Piecewise Smooth Solutions to the BurgersHilbert Equation


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1 Piecewise Smooth Solutions to the BurgersHilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA s: zhang December 7, 5 Abstract The paper is concerned with the BurgersHilbert equation u t + u / H[u], where the right hand side is a Hilbert transform Unique entropy admissible solutions are constructed, locally in time, having a single shock In a neighborhood of the shock curve, a detailed description of the solution is provided Introduction Consider the balance law obtained from Burgers equation by adding the Hilbert transform as a source term: u u t + H[u] Here H[f] lim ε + y >ε f y y dy denotes the Hilbert transform of a function f L IR The above equation was derived in [] as a model for nonlinear waves with constant frequency For initial data u, ū, 3 in H IR, the local eistence and uniqueness of the solution to was proved in [7], together with a sharp estimate on the time interval where this solution remains regular See also [8] for a shorter proof For general initial data ū L IR, the global eistence of entropy weak solutions was recently proved in [4] together with a partial uniqueness result We remark that, in this general setting, the wellposedness of the Cauchy problem remains a largely open question In the present paper we consider an intermediate situation Namely, we construct solutions of which are piecewise continuous, with a single shock Our solutions have the form ut, ϕ yt + w t, yt,
2 where t yt denotes the location of the shock Here w H ], [ ], + [, while ϕ ln, for near the origin In Section we write in an equivalent form, and state an eistenceuniqueness theorem, locally in time The key a priori estimates on approimate solutions, and a proof of the main theorem, are then worked out in Sections 3 to 5 The present results can be easily etended to the case of solutions with finitely many, noninteracting shocks An interesting open problem is to describe the local behavior of a solution in a neighborhood of a point t, where either i a new shock is formed, or ii two shocks merge into a single one Motivated by the analysis in [] we conjecture that, for generic initial data ū H IR C 3 IR, the corresponding solution of remains piecewise smooth with finitely many shock curves on any domain of the form [, T ] IR We thus regard the present results as a first step toward a description of all generic singularities For other eamples of hyperbolic equations where generic singularities have been studied we refer to [, 3, 5, 6, 9] The possible emergence of singularities, for more general dispersive perturbations of Burgers equation, has been recently studied in [] Statement of the main result Consider a piecewise smooth solution of with one single shock Calling yt the location of the shock at time t, by the RankineHugoniot conditions we have ẏt u t + u + t where u, u + denote the left and right limits of ut, as yt Here and in the sequel, the upper dot denotes a derivative wrt time It is convenient to shift the space coordinate, replacing with yt, so that in the new coordinate system the shock is always located at the origin In these new coordinates, the equation takes the equivalent form u u t + ẏ u H[u] We shall construct solutions to in a special form, providing a cancellation between leading order terms in the transport equation and the Hilbert transform Consider a smooth function with compact support η Cc IR, with η η, and such that η if, Moreover, define η if, η if [, ] ϕ ln 3 η 4
