + R of the base measure Notice that here the factor 1 + R cancels out with the derivative of the arctangent I 3 accounts for the changes in u Observe that ε 1 [u ε x + εwx ux] vx + u x xwx vx + Rx Sx wx cux This can be written in the form v + R S w c v + Rc w Sc z Sz w c Notice that the last term on the right hand side of 317 does not appear in I 3 In fact, the last term Sz w c is the relative shift term coming from the equation 4 Subsequent computations will show that this term is inessential, because its contribution can be bounded by the decrease in the interaction potential In an entirely similar way we obtain ε 1 [u ε x + εzx ux] vx + u x xzx vx + 9 Rx Sx zx, cux

10 v + R S z c v + Rc w Sc z Rz w + c I 6 accounts for the change in base measure with densities R and S, produced by the shifts w, z To see this, assume that the mass with density R is transported from x to x + εwx If the mass were conserved, the new density should be R ε x + εwx R x εwxr x x εw x xr x + oε 318 In addition, if the mass with density S is transported from x to x + εzx, by 7 the crossing between forward and backward waves yields the source term c c R S RS z w 319 c On the other hand, if we shift the graph of R horizontally by εw and then vertically by ε r, the new density will be R ε x + εwx R x εwxr x x + εrx rx + oε 30 Subtracting from 30 we obtain the expression Rr + wr x + R w x + c R S RS w z 31 The integrals I 4 and I 5 does not seem to have a clear geometric interpretation I 4 is somewhat related to the change in Lebesgue measure produced by the shifts w, z, while I 5 is related to the change in base measure with densities R and S, produced by the shifts w, z As shown by our subsequent computations, these two additional terms must be included in the definition 315, in order to estimate the time derivatives of I 3 and I 6 Our goal is to prove Proposition 1 Let u, R, S be a smooth solution to 1 and 6, and assume that the first order perturbations v, r, s satisfy the corresponding linear equations Then for any τ 0 one has { τ } vτ, rτ, sτ exp Cτ + asds v0, r0, s0, uτ,rτ,sτ 0 u0,r0,s0 3 with a constant C depending only on the total energy Toward the proof, the main argument goes as follows At time t 0 let a tangent vector v0, r0, s0 be given By the definition 313, for any ɛ > 0 we can find shifts w 0, z 0 and perturbations r 0, s 0 satisfying r 0, w 0, s 0, z 0 ɛ + v0, r0, s0 33 u0,r0,s0 u0,r0,s0 10

11 together with the constraints r0 r 0 w 0 R x 0 + c 8c w 0 z 0 S 0, s0 s 0 z 0 S x 0 + c 8c w 0 z 0 R 0 34 In order to prove 3, for any t [0, τ] it suffices to find shifts wt, zt, together with rt, st satisfying 314 and the initial condition 34, so that d rt, wt, st, zt C + at rt, wt, st, zt dt ut,rt,st ut,rt,st 35 These shifts wt, zt will be obtained by propagating along characteristics the shifts w 0, z 0 in the initial data More precisely, we choose w, z to be the solutions of the linearized system w t cuw x c u v + u x w, 36 z t + cuz x c uv + u x z, with initial data w0, x w 0 x, z0, x z 0 x By 38 and the identities 314, this determines the evolution equation for r, s 37 In the next section, by carefully estimating the time derivatives of all terms in 315, we shall prove that 35 holds In turn, this will yield 3 4 Estimates on the norm of tangent vectors The first part of the proof of 35 is largely computational Using the evolution equations 1, 4, 6 for u, R, S, and 38, 36 for r, s, w, z, together with the identities 314, we estimate the time derivative of each integral in To estimate the time derivative of I 1 shift in the base measure, using 36 we first compute w1 + R t cw1 + R x w t cw x 1 + R + w [ R t cr x ] wcx c v + R S w 1 + R + c c c wr S RS R + S c v + R c w S c z 1 + R + c c wr S RS R + S c c zs1 + R 11

12 Thanks to 31 we obtain d w 1 + R W dx O1 w 1 + R S + RS + R + S W dx dt +O1 z S + R S W + v Rw dx + O1 + c Sz 1 + R W dx c +at w 1 + R W dx c 0 w 1 + R S W dx 41 To estimate the time derivative of I change in arctan, using 38 we first compute r + wrx Next, c t cr + wr x x [ r t cr x ] + wt cw x R x + w [ R x t cr x x ] c R S c c r + c R x v + c c c R S v + c c [ c c c v + R S c w R x + w R S c Rr Ss R S + c RR x SS x c c SwR x S x + c c Sr s + c c c R S v + R S w c w zs 8c t c c w zs 8c x c c c 8c 3 u t cu x w zs + c 8c w t cw x S c 8c z t + cz x S + c 8c cz xs + c 8c w z [ S t + cs x ] c 8c w zcs x c c c 8c 3 w zs 3 c 8c v + R S w S c c 8c v + R S z c S ] 4 + c z xs c 16c 3 w zr S RS c 8c 3 w zrs S 3 c c w zss x 43 1

