Shock interactions for the Burgers-Hilbert equation

Abstract This paper provides an asymptotic description of a solution to the Burgers-Hilbert equation in a neighborhood of a point where two shocks interact. The solution is obtained as the sum of a function with H 2 regularity away from the shocks plus a corrector term having an asymptotic behavior like close to each shock. A key step in the analysis is the construction of piecewise smooth solutions with a single shock for a general class of initial data.


Introduction
Consider the balance law obtained from Burgers' equation by adding the Hilbert transform as a source term 1 π |y|>ε f (x − y) y dy denotes the Hilbert transform of a function f ∈ L 2 (R). It is well known [8] that H is a linear isometry from L 2 (R) onto itself. Given any initial data u(0, ·) =ū(·) (1.2) withū ∈ H 2 (R), the local existence and uniqueness of solutions to (1.1) was proved in [6], together with a sharp estimate on the time interval where this solution remains smooth. For a general initial dataū ∈ L 2 (R), the global existence of entropy weak solutions to (1.1) was proved in [3], together with a partial uniqueness result. We remark that the well-posedness of the Cauchy problem for (1.1) remains a largely open question.
More recently, piecewise continuous solutions with a single shock have been constructed in [4]. As shown in Fig. 1, these solutions have the form u(t, x) = w t, x − y(t) + ϕ x − y(t) , (1.3) where y(t) denotes the location of the shock at time t, and w(t, ·) ∈ H 2 ] − ∞, 0[ ∪ ]0, +∞[ for all t ≥ 0. Moreover, ϕ is a fixed function with compact support, describing the asymptotic behavior of the solution near the shock. It is smooth outside the origin and satisfies ϕ(x) = 2 π |x| ln |x| for |x| ≤ 1 .
( 1.4) Remarkably, this "corrector term" ϕ is universal, i.e., it does not depend on the particular solution of (1.1). The same analysis applies to solutions with finitely many, noninteracting shocks. In addition, the local asymptotic behavior of a solution up to the time when a new shock is formed was investigated in [9]. The aim of the present note is to describe the asymptotic behavior of a solution in a neighborhood of a point where two shocks interact. Calling T > 0 the time when the interaction takes place, our analysis splits into two parts. We first describe the behavior of the solution as t → T −, i.e. as the two shocks approach each other. In a second step, to construct the solution for t > T , we solve a Cauchy problem with initial data given at t = T .
As it turns out, the profile u(T, ·) is not "well prepared", in the sense that it cannot be written in the form (1.3). To explain the difficulty, we recall that the solutions constructed in [4] had initial data of the form u(0, x) = w(x − y 0 ) + ϕ(x − y 0 ), (1.5) for some w ∈ H 2 R \ {0} and y 0 ∈ R. These data are "well prepared", in the sense that they already contain the corrector term ϕ. A natural class of initial data, not considered in [4], is (1. 6) By assumption, at time t = 0 the derivative u x (0, x) = w x (x − y 0 ) is piecewise continuous and uniformly bounded. However, in the solution to (1.1), (1.6), at each time t > 0 we expect that u x (t, x) → ±∞ as x → y(t)∓. For this reason, the local construction of this solution requires a careful analysis. A more general class of initial data, containing both (1.5) and (1.6), as well as all profiles u(T, ·) emerging from our shock interactions, will be studied in Section 2.
We recall here the definition of entropy weak solutions used in [3]. (i) The map t → u(t, ·) is continuous with values in L 2 (R) and satisfies the initial condition (1.2).
(ii) For any k ∈ R and every nonnegative test function φ ∈ C 1 c (]0, ∞[ ×R) one has The present paper will be concerned with a more regular class of solutions, which are piecewise continuous and can be determined by integrating along characteristics. These correspond to the "broad solutions" considered in [2,7]. Throughout the sequel, the upper dot denotes a derivative w.r.t. time.
Definition 1.2 An entropy weak solution u = u(t, x) of (1.1)-(1.2), defined on the interval t ∈ [0, T ], will be called a piecewise regular solution if there exist finitely many shock curves y 1 (t), . . . , y n (t) such that the following holds.
(ii) For each i = 1, . . . , n, the Rankine-Hugoniot conditions hold: . (1.9) (iii) Along every characteristic curve t → x(t) such thaṫ x(t) = u t, x(t) , (1.10) one has d dt u t, x(t) = H[u](x(t)). (1.11) In the above setting, the Hilbert transform of the piecewise regular function u(t, ·) can be computed using an integration by parts: (1.12) The remainder of the paper is organized as follows. In Section 2 we state a local existence and uniqueness theorem for solutions to (1.1), valid for a class of initial data containing one single shock, but more general than in [4]. Towards the proof of Theorem 2.1, Section 3 develops various a priori estimates, while in Section 4 the local solution is constructed as a limit of a convergent sequence of approximations. As in [4], these are obtained by iteratively solving a sequence of linearized problems.
In the second part of the paper we study solutions of (1.1) with two shocks, up to the time of interaction. In Section 5 we perform some preliminary computations, motivating a particular form of the corrector term. In Section 6 we state and prove the second main result of the paper, Theorem 6.1, providing a detailed description of solutions up to the interaction time. This is achieved by a change of both time and space coordinates, so that the two shocks are located at the two points x 1 (t) = t < 0 = x 2 (t), and interact at time t = 0. Our analysis shows that, at the interaction time, the solution profile contains a single shock and lies within the class of initial data covered by Theorem 2.1. Combining our two theorems, one thus obtains a complete description of the solution to (1.1) in a neighborhood of the interaction time.
2 Solutions with one shock and general initial data Consider a piecewise regular solution of the Burgers-Hilbert equation (1.1), with one single shock. By the Rankine-Hugoniot condition, the location y(t) of the shock at time t satisfieṡ As in [4], we shift the space coordinate, replacing x with x−y(t), so that in the new coordinate system the shock is always located at the origin. In these new coordinates, (1.1) takes the form In [4], given a "well prepared" initial data (1.5), a unique piecewise smooth entropy solution to (2.2) of the form was constructed. Here w(t, ·) ∈ H 2 R\{0} , while η ∈ C ∞ (R) is an even cut-off function, For future use, it will be convenient to introduce the function Our main goal in this section is to solve the Cauchy problem for (2.2) with initial data for some constants c 1 , c 2 ∈ R. Note that this reduces to (1.5) in the case c 1 = c 2 = 1.
To handle the more general initial data (2.6)-(2.7), we write the solution of (2.2) in the form where the corrector term ϕ (w) (t, x) now depends explicitly on time t and on the strength of the jump To make an appropriate guess for the function ϕ (w) , we observe that, by (1.12), the equation (2.2) can be approximated by the simpler equation Indeed, we expect that the solutions of (2.2) and (2.10) with the same initial data will have the same asymptotic structure near the origin. Their difference will lie in the more regular space H 2 R\{0} . With this in mind, we thus make the ansatz (2.11) Inserting (2.8) into (2.2), we obtain an equation for the remaining component w(t, ·). Namely where a and F are given by We observe that, in the present case of a solution with a single shock, by (2.5) the entropy admissibility condition (1.8) reduces to Moreover, Definition 1.2 is satisfied provided that, along every characteristic curve t → x(t; t 0 , x 0 ) = 0 obtained by solvinġ The first main result of this paper provides the existence and uniqueness of an entropic solution, locally in time. The solution to the equivalent equation (2.12) will be obtained as a limit of a sequence of approximations. Namely, consider a sequence of linear approximations constructed as follows.
As a first step, define By induction, let w n be given. We define w n+1 to be the solution of the linear, nonhomogeneous Cauchy problem The induction argument requires three steps: (i) Existence and uniqueness of solutions to each linear problem (2.19).
(iii) Convergence in a weak norm. This will follow from the bound These steps will be worked out in the next two sections.