3 Notice that ϕ has support contained in the interval [, ] and is smooth separately on the domains { < } and { > } In addition, we consider the space of functions H H ], [ ], + [ 5 Every function w H is continuously differentiable outside the origin The distributional derivative of w is an L function restricted to the half lines ], [ and ], + [ However, both w and w can have a jump at the origin It is clear that the traces { u w, u + w+, are continuous linear functionals on H u _ { b w, b + w +, 6 + u u ϕ + w w ϕ Figure : Decomposing a piecewise regular function u ϕ + w as a sum of the function ϕ defined at 4 and a function w H IR \ {}, continuously differentiable outside the origin Solutions of will be constructed in the form ut, ϕ + wt, 7 In order that the shock be entropy admissible, the function w should range in the open domain D { w H IR \ {} } ; w > w+ 8 By 68, for this solution has the asymptotic behavior u t + b ln t + + O 3/ if <, ut, u + t + b + t + ln + O 3/ if >, 9 for suitable functions u ±, b ± Here and throughout the sequel, the Landau symbol O denotes a uniformly bounded quantity Inserting 7 in the equation and recalling 6, one obtains w t + ϕ + w u + u + ϕ + w H[ϕ] + H[w] To derive estimates on the Hilbert transform, the following observation is useful Consider a function f with compact support, continuously differentiable for < and for >, with a 3
4 jump at the origin Then, for any, an integration by parts yields H[f] f y ln y dy + [ f+ f ] ln A similar computation shows that, to leading order, the Hilbert transform of w near the origin is given by H[w] u+ u ln + O, with u, u + as in 6 On the other hand, for one has w + w+ ϕ + w ϕ sign u+ u u+ u ln + O + O ln sign + ln 3 The identity between the leading terms in and 3 achieves a crucial cancellation between the two sides of It is thus convenient to write this equation in the equivalent form w t + ϕ + w u + u + w H[ϕ] ϕϕ + H[w] w u + u + ϕ 4 Definition By an entropic solution to the Cauchy problem with initial data we mean a function w : [, T ] IR IR such that w, w D, 5 i For every t [, T ], the norm wt, H IR\{} remains uniformly bounded As, the limits satisfy u t ut, > ut, + u + t 6 Indeed, if f Cc IR, then for a suitably large constant M we have f y f + y H[f] lim dy lim dy ε + y >ε y ε + y >ε y ε lim + ε + M ε lim + ε + M M ε M ε f + y f dy y f + y dy lim [f ε f] ln ε ε + + lim ε +[f + ε f] ln ε + [f M f] ln M [f + M f] ln M f + y dy f y ln y dy By approimating f with a sequence of smooth functions with compact support we obtain 4
5 ii The equation 4 is satisfied in integral sense Namely, for every t and, calling t t; t, the solution to the Cauchy problem one has with ẋ ϕ + wt, u t + u + t, t, 7 t wt, w; t, + F t, t; t, dt, 8 F H[ϕ] ϕϕ + H[w] w u + u + ϕ 9 A few remarks are in order: i The bound on the norm wt, H implies that the limits in 6 are well defined By requiring that the inequality in 6 holds we make sure that the shock is entropy admissible ii Since wt, H IR \ {}, the right hand side of the ODE in 7 is continuously differentiable wrt Combined with the inequalities in 6, this implies that the backward characteristic t t; t, is well defined for all t [, t ] iii In [], a function satisfying the integral equations 8 was called a broad solution The regularity assumption on wt, and the fact that the source term F in 9 is continuous outside the origin imply that w wt, is continuously differentiable wrt both variables t,, for Therefore, the identity in 4 is satisfied at every point t,, with The main result of this paper provides the eistence and uniqueness of an entropic solution, locally in time Theorem For every w D there eists T > such that the Cauchy problem, 5 admits a unique entropic solution, defined for t [, T ] In turn, Theorem yields the eistence of a piecewise regular solution to the BurgersHilbert equation, locally in time, for initial data of the form with w D u, ϕ + w, The solution w wt, of 4 will be obtained as a limit of a sequence of approimations More precisely, for n, we define w t, w for all t 5