13 By 314, combining 4 with 43 we obtain r t c r x [ r + wr x t cr + wr x x ] [ c w zs 8c c c SwR x S x + c c Sr s + c c c R S t ] c c w zs 8c x v + R S w c c c c 8c 3 w zs 3 + c 8c v + R S w + z S c c z xs + c 16c 3 w zr S RS + c 8c 3 w zrs S 3 + c c w zss x c c wsr x zss x c z xs + r c c S s wr x zs x + c 8c w zs R + c c c R S v + R S w c c c c 8c 3 w zs 3 + c 8c v + R S w + z c S c c S r S s + S z x + c c S Rw z + c c c R v + Rw c Sz c c c c S v + Rw c Sz c +O1 w + z 1 + R S + RS We thus conclude d r W dx O1 dt +O1 +O1 S r W + O1 S s + S z x + c S Rw z W+ dx S v + Rw c Sz c W+ dx + O1 w 1 + R S + RS W dx + O1 R v + Rw c Sz c W dx z 1 + R S + RS W + dx 44 +at r W dx c 0 r S W dx 45 13

14 3 To estimate the time derivative of I 3 change in u, using the identities in for v t and v x, we first compute v t cv x s + c R Sv c 46 Next, by 4 and 36 we obtain Rw c Sz c t c Rw c Sz c x 1 c wr t cr x 1 c zs t + cs x + zs x + R c w t cw x S c z t + cz x + Sz x c c RSw S z 47 c 8c wr S c 8c zs R + zs x c c R Finally, by 6 it follows v + R S c w c c S v + R S c z + Sz x c c RSw S z, 1 + R t c1 + R x c c R S RS c R S c Putting together and using 314 one obtains [ v + Rw c Sz ] [ 1 + R c v + Rw c c Sz ] 1 + R c t [ Rw v t cv x + c Sz Rw c c t c Sz ] 1 + R c x + v + Rw c Sz [ 1 + R t c1 + R ] c x x [s + c c vr S + c 8c wr S c 8c zs R + zs x c c R v + R S c w c c S v + R S c + v + Rw c Sz [ c ] c c R S RS c R S c ] z + Sz x c c SRw S z 1 + R [ s + c 8c z ws c c S v + Rw c Sz + Sz x + c ] w zrs 1 + R c + v + Rw c Sz [ c ] c c R S RS c R S c 48 14

15 We thus conclude d v + Rw dt c Sz c 1 + R W dx s 1 + R W + dx + O1 v + Rw c Sz c 1 + R + S + R S + RS W dx +O1 w S 1 + R W dx + O1 +O1 Sz x + c w zrs 1 + R W dx z S 1 + R W + dx +at v + Rw c Sz c 1 + R W c 0 v + Rw c Sz c 1 + R S W dx 49 4 To estimate the time derivative of I 4, recalling 36 we first compute w x t cw x x c c R S v + R S c w c [ R Sc c v + r s ] c c 3 R S w 410 c c R xw S x w c c R Sw x Moreover, by 4 and 36, one has c ws c c ws t c ws c v + R S w c c zs t x c c zs x c S c zs + c 16c 3 wr S c c R SwS 8c3 c ws x, 411 v + R S z S + c c 16c 3 zs R c c R SzS 8c3 c zs x c c z xs 41 15

16 Combining the identities and recalling 314, we obtain w x + c [ w zs c w x + c t ]t w zs c c c s r + c c c S w x + c w zs + c Sz x + c c w zsr c Rw x + c c w zsr c c c c R v + Rw c Sz c + c c c c S v + Rw c Sz c c c c 3 RSw z c 8c 3 S w z By the previous analysis, thanks to the uniform bounds 31 on the weights, we conclude d w x + c w zs dt W dx O1 r W dx + O1 s W + S wx dx + O1 + c w zs W dx Szx +O1 + c w zrs W + Rwx dx + O1 + c w zrs W dx v Rw +O1 + c Sz R W v Rw dx + O1 + c c Sz S W + dx c +O1 w RS + S W dx + O1 +at w x + c w zs W dx c 0 z RS + S W + dx w x + c w zs S W dx

17 5 To estimate the time derivative of I 5, using 413 we compute [ [ R w x + c ]t w zs + Rc w x + c ]x w zs c c c R s R r + c c c RS w x + c w zs + c c R Sz x + c w zsr c c R Rw x + c w zsr c c c c R v + Rw c Sz c + c c c c SR v + Rw c Sz c + c c c 3 R Sw z c + c R S w x + c w zs 8c 3 S Rw z 415 c c R s R r + R w x + c w zsr c + c c S Rw x + c w zrs + c c R Sz x + c w zsr c c c c R v + Rw c Sz c + c c c c SR v + Rw c Sz c + c c c 3 R Sw z c 8c 3 S Rw z c S w x + c w zs 17

