A Lipschitz metric for the Hunter-Saxton equation

(1)

Full Terms & Conditions of access and use can be found at

https://www.tandfonline.com/action/journalInformation?journalCode=lpde20

Communications in Partial Differential Equations

ISSN: 0360-5302 (Print) 1532-4133 (Online) Journal homepage: https://www.tandfonline.com/loi/lpde20

A Lipschitz metric for the Hunter–Saxton equation

José Antonio Carrillo, Katrin Grunert & Helge Holden

To cite this article: José Antonio Carrillo, Katrin Grunert & Helge Holden (2019) A Lipschitz metric for the Hunter–Saxton equation, Communications in Partial Differential Equations, 44:4, 309-334, DOI: 10.1080/03605302.2018.1547744

To link to this article: https://doi.org/10.1080/03605302.2018.1547744

Francis Group, LLC.

Published online: 15 Feb 2019.

Submit your article to this journal

Article views: 97

View Crossmark data

(2)

A Lipschitz metric for the Hunter–Saxton equation

Jose Antonio Carrillo^a, Katrin Grunert^b, and Helge Holden^b

aDepartment of Mathematics, Imperial College London, South Kensington Campus, London, UK;

bDepartment of Mathematical Sciences, NTNU–Norwegian University of Science and Technology, Trondheim, Norway

ABSTRACT

We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter–Saxton (HS) equation.

Generically, the solutions of the HS equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this article is the construction of a Lipschitz metric that compares two solutions of the HS equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.

ARTICLE HISTORY Received 9 January 2017 Accepted 29 August 2018 KEYWORDS:

Conservative solution;

Hunter–Saxton equation;

Lipschitz metric 2010 MATHEMATICS SUBJECT

CLASSIFICATION Primary: 35Q53; 35B35;

Secondary: 35B60

1. Introduction

In this article, we consider the Cauchy problem for conservative solutions of the (integrated) Hunter–Saxton (HS) Equation [1]

u_tþuu_x ¼1 4

ð_x

1u²_xð Þy dy1 4

ð₁

x

u²_xð Þy dy; uj_t¼0 ¼u₀: (1.1) The equation has been extensively studied, starting with studies of Hunter and Zheng [2,3]. The initial value problem is not well-posed without further constraints: Consider the trivial case u₀¼0 which clearly has as one solutionuðt;xÞ ¼0. However, as can be easily verified, also

u t;ð xÞ ¼ a

4tI _1;^a

8t²

ð Þð Þ þx 2x t I ^a

8t²;^a₈t²

ð Þð Þ þx a 4tI ^a

8t²;1

ð Þð Þ;x (1.2) is a solution for anya0. Here IA is the indicator (characteristic) function of the setA.

Furthermore, it turns out that the solutionu of the HS equation may develop singularities in finite time in the following sense: Unless the initial data is monotone increasing, we find

infð Þ ! 1u_x as t"t¼2=supu⁰₀

: (1.3)

CONTACT Helge Holden helge.holden@ntnu.no Department of Mathematical Sciences, NTNU – Norwegian University of Science and Technology, NO-7491 Trondheim, Norway.

Color versions of one or more of the figures in the article can be found online atwww.tandfonline.com/lpde.

ß2019 The Author(s). Published by Taylor & Francis Group, LLC.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/

licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2019, VOL. 44, NO. 4, 309–334

https://doi.org/10.1080/03605302.2018.1547744

(3)

Past wave breaking there are at least two different classes of solutions, denoted conservative (energy is conserved) and dissipative (where energy is removed locally) solutions, respectively, and this dichotomy is the source of the interesting behavior of solutions of the equation. We will consider in this article the so-called conservative case where an associated energy is preserved.

Zhang and Zheng [4–6] gave the first proof of global solutions of the HS equation on the half-line using Young measures and mollifications with compactly supported initial data. Their proof covered both the conservative case and the dissipative case.