Preliminary estimates
To achieve the above steps (i)-(iii), we establish in this section some key estimates on the right hand side of (2.19), by splitting it into three parts: where Consider the function where φ(x, b) is given by (2.4). For every b ∈ 0, 1 2e one checks that the function g b ∈ C ∞ (R\{0})∩C(R) is negative and decreasing on the open interval 0, 1 2e . Moreover, it satisfies The next lemma provides some bounds on the Hilbert transform of g b . As usual, by the Landau symbol O(1) we shall denote a uniformly bounded quantity. (3.6) Moreover, for every δ > 0 sufficiently small one has Proof. Fix b ∈ 0, 1 2e . By (1.12), one has Two cases are considered: and similarly Case 2: If 0 < x < 1 2e , then we split H[g b ](x) into three parts as follows: We first estimate and similarly By a similar argument, one obtains Finally, using the fact that g ′ b is concave, we obtain We thus achieve the same estimates as in Case 1, and this yields (3.6).
Finally, the function g b is continuous with compact support and smooth outside the origin. Therefore, the Hilbert transform H[g b ] is smooth outside the origin. As |x| → ∞, one has Thus, (3.6) yields (3.7).
for all x > 0. Since the same arguments used in the proof of Lemma 3.1 yield that, for 0 < |x| ≤ 1 2e , Moreover, for δ > 0 sufficiently small, The next lemma provides some a priori estimates on the function F = F (t, x, w) introduced at (2.14).
Then there exists a constant C 1 > 0, depending on M 0 , δ 0 , c 1 , and c 2 such that, for a.e. t ∈ [0, T ] and |x| < 1 2e , one has (3.8) Furthermore, for every δ > 0 sufficiently small Proof. According to (3.1), the function F can be decomposed as the sum of three terms, which will be estimated separately.
Our third lemma estimates the change in the function F = F (t, x, w) as w(·) takes different values. These estimates will play a key role in the proof of convergence of the approximations considered at (2.20).
Moreover, assume that σ (w i ) is locally Lipschitz on ]0, T ] and that there exists a function K(t) such that Then there exists a constant C 2 > 0, depending on M 0 , δ 0 , c 1 , and c 2 such that, for every x ∈ − 1 2e , 1 2e and a.e. t ∈ [0, T ], one has Proof. 1. For notational convenience, we set For every − 1 2e < x < 0, by a similar argument, we obtain the same bounds as in (3.21)-(3.23). Therefore (3.24) 3. To achieve bound on A (z) , for 0 < x < 1 2e we compute This yields On the other hand, for 0 < x < 1 2e we observe that Similarly, one gets the same estimate for −