18 We thus conclude d dt Rwx + c w zrs W dx O1 R s W dx + O1 R r + R w x + c w zsr W dx Rwx +O1 + c w zrs S W dx Szx +O1 + c w zsr R W dx v Rw +O1 + c Sz R W dx + O1 c v + Rw w +O1 + z 1 + R 1 + S W dx + O1 +at Rw x + c w zrs W dx c 0 c Sz c Rw x + c RS W + dx S wx + c w zs W dx w zrs S W dx

19 6 Finally, to estimate the time derivative of I 6 change in base measure with density R, we compute R r + R w x + c w zsr + c R r + R w x + c w t zsrx R t cr x r + Rw x + c w zsr [ c ] c +R r t c r x + Rw x t crw x x + w zsr t x w zsr c R S r + Rw x + c w zsr + c c c c R 3 v + Rw c Sz c + c c c 3 R 3 Sz w + c c c c c 8c 3 R S w z c c R S s + S z x + c S Rw z + c c SR r S R v + Rw c Sz c + c c R s c R R r + R w x + c w zsr + c c SR Rw x + c w zrs + c c R Sz x + c w zsr c c c c R 3 v + Rw c Sz c + c c c c SR v + Rw c Sz c + c c c 3 R 3 Sw z c 8c 3 S R w z c S R w x + c w zs c c R Sz x + c w zrs c c S Rw x + c w zsr + c c c c v + Rw c Sz R S RS c c c S r + c c R s + R r c c S + R w x + c R Sw z S s c c R + S z x + c RS w z

20 This yields the estimate d dt R r + R w x + c w zsr W dx O1 +O1 R Szx + c w zrs W + dx S Rwx + c w zrs W dx v Rw +O1 + c Sz R S RS W dx c +O1 S r W dx + O1 R s W + dx 418 +O1 +O1 S R r + R w x + c R Sw z W dx R S s + S z x + c RS w z W+ dx at + c0 R r + R w x + c w zsr S W dx I I I I I I I I I I I 5 I 6 Figure 3: A graphical summary of all the a priori estimates If a lower box I k is connected to an upper box I l, this means that the integral I l is used in order to control the time derivative I k d dt I k If l F k, then I k and I l are connected by a solid line If l Fk, then I k and I l are connected by a dashed line 0

21 7 We keep track of all the above estimates by the diagram in Fig 3 Recalling 315, the weighted norm of a tangent vector can be written as r, w, s, z κ 1 I 1 + κ I + κ 3 I 3 + κ 4 I 4 + κ 5 I 5 + κ 6 I 6 u,r,s 6 κ k k1 J k W dx + J + k W+ dx, 419 where J k, J + k are the various integrands According to the estimates 41, 45, 49, 414, 416, and 418, the time derivative of each I k can be estimated as I k O1 1 + S W dx R W + dx J l J + l l F k + O1 J l 1 + R W dx + J + l 1 + S W + dx 40 l F k +ati k c 0 S J k W dx + R J + k W+ dx Here Fk, F k {1,,, 6} are suitable sets of indices, illustrated in Fig 3 By direct inspection, we see that the set-valued map k F k has no cycles Indeed, the composition F k F k F k yields the empty set By choosing a constant δ > 0 small enough, we thus obtain a weighted norm r, w, s, z I1 + δi + δ 3 I 3 + δi 4 + δ I 5 + δ 3 I 6 41 u,r,s which satisfies the desired inequality 35 This completes the proof of Proposition 1 5 Tangent vectors in transformed coordinates Given any path θ u θ, θ [0, 1] of smooth solutions to 11, the analysis in the previous section has provided an estimate on how its weighted length increases in time However, even for smooth initial data, it is well known that the quantities u t, u x can blow up in finite time [19] When this happens, a tangent vector may no longer exist; even if it does exist, it is not obvious that our earlier estimates should remain valid Aim of this section is to address these issues Roughly speaking, we claim that i Every path of solutions θ u θ can be uniformly approximated by a second path θ ũ θ such that, for all but finitely many values of θ [0, 1], the solution ũ θ is piecewise smooth, with generic singularities ii If all solutions u θ are piecewise smooth, with generic singularities along finitely many points and finitely many curves in the t-x plane, then the tangent vectors are still well defined and their norms can be estimated as before 1

22 A precise formulation of i was recently proved by the authors in [7] The proof is based on the representation of solutions to 11 in terms of a semilinear system with smooth coefficients [17], followed by an application of Thom s transversality theorem We review here this basic construction, and the characterization of generic structurally stable singularities [16] To deal with possibly unbounded values of R, S in 4, following [17] it is convenient to introduce a new set of dependent variables: Using 6, we obtain the equations α arctan R, β arctan S 51 α t c α x β t + c β x 1 + R R t c R x c R S c 1 + R, S S t + c S x c S R c 1 + S 53 We now perform a further change of independent variables Consider the equations for the backward and forward characteristics: ẋ cu, ẋ + cu, 54 where the upper dot denotes a derivative wrt time The characteristics passing through the point t, x will be denoted by s x s, t, x, s x + s, t, x, respectively We shall use a set of coordinates X, Y on the t-x plane such that X is constant along backward characteristics and Y is constant along forward characteristics, namely X t cux x 0, 55 Y t + cuy x 0 For example, one can define X, Y to be the intersections with the x-axis, of the characteristics through the point t, x, ie Xt, x x 0, t, x, Y t, x x + 0, t, x 56 More generally, one can consider strictly increasing functions x Xx and x Y x and define Xt, x X x 0, t, x, Y t, x Y x + 0, t, x 57 For any smooth function f, using 55 one finds f t + cf x f X X t + f Y Y t + cf X X x + cf Y Y x X t + cx x f X cx x f X, f t cf x f X X t + f Y Y t cf X X x cf Y Y x Y t cy x f Y cy x f Y 58 We now introduce the further variables p 1 + R X x, q 1 + S Y x 59