Subsequently, Bressan and Constantin [7] using a clever rewrite of the equation in terms of new variables, showed global existence of conservative solutions without the assumption of compactly supported initial data. The novel variables turned the partial differential equation into a system of linear ordinary differential equations taking values in a Banach space, and where the singularities were removed. A similar, but considerably more complicated, transformation can be used to study the very closely related Camassa–Holm equation, see Bressan and Constantin [8] and Holden and Raynaud [9]. Bressan and Fonte [10]

introduce a metric based on optimal transport for the Camassa–Holm equation. However, the approach is very different from the present, and is based on approximations by multi- peakons. Furthermore, Bressan et al. [11] analyze the more general equationutþfðuÞ_x¼

1 2

Ð_x

0 f⁰⁰ðuÞu²_xdx (the standard HS equation corresponds to fðuÞ ¼u²=2), and derive an explicit solution [11, Eqs. (32) and (33)] that is equivalent to (1.8). The convergence of a numerical method to compute the solution of the HS equation can be found in [12].

We note in passing that the original form of the HS equation is u_tþuu_x

ð Þ_x¼1 2u²_x;

and like most other researchers working on the HS equation, we prefer to work with an integrated version. However, in addition to (1.1), one may study, for instance,

utþuux¼1 2

ð_x

0

u²_xð Þy dy;

and while the properties are mostly the same, the explicit solutions differ.

Our aim here is to determine a Lipschitz metric d that compares two solutions u₁ðtÞ;u₂ðtÞat timetwith the corresponding initial data, i.e.,

d uð ₁ð Þt ;u₂ð Þt Þ C tð Þd uð ₁ð Þ0 ;u₂ð Þ0 Þ;

where C(t) denotes some increasing function of time. The existence of such a metric is clearly intrinsically connected with the uniqueness question, and as we could see from the example where (1.2) as well as the trivial solution both satisfy the equation, this is not a trivial matter. Unfortunately, none of the standard norms in H^s or L^p will work. A Lipschitz metric was derived by Bressan et al. [13], and we here offer an alternative metric that also provides a simpler and more efficient way to solve the initial value problem.

Let us be now more precise about the notion of solution. We consider the Cauchy problem for the integrated and augmented HS equation, which, in the conservative case, is given by

u_tþuu_x¼1 4

ð_x

1dl1 4

ð₁

x

dl; (1.4a)

(4)

l_tþð Þul _x¼0: (1.4b) In order to study conservative solution, the HS Eq. (1.4a) is augmented by the secondEq.

(1.4b) that keeps track of the energy. A short computation reveals that if the solution u is smooth and l¼u²_x, then Eq. (1.4b) is clearly satisfied. In particular, it shows that the energy lðt;RÞ ¼lð0;RÞ is constant in time. However, the challenge is to treat the case without this regularity, and the proper way to do that is to let l be a non-negative and finite Radon measure. When there is a blow-up in the spatial derivative of the solution (cf.

(1.3)), energy is transferred from the absolutely continuous part of the measure to the singular part, and, after the blow-up, the energy is transferred back to the absolutely continuous part of the measure. Thus, we will consider the solution space consisting of all pairs ðu;lÞ such that

u t;ð Þ 2L¹ð ÞR ; uxðt;Þ 2L²ð ÞR ; lðt;Þ 2 Mþð ÞR and dl_ac¼u²_xdx;

where MþðRÞ denotes the set of all non-negative and finite Radon measures onR.

We would like to identify a natural Lipschitz metric, which measures the distance between pairs ðu_i;l_iÞ,i¼1, 2, of solutions. The Lipschitz metric constructed by Bressan et al. [13] (and extended to the two-component HS equation by Nordli [14,15]) is based on the reformulation of the HS equation in Lagrangian coordinates which at the same time linearizes the equation. However, there is an intrinsic non-uniqueness in Lagrangian coordinates as there are several distinct ways to parametrize the particle trajectories for one and the same solution in the original, or Eulerian, coordinates. This has to be accounted for when one measures the distance between solutions in Lagrangian coordinates, as one has to identify different elements belonging to one and the same equivalence class. We denote this as relabeling. In addition, for this construction one not only needs to know the solution in Eulerian coordinates but also in Lagrangian coordinates for allt.