6 Net, let the nth approimation w n t, be constructed By induction, we then define w n+ t, to be the solution of the linear, nonhomogeneous Cauchy problem w t + ϕ + w n u n + u + n w H[ϕ] ϕϕ + H[w] w u + u + ϕ with initial data 5 The induction argument requires three steps: i Eistence and uniqueness of solutions to the linear problem with initial data 5 ii A priori bounds on the strong norm w n t H IR\{}, uniformly valid for t [, T ] and all n iii Convergence in a weak norm This will follow from the bound w n+ t w n t H IR\{} < n In the following sections we shall provide estimates on each term on the right hand side of, and complete the above steps i iii 3 Estimates on the source terms To estimate the right hand side of, we consider again the cutoff function η in 3 and split an arbitrary function w H IR \ {} as a sum: where v w v + v + v 3, 3 { w η if <, v { w η if <, w+ η if >, w + η if >, 3 v 3 w v v 33 The right hand side of can be epressed as the sum of the following terms: A H[ϕ], B ϕϕ, C H[v +v 3 ], D H[v ] w u + u + ϕ 34 The goal of this section is to provide a priori bounds of the size of these source terms and on their first and second derivatives Lemma There eists constants K, K such that the following holds For any δ ], /] and any w H IR \ {}, the source terms in 34 satisfy A H IR\[ δ,δ] + B H IR\[ δ,δ] K δ /3, 35 6
7 C H IR\[ δ,δ] + D H IR\[ δ,δ] K δ /3 w H IR\{} 36 Proof We begin by observing that the function ϕ is continuous with compact support, smooth outside the origin Therefore, the Hilbert transform A H[ϕ] is smooth outside the origin As one clearly has A O, A O, A O 3 37 In addition, as, we claim that A O ln, A O ln, A O Indeed, to fi the ideas, let < < / By we have ln 38 H[ϕ] ϕ y ln y dy I + I + I 3, 39 where: I and moreover, I + ln y dy + ln y dy ϕ y ln y dy O, 3 + dy O ln, 3 I 3 / / ln y dy + ln y dy I 3 + I 3 + I / ln y dy + ln y dy ln y ln y + dy / 3 We now have I 3 ln I 3 ln / / dy O ln, ln y dy O ln, 33 I 33 ln ln + dy O 3 Hence H[ϕ] O ln This yields the first estimate in 38 7
8 Net, we estimate the derivative H[ϕ] I + I + I 3 The term I is uniformly bounded, while I y dy + y dy dy O ln 34 y Differentiating I 3 wrt we obtain / I 3 + / ɛ + lim + ɛ / / y dy + + 3/ 3/ +ɛ ln y y dy y dy 35 Assuming < < /, we obtain / y dy / y dy O ln, / y dy / dy O ln, / 3/ y dy y dy / ln 3 / The remaining term is estimated as ɛ 3/ ln y ɛ + y dy + / ɛ / 3/ dy O ln, 3/ ɛ y dy O ln ln y ln y dy ɛ Combining the previous estimates we obtain H[ϕ] O ln This gives the second estimate in 38 Finally, we estimate the second derivative of the Hilbert transform H[ϕ] By 3 and 34 we obtain I 3 I / + ln / I y dy + + / + 3 ln 3/ O, 3 I i i ln dy O y 36 / y dy + 3/ + 3/ / y dy y dy 37 8
9 Assuming < < /, we obtain / / / 3/ y dy ln y dy y dy / / y dy 3 ln The remaining term is estimated by 3/ y where / / y y dy / / / / / / ln dy O y, dy O ln, ln dy O /, 3/ dy / ln dy O y / ln y y dy ln / ln 3/ dy +, y y y y dy / / Therefore, by 36 and 38 3, we have H[ϕ] O y dy ln The function B ϕϕ is smooth outside the origin and vanishes for As, the following estimates are straightforward: B O ln, B O ln, B O ln 3 3 Net, we observe that v 3 H IR Moreover, there eists a constant C η such that v 3 H IR C η w H IR\{} Clearly, the Hilbert transform H[v 3 ] satisfies the same bounds Hence H[v3 ] H IR O w H IR\{} 3 We observe that v is Lipschitz continuous, has compact support and is continuously differentiable outside the origin Since v has better regularity properties than ϕ, the same arguments used to estimate the Hilbert transform of ϕ also apply to H[v ] More precisely, as in 37 for we have H[v ] O, H[v ] O, H[v ] O