23 Notice that the above definitions imply 1 X x p 1 + cos αp, 1 + R 1 Y x q 1 + cos βq S Starting with the nonlinear equation 1, using X, Y as independent variables one obtains a semilinear hyperbolic system with smooth coefficients for the variables u, α, β, p, q, namely u X sin α p, u Y sin β q, α Y c 8c cos β cos α q, β X c 8c cos α cos β p, p Y c 8c sin β sin α pq, q X c 8c sin α sin β pq The map X, Y t, x can be constructed as follows Setting f x, then f t in the two equations at 58, we find { { c cxx x X, 1 c Xx t X, c cy x x Y, 1 c Y x t Y, respectively Therefore, using 510 we obtain 1 x X X x 1 x Y Y x 1 t X cx x t Y 1 cy x 1+cos α p 4, 1+cos β q 4, 1+cos α p, 1+cos β q Given the initial data, one particular way to assign the corresponding boundary data for is as follows In the X-Y plane, consider the line parameterized as x Xx, Y x u, α, β, p, q by setting γ 0 {X + Y 0} R 516 x, x Along γ 0 we can assign the boundary data u u 0 x, { α arctan R0, x, β arctan S0, x, { p 1 + R 0, x, q 1 + S 0, x, 517 at each point x, x γ 0 We recall that, at time t 0, by one has R0, x S0, x u t + cuu x 0, x u 1 x + cu 0 xu 0,x x, u t cuu x 0, x u 1 x cu 0 xu 0,x x 3

24 Remark 3 The above construction is by no means the unique way to prescribe initial values One should be aware that many distinct solutions to the system can yield the same solution u ut, x of 1- Indeed, let u, α, β, p, q, x, tx, Y be one particular solution Let φ, ψ : R R be two C bijections, with φ > 0 and ψ > 0 Introduce the new independent and dependent variables X, Ỹ and ũ, α, β, p, q, x, t by setting X φ X, Y ψỹ, 518 ũ, α, β, x, t X, Ỹ u, α, β, x, tx, Y, 519 p X, Ỹ px, Y φ X, q X, 50 Ỹ qx, Y ψ Ỹ Then, as functions of X, Ỹ, the variables ũ, α, β, p, q, x, t provide another solution of the same system Moreover, by 519 the set { t X, Ỹ, x X, Ỹ, ũ X, Ỹ ; X, Ỹ R} 51 coincides with the set in 53 Hence it is the graph of the same solution u of 1 One can regard the variable transformation 518 simply as a relabeling of forward and backward characteristics, in the solution u In connection with first order wave equations, relabeling symmetries have been studied in [14, 1] Remark 4 The system is clearly invariant wrt the addition of an integer multiple of π to the variables α, β Taking advantage of this property, in the following we shall regard α, β as points in the quotient manifold T R/πZ As a consequence, we have the implications α π cos α > 1, 5 β π cos β > 1 Remark 5 Since the semilinear system has smooth coefficients, for smooth initial data all components of the solution remain smooth on the entire X-Y plane As proved in [17], the quadratic terms in 513 containing the product pq account for transversal wave interactions and do not produce finite time blowup of the variables p, q Moreover, if the values of p, q are uniformly positive and bounded on the line γ 0, then they remain uniformly positive and bounded on compact sets of the X-Y plane Throughout this paper, we always consider solutions of where p, q > 0 The main results in [8, 17] can be summarized as Theorem 3 Let c cu be a smooth, uniformly positive function Let t, x, u, α, β, p, qx, Y be a smooth solution of the semilinear system with boundary data as in 517 Then the function u ut, x whose graph is Graphu { tx, Y, xx, Y, ux, Y ; X, Y R } 53 provides the unique conservative solution to the Cauchy problem 1-4