The present approach is based on the fact that a natural metric for measuring distan- ces between Radon measures (with the same total mass) is given through the Wasserstein (or Monge–Kantorovich) distance d_W, which in one dimension is defined with the help of pseudo inverses, see Villani [16]. This tool has been used extensively in the field of kinetic equations [17,18], conservation laws [19,20] and non-linear diffusion equations [21–23]. To be more precise, given two positive and finite Radon measuresl1

andl₂, where we for simplicity assume thatl₁ðRÞ ¼l₂ðRÞ ¼C, let

F_ið Þ ¼x l_iðð1;xÞÞ i¼1;2; (1.5) and define their pseudo inverses v_i:½0;C !R as follows

v_ið Þ ¼n supxjFið Þx <n : Then, we define

d_Wðl₁;l₂Þ ¼ kv₁v₂k_L1ð½0;CÞ:

As far as the distance between u1 and u2 is concerned, we are only interested in measuring the“distance in theL¹ norm”. Thus we introduce the distance das follows:

d uðð ₁;l₁Þ;ðu₂;l₂ÞÞ ¼ ku₁ðv₁ð Þ Þ u₂ðv₂ð Þ Þk_L¹ð½_0;CÞþd_Wðl₁;l₂Þ:

(5)

For this to work, it is necessary that this metric behaves nicely with the time evolution. Thus as a first step, we are interested in determining the time evolution of both vðt;xÞ, the pseudo inverse oflðt;xÞ, anduðt;vðt;xÞÞ.

Let ðuðtÞ;lðtÞÞ be a weak conservative solution to the HS equation with total energy lðt;RÞ ¼C. To begin with, we assume that F(t,x) is strictly increasing and smooth, which greatly simplifies the analysis. Recall that vðt; Þ:½0;C !R is given by

vðt;gÞ ¼supxjlðt;ð1;xÞÞ<g

¼supxjF t;ð xÞ<g :

According to the assumptions on F(t, x), we have that Fðt;vðt;gÞÞ ¼g for all g2

½0;C andvðt;Fðt;xÞÞ ¼x for allx2R. Direct formal calculations yield that

v_tðt;F t;ð xÞÞ þv_gðt;F t;ð xÞÞFtðt;xÞ ¼0; (1.6a) v_gðt;F t;ð xÞÞFxðt;xÞ ¼1: (1.6b) Recalling (1.4b) and the definition ofF(t,x), we have

F_xðt;xÞ ¼lðt;xÞ; (1.7a)

F_tðt;xÞ ¼ ð_x

1dl_tð Þ ¼ t ð_x

1dðu tð Þlð Þt Þ_x¼ u t;ð xÞlðt;xÞ: (1.7b) Thus combining (1.6) and (1.7), we obtain

v_tðt;F t;ð xÞÞ ¼ vgðt;F t;ð xÞÞFtðt;xÞ ¼v_gðt;F t;ð xÞÞu t;ð xÞlðt;xÞ

¼v_gðt;F tð;xÞÞFxðt;xÞu tð;xÞ ¼u tð;xÞ:

Introducingg¼Fðt;xÞ, we end up with

v_tðt;gÞ ¼u tð;vðt;gÞÞ;

where we again have used that vðt;Fðt;xÞÞ ¼x for all x2R. As far as the time evolution ofUðt;gÞ ¼uðt;vðt;gÞÞis concerned, we have

U_tðt;gÞ ¼u_tðt;vðt;gÞÞ þu_xðt;vðt;gÞÞv_tðt;gÞ

¼u_tðt;vðt;gÞÞ þuu_xðt;vðt;gÞÞ

¼1 4

ðvð Þt;g

1 dlðt;rÞ1 4

ð₁

vð Þt;g dlðt;rÞ

¼1 2

ðvð Þt;g

1 dlðt;rÞ1 4C

¼1

2F t;ð vðt;gÞÞ1 4C

¼1 2g1

4C:

Thus we get the very simple system of ordinary differential equations

v_tðt;gÞ ¼ Uðt;gÞ; (1.8a) Utðt;gÞ ¼1

2g1

4C: (1.8b)

The global solution of the initial value problem is simply given by vðt;gÞ;t;Uðt;gÞ

ð Þ 2R³jt2ð0;1Þ;g2½0;C

: (1.9)

(6)

The above derivation is only of formal character, and this derivation is but valid if F(t,x) is strictly increasing and smooth. However, it turns out that the simple result (1.8) also persists in the general case, but the proof is considerably more difficult, and is the main result of this article.