Proof of Theorem 2.1
In this section we give a proof of Theorem 2.1 by constructing a solution to the Cauchy problem (2.2) with general initial data of the form (2.6)-(2.7), locally in time. This solution will be obtained as limit of a Cauchy sequence of approximate solutions w n (t, x), following the steps (i)-(iii) outlined at the end of Section 2.
Step 1. Consider any initial profile w ∈ H 2 (R\{0}). Let δ 0 , M 0 > 0 be the constants defined by the identities Given two constants c 1 , c 2 ∈ R, the corresponding initial data of the form (2.6)-(2.7) is Set σ n (t) . = w n (t, 0−)−w n (t, 0+). As in (2.11), the correction term associated to w n is denoted by In this step, we will establish the existence and uniqueness of solutions to the linear problem (2.19).
We begin by observing that the speed of all characteristics for (2.19) is where ϕ n (t, x) . = ϕ (wn) (t, x), the correction term associated to w n . From (4.3) and (4.2) it follows that ϕ n (t, 0) = 0 and Furthermore, for any given (t, x) ∈ [0, T ]× ]0, 1 2e ], we estimate, using (3.5), Similarly, we also have In particular, setting and denote by t → x(t; t 0 , x 0 ) the solution to the Cauchy probleṁ By (4.5) it follows The next lemma provides the Lipschitz continuous dependence of the characteristic curves considered at (4.7).
Step 2. Consider a sequence of approximate solutions w (k) to (2.19), inductively defined as follows.
The following lemma provides a priori estimates on w (k) , uniformly valid for all k ≥ 1.
Thanks to the above estimates, we can now prove that the sequence of approximations w (k) is Cauchy and converges to a solution w of the linear problem (2.19). This is a key step toward the proof of Theorem 2.1.  The function w provides a solution to the Cauchy problem (2.19) and satisfies Moreover, σ(t) .
We are now ready to complete the proof of our first main result.
Proof of Theorem 2.1. As outlined at the end of Section 2, we construct, by induction, a sequence of approximate solutions (w n ) n≥1 where each w n is the solution to the linear problem (2.19). For some T > 0 small enough, depending only on M 0 , δ 0 , c 1 , and c 2 , we claim that For a fixed n ≥ 2, we define      W n . = w n − w n−1 , a n (t, x) . = a(t, x, w n ), A n (t, x) . = a n (t, x) − a n−1 (t, x), Set Z n = W n + V n . From the above definitions, by (2.19) it follows Recalling the first inequality in (3.19) and (4.15), we estimate for some constant C 7 > 0 depending only on M 0 , δ 0 , c 1 , and c 2 . Hence, choosing T > 0 sufficiently small, we have, using Duhamel's formula, for some constant C 8 > 0. On the other hand, (2.11), (4.21), and (3.25) imply

and (4.23) yields
for some constant C 9 > 0 depending only on M 0 , δ 0 , c 1 , and c 2 . In particular, for T > 0 sufficiently small, one has that Thus, (4.20) holds and for every t ∈ [0, T ] the sequence of approximations w n (t, ·) is Cauchy in the space H 1 R\{0} , and hence it converges to a unique limit w(t, ·).
Finally, to prove uniqueness, assume that w, w are two entropic solutions. We then define The arguments used in the previous steps now yield the inequality which implies β(τ ) = 0 for all τ ∈ [0, T ] and completes the proof.