25 α π Y γ 0 Q 0 P X β π Figure 4: The level sets {α π} and {β π} in a solution with generic singularities In the X-Y plane these are smooth curves which are structurally stable wrt small C perturbations Throughout the following, we shall be interested not in a single solution but in a continuous path of solutions θ u θ, θ [0, 1] We introduce suitable regularity conditions, allowing us to compute the length of this path by integrating a suitable norm of its tangent vector du θ t, /dθ Definition 1 We say that a solution u ut, x of 1 has generic singularities for t [0, T ] if it admits a representation of the form 53, where i the functions u, α, β, p, q, x, tx, Y are C, and ii on the domain where tx, Y [0, T ] the following generic conditions hold: G1 α π, α X 0 α Y 0, α XX 0, G β π, β Y 0 β X 0, β Y Y 0, G3 α π, β π, α X 0, β Y 0 t q Figure 5: The set of singular points where u x + in a solution ut, x These are the images of the sets {α π} and {β π} in Fig 4 By structural stability, every small perturbation will yield anther solution with the same type of singularities Some words of explanation are in order Even if the solution X, Y x, t, u, α, β, p, qx, Y of the semilinear system remains smooth on the entire X-Y plane, the function u ut, x in 53 can have singularities because the coordinate change Λ : X, Y x, t p x 5

26 is not smoothly invertible Indeed, by , the Jacobian matrix is computed by 1+cos α p 1+cos β q xx x DΛ Y t X t Y 1+cos α p u 1+cos β q u We recall that p, q remain uniformly positive and uniformly bounded on compact subsets of the X-Y plane By Remark 3, at a point X 0, Y 0 where α π and β π, this matrix is invertible, having a strictly positive determinant The function u ux, t considered at 53 is thus smooth on a neighborhood of the point t 0, x 0 tx 0, Y 0, xx 0, Y 0 To study the set of points in the x-t plane where u is singular, we thus need to look at points where either w π or β π The generic conditions G1 G guarantee that these level sets are smooth curves in the X-Y plane Condition G3 implies that the level sets where {α π} and {β π} intersect transversally because α Y β X 0 when α β 0 As observed in [7], the conditions G1 G3 are invariant wrt smooth variable transformations X, Y X, Ỹ We also remark that, if a solution U u, α, β, p, q of satisfies the generic conditions G1 G3, then by the implicit function theorem the same is true for every perturbed solution Ũ ũ, α, β, p, q sufficiently close to U In other words, generic singularities are structurally stable An example of structurally unstable solution, corresponding to a change of topology in the singular set, is shown in Fig 6 Definition We say that a path of initial data γ : θ u θ 0, uθ 1, θ [0, 1] is a piecewise regular path if the following conditions are satisfied i There exists a continuous map X, Y, θ u, α, β, p, q, x, t such that, for each θ [0, 1] the semilinear system is satisfied Moreover, the function u θ x, t whose graph is { Graphu θ t, x, ux, Y, θ; X, Y R } 55 provides the conservative solution of 11 with initial data u θ 0, x u θ 0x, u θ t 0, x u θ 1x ii There exist finitely many values 0 θ 0 < θ 1 < < θ N 1 such that the following holds For θ ]θ i 1, θ i [, the map X, Y, θ u, α, β, p, q, x, t is C Moreover, the solution u θ u θ t, x has generic singularities at time t 0 In addition, if for all θ [0, 1] \ {θ 1,, θ N }, the solution u θ has generic singularities for t [0, T ], then we say that the path of solutions γ : θ u θ is piecewise regular for t [0, T ] Remark 6 According to Remark 3, there are infinitely many parameterizations of the variables X, Y that yield the same solution u ut, x However, as shown in [7], the property of having generic singularities is independent of the particular representation used in 55 6

27 Remark 7 The above definition has a simple motivation If γ is a piecewise regular path, then we can compute its length as an integral of the norm of a tangent vector In addition, if γ is piecewise regular for t [0, T ], then the length of the path of solutions γ t : θ u θ t,, u θ t t, is well defined not only at t 0 but for all t [0, T ] See Definition 3 in Section 6 for details Remark 8 In Definition, the finitely many values of θ where u θ does not have structurally stable singularities correspond to bifurcation values As θ crosses one of these values, the topological structure of the singular set where u θ x ± usually changes, as shown in Fig 6 _ t θ < θ t θ θ t θ > θ x x x Figure 6: Here the solution u θ has generic ie, structurally stable singularities for θ < θ and for θ > θ However, when the parameter θ crosses the critical value θ, the topology of the singular set changes Following [7], on the wave speed c we assume A The map c : R R + is smooth and uniformly positive The quotient c u/cu is uniformly bounded Moreover, the following generic condition is satisfied: c u 0 c u 0 56 Notice that, by 56, the derivative c u vanishes only at isolated points The following result, proved in [7], shows that the set of piecewise regular paths is dense Theorem 4 Let the wave speed cu satisfy the assumptions A and let T > 0 be given Let θ t θ, x θ, u θ, α θ, β θ, p θ, q θ, θ [0, 1], be a smooth path of solutions to Then there exists a sequence of paths of solutions θ t θ n, x θ n, u θ n, α θ n, β θ n, p θ n, q θ n with the following properties i For each n 1, the path of corresponding solutions of 1 θ u θ n is regular for t [0, T ], according to Definition ii For any bounded domain Ω in the X-Y plane, as n the functions t θ n, x θ n, u θ n, α θ n, β θ n, p θ n, q θ n converge to t θ, x θ, u θ, α θ, β θ, p θ, q θ uniformly in C k [0, 1] Ω, for every k 1 Thanks to this density result, to construct a Lipschitz metric it now remains to show that the weighted length of a regular path satisfies the same estimates as the smooth paths considered 7