We prove two results. The first result, Theorem 2.9, describes a simple and explicit formula for conservative solutions of the Cauchy problem. Let u02H¹ðRÞ andl0 be a non-negative, finite Radon measure with C¼l₀ðRÞ. Define

v₀ð Þ ¼g supxjl₀ðð1;xÞÞ<g

; (1.10a)

U₀ð Þ ¼g u₀ðv₀ð Þg Þ: (1.10b) If limg!0v₀ðgÞ ¼ limg!Cv₀ðgÞ ¼ 1 (all other cases are treated in Theorem 2.9), we define

vðt;gÞ ¼t²

4 gC

2

þtU0ð Þ þg v₀ð Þ;g if g2ð0;CÞ; (1.11a) Uðt;nÞ ¼t

2 gC

2

þ U0ð Þ;g if g2ð0;CÞ: (1.11b) Then we have

x;t;u t;ð xÞ

ð Þ 2R³jt2½0;1Þ; x2R

¼ðvðt;gÞ;t;Uðt;gÞÞ 2R³jt2½0;1Þ; g2ð0;CÞ

; where u¼uðt;xÞ denotes the conservative solution of the HSEq. (1.4).

The second result, Theorem 2.11, describes the Lipschitz metric. Let u_0;j2H¹ðRÞand l_0;j be a non-negative, finite Radon measure with C_j¼l_0;jðRÞ for j¼1, 2, and define v_jðt;gÞ and Ujðt;gÞ by (1.10) and (1.11) for j¼1, 2 where u0 is replaced by u_0;j and l0

is replaced by l_0;j, respectively. Next introduce ^v_jðt;gÞ ¼v_jðt;C_jgÞ and U^_jðt;gÞ ¼ Ujðt;CjgÞforj¼1, 2. Define

d u ₁ð Þ;t l₁ð Þt

;u₂ð Þ;t l₂ð Þt

¼ kU^1ðt;Þ U^2ðt;Þk_L¹_ð_½_0;1_Þ

þ k^v₁ðt;Þ ^v₂ðt; Þk_L¹_ð_½_0;1_Þþ jC1C2j:

Then we have, see Theorem 2.11, that d u ₁ð Þ;t l₁ð Þt

1þtþ1 8t²

d u ₁ð Þ;0 l₁ð Þ0

;u₂ð Þ;0 l₂ð Þ0

:

2. The Lipschitz metric for the Hunter–Saxton equation Let us study the calculations (1.5)–(1.9) on two explicit examples.

Example 2.1. (i) Let u₀ð Þ ¼x p

2

1=2

erf x ffiffiffi2

p

; l₀ð Þ ¼x u²_0;xð Þdxx ¼e^x²dx; where erfðxÞ ¼^p²ﬃﬃ_pÐ_x

0 e^t²dtis the error function. We find that

(7)

F0ð Þ ¼x l₀ðð1;xÞÞ ¼ ffiffiffip p

2 ð1þerfð Þx Þ as well as C¼F0ð1Þ ¼ ffiffiffi

pp

. This implies that v₀ð Þ ¼g erf¹ 2

ffiffiffip p g1

; g2 0; ffiffiffi

pp

; U0ð Þ ¼g ^p₂ ¹⁼²erf 1

ffiffiffi2

p erf¹ 2 ffiffiffip p g1

; g2 0; ffiffiffi

pp

:

Considering the system of ordinary differential Equation (1.8) with initial data ðv;UÞj_t¼0¼ ðv₀;U₀Þ, we find

vðt;gÞ ¼t²

4 g1

2C

þ U0ð Þg tþv₀ð Þ;g Uðt;gÞ ¼t

2 g1 2C

þ U0ð Þ:g

See Figure 1. Observe that it is not easy to transform this solution explicitly back to the original variable u.