10 As in 38, for we have H[v ] O ln, H[v ] O ln ln, H[v ] O 34 The only difference is that in by O we now denote a quantity such that for some constant C independent of w O C w H IR\{}, 35 4 Finally, observing that the the function v in 3 has compact support, for we have the bounds D H[v ] O D O, D O 3 36 On the other hand, for we claim that D O, D O ln, D O, 37 where O is a quantity satisfying 35 Indeed, without loss of generality we can assume < < / Recalling the construction of w and ϕ, we have w u + u + ϕ The Hilbert transform of v is computed by H[v ] + + v y y dy v y y dy + u+ u ln u / y dy + + 3/ + O 38 u + 3/ y dy + / u + y dy The first term on the right hand side is bounded and the last term vanishes, in the principal value sense The second term is computed by u y dy u ln + ln + u ln + O, while the remaining integrals are estimated by / u + + 3/ y dy u+ ln ln u + ln + O Combining the previous estimates we obtain H[v ] u+ u ln + O 39 Net, we estimate the derivative D We have w u+ + u ϕ O ln, w u+ + u ϕ u+ u +O 33
11 To estimate the derivative of H[v ] we write H[v ] + + / v y y dy + 3/ u y dy v y 3/ y dy + v y / y dy 33 The first term on the right hand side of 33 is uniformly bounded The second term is estimated by u y dy u + O Furthermore, we have / + 3/ v y y dy / 3u+ Lastly, since v u + for ], ], we have 3/ / v y y dy Combining the previous estimates we thus obtain + 3/ + O + 4u+ / / u + y H[v ] u+ u + O v y 4u+ dy + y u+ + O 33 dy 333 Together with 33, as this yields the asymptotic estimate [ D H[v ] w u + u + ] ϕ O ln 334 The second derivative D is estimated in a similar way Indeed, by 3 33 and 33, we have w u + u + ϕ w u+ + u ϕ + w u+ + u ϕ + w u + u + ϕ ϕ + w u + u + ϕ ϕ 335 u+ u + O On the other hand, differentiating 33 and recalling 33 and 333 we have v y H[v ] + y 3 dy + u y 3 dy + / + 3/ v y / 4u+ dy y + / u + y dy 336
12 As before, the first term is uniformly bounded while the last term is zero The second term is computed by u u dy + O 337 y 3 The third term is estimated by / v y + 3/ y dy / 3u+ + 3/ + O Combining above estimates we obtain [ D H[v ] w u + u + ] ϕ v y u+ dy y 3 + 6u+ u + 3u+ 4u+ + u+ u + O O By the estimates 38, 3 it follows ln / / d A + B H IR\[ δ,δ] O δ d O δ 7/3 O δ 4/3 / O δ /3 34 Similarly, the estimates 36 follow from 3, and Construction of approimate solutions In this section, given an initial datum w D, we prove that all the approimate solutions w n at  are well defined, on a suitably small time interval [, T ] As in 6, we define { ū w, ū + w+, { u n t u + n t w n t,, w n t, + To fi the ideas, assume that the initial data w H IR \ {} satisfies ū ū + 6δ, w H IR\{} M, 4 for some possibly large constants δ, M > Choosing a time interval [, T ] sufficiently small, we claim that for each n the approimate solution w n satisfies the a priori bounds { u n t ū δ, u + n t ū + w δ, n t H IR\{} M, for all t [, T ] 4