28 in the previous section Toward this goal, we first derive an expression for the norm of a tangent vector as a line integral in the X-Y coordinates Consider a reference solution u 1 and a family of perturbed solutions u ε, ε [0, ε 0 ] We assume that, in the X-Y coordinates, these define a smooth family of solutions of , say t ε, x ε, u ε, α ε, β ε, p ε, q ε For each ε, the curves where X constant and Y constant correspond respectively to backward and forward characteristics of the solutions u ε t, x We remark that, at time t 0, we have considerable freedom in choosing these parameterizations We can take advantage of this in the following way Let w, z be the shifts in 36 At time t 0 we choose the parameterizations according to X ε 0, x + ε w0, x x, Y ε 0, x + ε z0, x x 57 Consider the curve in X-Y space Γ τ {X, Y, tx, Y τ} {X, Y τ, X ; X R} {Xτ, Y, Y ; Y R}, 58 and denote by Γ ε τ {X, Y, t ε X, Y τ} {X, Y ε τ, X ; X R} {X ε τ, Y, Y ; Y R} 59 the perturbed curve We can write the perturbed solutions as t ε, x ε, u ε, α ε, β ε, p ε, q ε t, x, u, α, β, p, q + εt, X, U, A, B, P, Q + oε 530 Since the system has smooth coefficients, the first order perturbations satisfy a linearized system and are well defined for all X, Y R We observe that the quantities v, r, s, w, z appearing in 315 can be expressed in terms of the first order perturbations T, X, U, A, B, P, Q Indeed, Notice that, by definition, 1 + R dx p dx, 1 + S dx q dy t ε X, Y ε τ, X t ε X ε τ, Y, Y τ Hence by the implicit function theorem, at ε 0: X ε tε t 1 ε ε T X 1 + cos αp and Y ε ε tε t 1 ε T Y 1 + cos βq 1 The shift in x is computed by In a similar way, x w lim ε X,Y ε τ,x xx,y τ,x ε 0 ε X X, Y τ, X + x Y Y ε ε X + ct X, Y τ, X ε0 x z lim ε X ε τ,y,y xxτ,y,y ε 0 ε X Xτ, Y, Y + x X Xε ε X ct Xτ, Y, Y, ε0 8

29 We now derive an expression for r, s One has r + wr x d dε tan αε X, Y ε τ, X 1 ε0 A T 1 + cos βq α Y sec α 531 and s + zs x d dε tan βε X ε τ, Y, Y 1 ε0 B T 1 + cos αp β X sec β 53 By 314 it thus follows r 1 A T 1 + cos βq α Y sec α c T β tan 533 and s 1 B T 1 + cos αp β X sec β c T α tan By 511 one has v + u x w d dε uε X, Y ε τ, X ε0 U u Y T 1 + cos βq U T tan α Therefore v + Rw c Sz c U tan α + tan β T We now calculate the terms I 4 I 6 in 315 The change in base measure with density 1 + R is given by lim ε 0 p ε X, Y ε τ, X px, Y τ, X ε P X, Y + p Y Y ε ε The change in base measure with density 1 + S is given by lim ε 0 q ε X ε τ, Y, Y qxτ, Y, Y ε QX, Y + q X Xε ε ε0 ε0 P T Q T 1 + cos βq p Y cos αp q X 537 The change in base measure with density R the integrand in I 6 is estimated by d p ε sin αε dε P T X, Y ε τ, X ε0 1 + cos βq p Y sin α + p sin α A T 1 + cos βq α Y 538 The difference between 536 and 538 shows that the change in base measure with density 1 the integrand in I 4 is computed by P T 1 + cos βq p Y cos α p sin α A T cos βq α Y 539

30 Combining the previous computations, the weighted norm of a tangent vector 315 can be written as a line integral over the line Γ τ defined at 58: where r, w, s, z u,r,s J 1 X ct p J 1 J 3 Ap T 6 l1 p 1+cos βq α Y c pt tan β cos α U tan α + tan β T p J 4 P cos α T c q p cos α Y cos β J 5 J 4 tan α J 6 1 c P sin α T q p sin α cos α Y P T 1+cos βq p Y { } κ l J l W dx + H l W + dy, 540 Γ τ p sin α A + cp q T α Y sin α 1+cos β + c c pt tan β cos α sin α cos β cos β pa sin α + cp q T α Y sin α + p sin α A cp q T α Y sin α 1+cos β + c c sin α tan β tan β sin α cos α T p Using 513 and 51, the above expression can be simplified as J 1 X ct p + c pt tan β sin α J 1 c Ap pt sin α J 3 U tan α + tan β T p J 4 P cos α p sin α A + c T p sin α 541 J 5 1 P sin α pa sin α + c c T p sin α J 6 P sin α + p sin α A In a similar way, we obtain H 1 X + ct q H 1 c Bq qt sin β H 3 U tan α + tan β T q H 4 Q cos β q sin β B + c T q sin β H 5 1 Q sin β qb sin β + c c T q sin β H 6 Q sin β + q sin β B 30 54