(ii) Let

u₀ð Þ ¼x arcsinhð Þ;x l₀ð Þ ¼x u²_0;xð Þdxx ¼ dx 1þx²:

Note that u0is not bounded, yet the same transformations apply. We find that F₀ð Þ ¼x l₀ðð1;xÞÞ ¼arctanð Þ þx p

2 as well as C¼F₀ð1Þ ¼p. This implies that

v₀ð Þ ¼g tan gp 2

and U₀ð Þ ¼g arcsinh tan gp 2

for g2ð0;pÞ:

Figure 1. The surfacefðv;t;UÞ jt2 ½0;2:5;g2 ð0; ffiffiffi pp

Þgdiscussed in Example 2.1 (i).

(8)

Here we find

vðt;gÞ ¼t²

4 g1

2C

þ U₀ð Þg tþv₀ð Þ;g Uðt;gÞ ¼t

2 g1 2C

þ U₀ð Þ:g

SeeFigure 2. Again it is not easy to transform this solution explicitly back to the original variable u.

Let us next consider an example where the initial measure is a pure point measure.

Example 2.2. This simple singular example shows the interplay between measures l and their pseudo inverses vðxÞ better.¹ Consider the example u0¼0 and l₀¼ad0, where d0is the Dirac delta function at the origin, anda0. ThenF₀ :R! ½0;areads

F₀ð Þ ¼x 0; if x0;

a; if x>0:

The corresponding pseudo inverse v₀:½0;a !R is then given by v₀ð Þ ¼g 1; if g¼0;

0; if g2ð0;a: Thus²

v₀ðF₀ð Þx Þ ¼ 1; if x0;

0; if x>0; and F₀ðv₀ð Þg Þ ¼0 for all g2½0;a: In general, one observes that jumps in F₀ðxÞ are mapped to intervals where v₀ðgÞ is constant and vice versa. This means in particular that intervals where F₀ðxÞ is constant shrink to single points. Moreover, if F0ðxÞ is constant on some interval, then u0ðxÞ is also constant on the same interval.

Figure 2. The surfacefðv;t;UÞ jt2 ½0;2:5;g2 ð0;pÞgdiscussed in Example 2.1 (ii).

1The solution in (1.2) comes from this example.

2Note that in the smooth case bothvðFÞandFðvÞare the identity function!

(9)

Next, we compute the time evolution of both vðt;gÞ and Uðt;gÞ ¼uðt;vðt;gÞÞ.

Following the approach by Bressan et al. [13], we obtain that the corresponding solution in Eulerian coordinates reads for tpositive

u t;ð xÞ ¼ a

4t; if x a 8t²; 2x

t ; if a

8t² xa 8t²; a

4t; if xa 8t²; 8>

>>

<

>>

>:

(2.1a)

lðt;xÞ ¼u²_xðt;xÞdx¼ 4

t²I½at²=8;at²=8ð Þdx;x (2.1b)

F t;ð xÞ ¼

0; if x a

8t²; 4x

t² þa

2; if a

8t²xa 8t²; a; if xa

8t²: 8>

>>

><

>>

:

(2.1c)

Calculating the pseudo inverse vðt;gÞ and Uðt;gÞ ¼uðt;vðt;gÞÞ for each t then yields

vðt;gÞ ¼t² 4 ga

2

; g2ð0;a; (2.2a)

Uðt;gÞ ¼t

2 ga 2

; g2½0;a; (2.2b)

and, in particular, that

U_tðt;gÞ ¼1 2 ga

2

; g2½0;a:

Thus we still obtain the same ordinary differential Equation (1.8) as in the smooth case! In addition, note that v_tð0;gÞ ¼0 for all g2 ð0;a, and hence the important infor- mation is encoded in U_tðt;gÞ.