13 This will be proved by induction For n these bounds are a trivial consequence of the definition In the following, we assume that the function w n w n t, satisfies 4, and show that the same bounds are satisfied by w n+ We recall that w n+ is defined as the solution to the linear equation, with initial data 5 A sequence of approimate solutions w k to the linear equation will be constructed by induction on k,, For notational convenience we introduce the function at, ϕ + w n t, u n t + u + n t 43 As in 7, call t t; t, the solution to the Cauchy problem ẋ at, t, t 44 We begin by defining w t, w 45 By induction, if w k has been constructed, we then set t w k+ t, w; t, + F k t, t; t, dt, 46 where F k is defined as in 9, with w replaced by w k and u ± t wt, ± replaced by w k t, ±, respectively Assuming that w n satisfies 4, we will show that every approimation w k to the linear Cauchy problem, 5 satisfies the same bounds, on a sufficiently small time interval [, T ] Our first result deals with solution to the linear transport equation 47 We show that, within a sufficiently short time interval, the H norm of the solution can be amplified at most by a factor of 3/ Lemma Let w n w n t, be a function that satisfies the bounds 4 for all t >, and define a at, as in 43 Then there eists T > small enough, depending only on δ, M, so that the following holds For any τ [, T ] and any solution w of the linear equation w t + at, w 47 with initial datum one has w w H IR \ [ δ τ, δ τ], wτ H IR\{} 3 w 48 H IR\[ δ τ, δ τ] Proof The equation 47 can be solved by the method of characteristics, separately on the regions where < and > We observe that characteristics move toward the origin from both sides In this first step we prove that all characteristics starting at time t inside the interval [ δ τ, δ τ] hit the origin before time τ see Fig Hence the profile wτ, does not depend on the values of w on this interval 3
14 We claim that there eists δ > such that { at, δ for all ], δ ], at, δ for all [ δ, [ 49 Indeed, 4 and 4 imply Moreover, for > we have at, at, + at, + u+ n t u n t ln + δ 4 w n, t, y dy C /, 4 for some constant C depending only on the norm w n t, H, hence only on M in 4 Choosing δ > small enough so that C δ / < δ, from 44 we obtain the first inequality in 49 The second inequality is proved in the same way In addition, by choosing the time interval [, T ] small enough, we can also assume δ T δ 4 Multiplying 47 by w one finds w t + aw a w 43 Integrating 43 over the domain Ω { } t, ; > δ τ t, t [, τ] 44 shown in Fig, we obtain w τ, d >δ τ τ w d + a w d dt 45 >δ τ t Indeed, by 49 and 4, for every τ ], T [ the flow points outward along the boundary of the domain Ω By 43 the derivative a satisfies a bound of the form a t, C a + ln, 46 where C a is a constant depending only on the norm w n H in 4 Taking the supremum of a t, over the set Ω t { ; > δ τ t}, 47 from 45 we thus obtain wτ L IR w L Ω + τ By Gronwall s lemma, this yields a bound on wτ L C a + lnδ τ t wt L Ω t dt 48 4
15 T τ Ω τ δ δ Figure : The norm wτ H IR\{} is estimated by using the balance laws for w, w, w on the shaded domain Ω By 49, along the boundary where δ τ t all characteristics move outward Hence no inward flu is present 3 Net, differentiating 47 wrt and multiplying by w we obtain w t + aw a w, w, w 49 w t + aw a w 4 Integrating 4 over the domain Ω in 44 and using the bound 46, by similar computations as before we now obtain τ w τ L IR w L Ω + C a + lnδ τ t w t L Ω t dt 4 By Gronwall s lemma, this yields a bound on w τ L 4 Differentiating 49 once again and multiplying all terms by w we find w t + aw a w a w, w, w, 4 w t + a w 3 a w a w w 43 Integrating 43 over the domain Ω in 44, we obtain τ wτ, d w y dy + 3 a w a w w d dt >δτ >δ τ t To estimate the right hand side of 44 we observe that, for small, 44 a ϕ +w n, O ln + w n H, a ϕ +w n, O + w n, Recalling that ϕ for, we have the bounds 45 E 3 a w + a w w O + ln w + O + w n, w H w, 46 5