31 It is clear that the integrands J l, H l are smooth, for l 1,, 4, 5, 6 We claim that the integrands J 3 and H 3 are continuous as well Indeed, using 535 we obtain U tan α + tan β T cv + Rw Sz c c R Sv + cv x + wr x + Rw x zs x Sz x r s + wr x + Rw x zs x Sz x dx r + Rw x c w zs dx s + zs 8c x c w zr dx 8c + wr x + c R Sw z dx Sz x + c R Sw z dx + c w zs R dx 8c The three terms on the right hand side correspond to the integrands in I, I 4 and I 1, respectively Hence they are continuous dx 6 Length of piecewise regular paths Let γ : θ u θ 0, uθ 1 be a piecewise regular path of initial data According to Definition there exists a smooth path of solutions of , say θ x θ, t θ, u θ, α θ, β θ, p θ, q θ X, Y, such that 55 holds for every θ [0, 1] At time t 0, an upper bound on the length of this path can be computed as follows For each θ [0, 1], consider the curve in the X-Y plane Γ θ 0 { X, Y ; t θ X, Y 0 The norm of the tangent vector is then determined by 540 Integrating wrt θ we obtain 1 6 { } κ l Jl θ W dx + Hl θ W+ dy dθ 61 0 Γ θ 0 l1 We recall that there exist infinitely many paths of solutions of which yields the same path of solutions to 1 Indeed, as shown in Remark 3, at time t 0 for each θ one can choose smooth, increasing functions φ θ, ψ θ smoothly depending on θ, and define the solutions x θ, t θ, ũ θ, α θ, β θ, p θ, q θ X, Ỹ as in On the other hand, different relabelings of the X, Y coordinates determine different values for the integral in 61 Indeed, these correspond to different choices of the shifts w, z in 313 To illustrate this point more clearly, fix a value of θ Then, for ε > 0 small, the family of solutions u θ+ε can be regarded as perturbations of the solution u θ At a given point τ, x, the shifts wτ, x and zτ, x are uniquely determined as follows Fig 7 Let X 0, Y 0 be the point in the X-Y plane such that x θ X 0, Y 0 x, t θ X 0, Y 0 τ For each ε > 0, define X ε and Y ε implicitly by setting } t θ+ε X 0, Y ε τ, t θ+ε X ε, Y 0 τ 31

32 The shifts are then uniquely defined by setting wτ, x, x θ+ε X 0, Y ε x θ X 0, Y 0 x θ+ε X ε, Y 0 x θ X 0, Y 0 lim, zτ, x lim ε 0 ε ε 0 ε 6 Y Yε Y 0 Γ θ+ε τ X 0 X ε θ Γ τ X Figure 7: Given a representation of the solutions u θ in terms of the variables X, Y, the shifts w, z are uniquely determined by 6 Here Γ θ τ {X, Y ; t θ X, Y τ} The above considerations lead to Definition 3 The length γ of the piecewise regular path γ : θ u θ 0, uθ 1 is defined as the infimum of the expressions in 61, taken over all piecewise smooth relabelings of the X-Y coordinates Based on the analysis in Section 3, we now give an estimate on how the length of a regular path can grow in time Theorem 5 Given any K, T > 0, there exist constants κ 1,, κ 6 in 61 and C K,T > 0 such that the following holds Consider a path of solutions θ u θ, u θ t of 11, which is piecewise regular for t [0, T ] and where each u θ has total energy K Then its length satisfies the estimates γ τ C K,T γ 0 for all 0 τ T 63 Proof 1 To fix the ideas, let u θ be structurally stable for every θ [0, 1] \ {θ 1,, θ N } Fix ε > 0 and choose a relabeling of the variables X, Y such that, at time t 0, 1 6 { } κ l Jl θ W dx + Hl θ W+ dy dθ γ 0 + ε 64 0 Γ θ 0 l1 Since the solution u is smooth in the X-Y variables and piecewise smooth in the x-t variables, the existence of the tangent vector is clear, for every θ [0, 1] and t [0, T ] We claim that, 3