We can of course also solve v_t ¼ U and Ut ¼^g₂^a₄ directly with initial data v₀¼ U₀¼0, which again yields (2.2). To return to the Eulerian variables u and l we have in the smooth region that

u tð;xÞ ¼ Uðt;gÞ; x¼vðt;gÞ; g2ð0;a; and we need to extend U andvto all ofR by continuity:

vðt;gÞ ¼ a

8t²þg; if g0;

t² 4 ga

2

; if g2½0;a; a

8t²þga; if ga;

t0;

8>

>>

><

>>

:

(10)

Uðt;gÞ ¼ a

4t¼ Uðt;0þÞ; if g0;

t 2 ga

2

; if g2½0;a;

a

4t¼ Uðt;aÞ; if ga;

t0:

8>

>>

<

>>

>:

Returning to the Eulerian variables we recover (2.1). We can also depict the full solution in the (x, t) plane in the new variables: The full solution reads

vðt;gÞ;t;Uðt;gÞ

ð Þ 2R³jt2ð0;1Þ;g2R

: SeeFigure 3.

The next example shows the difficulties that one has to face in the general case where the solution encounters a break down in the sense of steep gradients.

Example 2.3. Let

u0ð Þ ¼ xIx ½0;1ð ÞIx ½1;1Þð Þ;x l₀ð Þ ¼x u²_0;xð Þdxx ¼I½0;1ð Þdx:x Next, we find

F₀ð Þ ¼x xI½0;1ð Þ þx I½1;1Þð Þ;x v₀ð Þ ¼g 1; if g¼0;

g; if g2ð0;1; U0ð Þ ¼ g g; g2½0;1:

Figure 3. The surfacefðx;t;uÞ jt2 ½0; :5;x2 ½:05; :05gdiscussed in Example 2.2.

(11)

Assuming (1.8) holds also in this case, we find vðt;gÞ ¼t²

4 g1

2

tgþg; g2ð0;1; Uðt;gÞ ¼t

2 g1 2

g; g2½0;1: We extend the functions by continuity

vðt;gÞ ¼ t²

8þg; if g0;

t²

4 g1

2

tgþg; if g2½0;1; t²

8tþg; if g1;

8>

>>

><

>>

:

Uðt;gÞ ¼ t

4; if g0;

t 2 g1

2

g; if g2½0;1; t

41; if g1;

: 8>

>>

><

>>

:

which gives a well-defined global solution given by fðvðt;gÞ;t;Uðt;gÞÞjt0; g2Rg.

However, as we return to Eulerian variables, the time-development is more dramatic.

Solving the equation x¼vðt;gÞfor g2 ½0;1 yields g¼4xþt²=2

ðt2Þ² 2½0;1; which leads to the solution

u t;ð xÞ ¼ U t;4xþt²=2 ðt2Þ²

!

¼2xþt=2 t2 whenever

t²

8 <x< 1

4 ðt2Þ²t² 2

:

For t!2, we have that ux! 1atx¼ 1=2. The solution on the full line reads

u tð;xÞ ¼ 1

4t; if x 1

8t²; 2xþt=2

t2 ; if 1

8 t²x1

4 ðt2Þ²t² 2

; 1

4t1; if x1

4 ðt2Þ²t² 2

: 8>

>>

><

>>

:

The solution is illustrated inFigure 4.

(12)

The above examples already hint that the interplay between Eulerian and Lagrangian coordinates is going to play a major role in our further considerations. We assume a smooth solution of

u_tþuu_x¼1 4

ð_x

1u²_xð Þy dy ð₁

x

u²_xð Þy dy

; (2.3a)

u²_x _tþuu²_x

x ¼0: (2.3b)

Next, we rewrite the equation in Lagrangian coordinates. Introduce the characteristics y_tðt;nÞ ¼u t; y t;ð nÞ

: The Lagrangian velocity Ureads

U t;ð nÞ ¼u t; y t;ð nÞ :

Furthermore, we define the Lagrangian cumulative energy by H tð;nÞ ¼

ð_{y t;n}_{ð Þ}

1 u²_xðt;xÞdx: From (2.3a), we get that

U_t ¼u_tyþy_tu_xy¼1 4

ð_y

1u²_xdx ð₁

y

u²_xdx

!

¼1 2H1

4C where C¼H t;ð 1Þ is time independent, and

Figure 4. The solution discussed in Example 2.3.