16 δ τ t τ / w t, / d δ t s w LΩt w δ t s L Ωt, 47 τ Et, ddt O + ln δ τ t wt H Ω dt t >δ τ t +O τ τ [δ τ t] / wt H Ω dt + O t w n t H wt H Ω dt t 48 5 Calling Zt wt H Ω t, by the estimates 48, 4, and 48 we obtain an integral inequality of the form Z τ Z + C τ + ln δ τ t + [δ τ t] / + M Z t dt 49 By Gronwall s lemma, if τ > is sufficiently small this yields Zτ 3 Z, proving 48 The above estimate can be easily etended to the linear, nonhomogeneous problem w t + at, w F t,, w, w 43 Indeed, in the same setting as Lemma, using 48 and Duhamel s formula, for τ [, T ] we obtain wτ, H IR\{} 3 w 3 + H IR\[ δ τ, δ τ] τ F t, H dt IR\[ δ τ t, δ τ t] 43 Relying on Lemma we now prove uniform H bounds on all approimations w k, on a suitably small time interval [, T ] Lemma 3 Let w n w n t, be a function that satisfies the bounds 4 for all t >, and define a at, as in 43 Then there eists T > small enough, depending only on δ, M in 4, so that the following holds For every k and every τ [, T ], one has w k τ H IR\{} M, 43 w k τ, ū δ, w k τ, + ū + δ 433 Proof Recalling the constants K, K in Lemma, choose T > small enough so that T δ s /3 ds < M 6K + K M 434 6
17 The estimate 43 trivially holds for w τ w Assuming that it holds for w k t, t [, T ], by 43 for any τ [, T ] we have the estimate w k+ τ H IR\{} 3 w H IR\{} M + 3 τ 3 4 M + 3 K + K M < 3 4 M + 3 K + K M K [δ τ t] /3 dt + 3 τ δ s /3 ds By induction, this proves the bound 43 τ τ M 6K + K M M A + B + C + D H IR\[ δ τ t, δ τ t] ds K [δ τ t] /3 w k t H IR\{} dt To prove the two estimates in 433, we write w k+ τ, + ū + w; τ, + ū + + τ sup A + B + C + Dt L 436 t [,τ] The a priori bound on w k t, H IR\{} implies that the L norm in 436 is uniformly bounded By possibly choosing a smaller T >, both terms on the right hand side of 436 will be < δ / This yields the second inequality in 433 The first inequality is proved in the same way The net lemma shows that the sequence of approimations w k defined at converges to a solution to Lemma 4 For some T > sufficiently small, the sequence of approimations w k t, converges in H IR \ {} to a function w wt, The convergence is uniform for t [, T ] This limit function provides a solution to the initial value problem with initial data 5 Proof By the previous bounds, the difference between two approimations can be estimated by w k+ τ w k τ H IR\{} 3 τ If T > is small enough, so that [δ τ t] /3 K w k t w k t H IR\[ δ τ t, δ τ t] dt 3 T δ s /3 K ds, 437 then for every τ [, T ] the sequence w k τ, is Cauchy in H IR \ {}, hence it converges to a unique limit function wτ, It remains to prove that that w provides a solution to with initial data 5, in the sense that the integral identities 8 are satisfied for all t [, T ] and 7
18 This is clear, because for every ɛ > as k the source terms on the right hand side of converge uniformly on the set {t, ; t [, T ], ɛ} 5 Convergence of the approimate solutions By the analysis in the previous section, the sequence of approimate solutions w n of, 5 is well defined, on a suitably small time interval [, T ] Moreover, the uniform bounds 4 hold To complete the proof of Theorem, it remains to show that the w n converge to a limit function w, providing an entropic solution to the Cauchy problem, 5 Toward this goal we prove that on a suitably small time interval [, T ] the sequence w n n is Cauchy wrt the norm of H IR \ {}, hence it converges to a unique limit This will be achieved in several steps For a fied n, consider the differences { { W wn+ w n, U u n+ u n, W n wn w n, u n u n, U n { U + u n+ u + n, u + n u + n From we deduce W t + ϕ + w n u n u + n W + W n U n + U + n w n, H[W ] W U + U + ϕ 5 Multiplying both sides by W we obtain the balance law W [ϕ t + + w n u n u + ] n W ϕ + w n W W n U n + U n + W w n, + H[W ] W W U + U + 5 W ϕ Integrating over the domain Ω in 44 and observing that ϕ O + ln, we obtain τ { W τ, d ϕ + w n W >δ τ t W n U n + U n + W w n, + H[W ] W U + n W U + U + W ϕ } ddt τ { O lnτ t W s L + W n t H W t L + W t L + lnτ t } W t H W t L dt τ C 3 W t L W n t H + lnτ t W t H dt, 53 8