33 for every θ / {θ 1,, θ N }, an estimate such as 3 holds Namely v θ τ, r θ τ, s θ τ u θ τ,r θ τ,s θ τ { τ } exp C 0 τ + a θ sds 0 65 v θ 0, r θ 0, s θ 0 u θ 0,R θ 0,S θ 0 Here the constant C 0 and the integral of a θ depend only on T and on an upper bound on the total energy Integrating 65 over the interval θ [0, 1], one obtains an estimate of the form γ τ C γ 0 + ε for all 0 τ T This proves 63, because ε > 0 was arbitrary It now remains to prove the estimate 65 We observe that, if u θ were smooth for all x, t R [0, τ], the result follows directly from 35, proved by the computations in Section 4 We need to show that the same conclusion can be reached if u θ is piecewise smooth, with structurally stable singularities α π Y ε Yε Γ τ+ε Γτ ε X ε X ε Figure 8: Proving that the rate of change in the length of a tangent vector is not affected by the presence of a singularity Fix a time τ and call Γ τ {t θ X, Y τ} the level set in the X-Y plane Since the estimates of the previous section hold in regions where u θ is smooth, to obtain a bound on the weighted norm of the tangent vector it suffices to show that the effect of isolated singularities is negligible To lighten the notation, in the following the superscript θ will be omitted With reference to Fig 8, assume that the solution has a structurally stable singularity along a backward characteristic We claim that this singularity does not affect the estimate 35 In other words, the time derivative d dt 6 l1 is not affected by the presence of the singularity { } κ l J l W dx + H l W + dy Γ t 33

34 Y α π Γ t3 P 3 Γ t t t 3 β π P 0 t t X Figure 9: Here P is a singularity point of Type, where α π and α X 0, but α XX 0 and β π At P 3 the solution has a singularity of Type 3, where α β π, but α X 0 and β Y 0 The weighted norm of the tangent vector is continuous at the times t t and t t 3 For a given time τ, let X ε, Y ε be the point where the curve Γ τ ε {tx, Y τ ε} intersects the singular curve {αx, Y π} Similarly, let X ε, Y ε be the point where the curve Γ τ+ε {tx, Y τ + ε} intersects the singular curve {αx, Y π} Define the curves { σ + ε Γτ+ε {X [X ε, X ε ]}, Γ τ ε {x [X ε, X ε ]}, σ ε { η + ε Γτ+ε {Y [Y ε, Y ε ]}, Γ τ ε {Y [Y ε, Y ε ]} η ε To prove our claim, it suffices to show that lim ε 0 lim ε 0 1 ε σ ε + 1 ε η ε + 6 J l W dx 0, 66 σε l1 6 H l W + dy 0 67 ηε l1 The first limit holds because the integrand is a continuous function of X, Y and X ε X ε Oε The second limit holds because the integrand is a continuous function of X, Y and Y ε Y ε Oε The basic estimate 35 thus remains valid also in the presence of singular curves where α π or β π Finally, we analyze what happens in the presence of singular points of Type, where α π and α x 0, and of Type 3, where α β π Since the solution u θ is structurally stable, there can be at most finitely many such points, say Q j X j, Y j, j 1,, N To complete the proof of our claim, it thus suffices to show that, at each time τ j tx j, Y j, the map 1 6 { } t κ l Jl θ W dx + Hl θ W+ dy 68 0 Γ t l1 34

35 is continuous at t τ j But this is clear, because the path Γ t depends continuously on t and the integrands J l, H l are uniformly bounded Moreover, they are continuous everywhere with a possible exception of the finitely many singular points Q j 7 Construction of the geodesic distance A key result proved in [7] shows that every path of solutions to 11 can be approximated by a path which remains regular for t [0, T ] More precisely, an application of Thom s transversality theorem yields Theorem 6 Let the wave speed cu satisfy the assumptions A Let u θ, α θ, β θ, p θ, q θ, x θ, t θ X, Y be a path of C solutions to the semilinear system , depending smoothly on θ [0, 1] Then, for any T, ε > 0 and any integer k 1, there exists a perturbed path of solutions ũ θ, α θ, β θ, p θ, q θ, x θ, t θ X, Y such that u θ ũ θ, α θ α θ, β θ β θ, p θ p θ, q θ q θ, x θ x θ, t θ t θ < ε 71 C k Ω Here Ω R is a domain containing the set { } X, Y ; t θ X, Y [0, T ] or t θ X, Y [0, T ], for some θ [0, 1] Moreover, all except finitely many solutions ũ θ, α θ, β θ, p θ, q θ, x θ, t θ have structurally stable singularities inside Ω In other words, by slightly perturbing the initial data u θ 0, uθ 1, θ [0, 1], we can construct a one-parameter family of conservative solutions u θ u θ t, x which have structurally stable singularities, for all but finitely many values of θ This implies that for all t [0, T ] the length of the path θ u θ t, is well defined by the formula γ t 1 d dθ uθ t dθ 7 0 u θ t Here u is a weighted norm defined as in , or equivalently at 540 A geodesic distance d on the space H 1 R L R will be constructed in two steps i As proved in [7], there is an open dense set of initial data D C 3 R H 1 R C R L R, 73 such that, if u 0, u 1 D, then the solution of 1- has structurally stable singularities On D C c D we construct a geodesic distance, defined as the infimum among the weighted lengths of all piecewise regular paths connecting two given points ii By continuity, this distance can then be extended from D to a larger space, defined as the completion of D wrt the distance d In particular, this completion will contain the space H 1 W 1,1 L L 1 35