Two interacting shocks
In this section, denote by u(t, x) the solution to Burgers' equation We expect that v can provide a leading order correction term, in an ansatz describing the solution with two interacting shocks to the Burgers-Hilbert equation (1.1).

Constructing a solution with two interacting shocks
We consider here a solution of the Burgers-Hilbert equation (1.1), which is piecewise continuous and which has two shocks located at the points y 1 (t) < y 2 (t). By the Rankine-Hugoniot conditions, the time derivatives satisfẏ = u i t, y i (t)± denote the left and the right limits of u(t, x) as x → y i (t). Throughout the following, we assume that The function τ . = y 1 − y 2 is negative and monotone increasing. It will be useful to change the space and the time variables, so that in the new variablest,x the location of one shock is fixed, while the other moves with constant speed 1. For this purpose, we set As a consequence, the two shocks, in the new coordinate system, are located at Thus, by (6.1), (6.3), and (6.4), we can recast the original equation (1.1) in the following equivalent form .
yield the speeds of the two shocks in the original coordinates, as shown in Fig. 3.
We shall construct the solution of (6.5) in the form Here ϕ is a continuous function, which satisfies ϕ(t, t) = ϕ(t, 0) = 0, while According to (6.8), the function w(t, ·) is continuously differentiable outside the two points x = t and x = 0. Moreover, the distributional derivative D x w(t, ·) is an L 2 function restricted to each interval ] − ∞, t[ , ]t, 0[ and ]0, +∞[ . However, both w(t, ·) and w x (t, ·) can have a jump at x = t and at x = 0. At the points (t, t) and (t, 0), the following traces are well defined: = w x (t, t+), = w x (t, 0+).

(6.10)
For the shocks to be entropy admissible, the inequalities will always be assumed. Writing , (6.12) the equation (6.5) reads w t + a(t, x, w) · w x = F (t, x, w), (6.13) where a and F are given by (6.14) , (6.15) respectively. Here the function ϕ (w) (t, x) is chosen in such a way that a cancellation between leading order terms near to the location of the two shocks at x = t and at x = 0 is achieved. More precisely, in view of (5.14) and recalling (2.3) and (2.4), we set φ 0 (x) = φ(x, 0) = 2η(x) π · |x| ln |x|, (6.16) and define (6.17) The following theorem provides the existence of solutions to the Cauchy problem for (1.1) where the initial datum contains two shocks. In particular, the solution to (6.5) is constructed up to the time where the two shocks interact. Furthermore, the solution is of the form (6.7), where ϕ = ϕ (w) , the corrector function defined in (6.17).
Theorem 6.1 For any given constants b, M 0 , δ 1 , δ 2 > 0, there exists ε 0 > 0 small enough and a constant K such that the following holds.
Consider any τ 0 ∈ [−ε 0 , 0[ and any initial condition Then the Cauchy problem (6.13) with initial data admits a unique entropic solution, defined for t ∈ [τ 0 , 0]. Moreover, this solution satisfies for all t ∈ [τ 0 , 0[. Toward a proof of Theorem 6.1, solutions to (6.13) will be constructed by an iteration procedure. The main difference between this and the earlier case with a single shock is that the correction term ϕ now depends on time through the variable strengths σ 1 , σ 2 of the two shocks. Define By induction, let w (n) be given and satisfy (6.8) for every t ∈ [τ 0 , 0[ . Moreover, call σ (n) 1 (t) and σ (n) 2 (t) the strengths of the two shocks at x = t and x = 0 of w (n) , respectively. We construct the next iterate w = w (n+1) (t, x) by solving the linear equation F (t, x, w), (6.22) with initial data (6.19) and a as introduced in (6.14).
The induction argument requires the following steps: (i) Given w (n) , the equation (6.22) with the initial data w admits a unique solution w with (iii) Convergence in a weak norm. This will follow from the bound n≥1 w (n+1) (t, ·) − w (n) (t, ·) H 2 R\{t,0} < + ∞.
The next lemma estimates the change in the function F = F (t, x, w) as w(·) takes different values. These estimates will play a key role in the proof of convergence of the approximations inductively defined by (6.22).
Step 2. Let us now consider a sequence of approximate solutions w (k) to (6.22) inductively defined as follows.
Thanks to the above estimates, we can now prove that the sequence of approximations w (k) is Cauchy, and converges to a solution w of the linear problem (6.22). This will accomplish the inductive step, toward the proof of Theorem 6.1.