19 for some constant C 3 Net, differentiating 5 wrt we obtain W t + ϕ + w n u n u + n W + ϕ + w n, W + H[W ] W U + U + ϕ ϕ W Multiplying both sides by W we obtain the balance law [ W t + ϕ + w n u n u + ] n W ϕ + w n, W W n U n + U n + w n, + W n, w n, 54 W n U n + U n + W w n, w n, W n, W + H[W ] W W U + U + W ϕ ϕ W By the definition 4 one has 55 ϕ L IR\[ δ τ t,δ τ t] O τ t / 56 Integrating 55 over the domain Ω in 44 we obtain W t, d O τ + W t H W t L τ t / } dt { lnτ t W t L + W n t H W t L 57 3 Calling Zt W t H IR\{}, from 53 and 57 we obtain an integral inequality of the form τ Z τ C 4 Zt W n t H + Zt τ t / dt, 58 for some constant C 4 We now set ε sup W n t H IR\{} t [,T ] Since Z, calling τ the first time where Z ε / one has ε τ C ε 4 ɛ + ε τ t / dt 3 C 4ε τ Hence τ 3C 4 Choosing < T < 3C 4, we thus obtain Zt ɛ This establishes the desired contraction property: for all t [, T ] sup w n+ t w n t H IR\{} t [,T ] sup w n t w n t H IR\{} 59 t [,T ] 9
20 4 By 59, for every t [, T ] the sequence of approimations w n t, is Cauchy in the space H IR \ {}, hence it converges to a unique limit wt, It remains to check that this limit function w is an entropic solution, ie it satisfies the integral equation 8 But this is clear, because for every ɛ > the sequence of functions F n H[ϕ] ϕϕ + H[w n ] w n u n + u + n ϕ 5 converges to the corresponding function F in 9, uniformly for t [, T ] and ɛ 5 Finally, to prove uniqueness, assume that w, w are two entropic solutions Consider the differences W { U u w w, ũ, U + u + ũ +, and call Zt W t H IR\{} Since Z, the same arguments used to prove 58 now yield τ Z τ C 4 Zt [Zt + Zt ] τ t / dt For τ [, T ] sufficiently small, we thus obtain Zτ Theorem This completes the proof of Acknowledgment This research was partially supported by NSF, with grant DMS4786: Hyperbolic Conservation Laws and Applications References [] J Biello and J K Hunter, Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities Comm Pure Appl Math 63 9, [] A Bressan and G Chen, Generic regularity of conservative solutions to a nonlinear wave equation, Ann Inst HPoincaré, Anal Nonlin, submitted [3] A Bressan, T Huang, and F Yu, Structurally stable singularities for a nonlinear wave equation Bull Inst Math Acad Sinica, to appear [4] A Bressan and K Nguyen, Global eistence of weak solutions for the BurgersHilbert equation SIAM J Math Anal 46 4, [5] JG Dubois and JP Dufour, Singularités de solutions d équations au dérivées partielles J Differential Equations 6 985, 74 [6] J Guckenheimer, Catastrophes and partial differential equations Ann Inst Fourier 3 973, 3 59 [7] J K Hunter and M Ifrim, Enhanced life span of smooth solutions of a BurgersHilbert equation SIAM J Math Anal 44, 39 5
21 [8] J K Hunter and M Ifrim, D Tataru, and T K Wong, Long time solutions for a BurgersHilbert equation via a modified energy method Proc Amer Math Soc 43 5, [9] DX Kong, Formation and propagation of singularities for quasilinear hyperbolic systems Trans Amer Math Soc [] F Linares, D Pilod, and JC Saut, Dispersive perturbations of Burgers and hyperbolic equations I : local theory SIAM J Math Anal 46 4, [] B L Rozdestvenskii and N Yanenko, Systems of Quasilinear Equations, AMS Translations of Mathematical Monographs, Vol 55, 983 [] D Schaeffer, A regularity theorem for conservation laws, Adv in Math 973,
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