(13)

H_t ¼ ð_{y t;}ð Þn

1 ðu²_xðt;xÞÞ_tdxþy_tu²_xðt;yÞ ¼ ð_{y t;}ð Þn

1 ððu²_xÞ_tþðuu²_xÞ_xÞðt;xÞdx¼0

by (2.3b). In this formal computation, we require that u and u_x are smooth and decay rapidly at infinity. Hence, the HS equation formally is equivalent to the following system of ordinary differential equations:

y_t ¼U; (2.4a)

U_t ¼1 2H1

4C; (2.4b)

H_t ¼0: (2.4c)

Global existence of solutions to (2.4) follows from the linear nature of the system.

There is no exchange of energy across the characteristics and the system (2.4) can be solved explicitly. This is in contrast to the Camassa–Holm equation where energy is exchanged across characteristics. We have

y t;ð nÞ ¼ 1

4Hð0;nÞ 1 8C

t²þUð0;nÞtþyð0;nÞ;

U tð;nÞ ¼ 1

2Hð0;nÞ 1 4C

tþUð0;nÞ;

H t;ð nÞ ¼Hð0;nÞ:

We next focus on the general case without assuming regularity of the solution. It turns out that in addition to the variable u we will need a measure l that in smooth regions coincides with the energy density u²_xdx. At wave breaking, the energy at the point where the wave breaking takes place, is transformed into a point measure. It is this dynamics that is encoded in the measure lthat allows us to treat general initial data. An important com- plication stems from the fact that the original solution in two variables ðu;lÞ is transformed into Lagrangian coordinates with three variables (y, U, H). This is a well-known consequence of the fact that one can parametrize a particle path in several different ways, corresponding to the same motion. This poses technical complications when we want to measure the distance between two distinct solutions in Lagrangian coordinates that correspond to the same Eulerian solution, and we denote this as relabeling of the solution.

We will employ the notation and the results by Bressan et al. [13] and Nordli [14].

Define the Banach spaces

E1¼ f 2L¹ð Þ jR f⁰2L²ð ÞR ; lim

n!1fð Þ ¼n 0

n o

; E₂¼f 2L¹ð Þ jR f⁰2L²ð ÞR

; with norms

kfk_E

j ¼ kfk_L1þ kf⁰k_L2; f 2E_j;j¼1;2:

Let

B¼E₂ E₂ E₁;

(14)

with norm

kðf₁;f₂;f₃Þk_B ¼ kf1k_E₂þ kf2k_E₂þ kf3k_E₁; ðf₁;f₂;f₃Þ 2B:

We are given some initial dataðu₀;l₀Þ 2 D, where the setDis defined as follows.

Definition 2.4. The setD consists of all pairsðu;lÞsuch that (i) u2E₂;

(ii) l is a non-negative and finite Radon measure such thatl_ac¼u²_xdx, where lac

denotes the absolute continuous part of l with respect to the Lebesgue measure.

The Lagrangian variables are given by ðf;U;HÞ (withf¼yId), and the appropriate space is defined as follows.

Definition 2.5. The set F consists of the elements ðf;U;HÞ 2B¼E2 E2 E1

such that

(i) ðf;U;HÞ 2 ðW^1;1ðRÞÞ³, wherefðnÞ ¼yðnÞn;

(ii) yn0; Hn0 andynþHnc, almost everywhere, wherec is a strictly positive constant;

(iii) ynHn¼U_n² almost everywhere.

The key subspaceF0 F is defined by

F₀ ¼X¼ðy;U;HÞ 2 F jyþH¼Id :

We need to clarify the relation between the Eulerian variables ðu;lÞ and the Lagrangian variables ðf;U;HÞ. The transformation

L:D ! F0; X¼L u;ð lÞ is defined as follows.

Definition 2.6. For anyðu;lÞin D, let

yð Þ ¼n supxjlðð1;xÞÞ þx < n

; (2.5a)

Hð Þ ¼n nyð Þ;n (2.5b) Uð Þ ¼n uyð Þ:n (2.5c) Then X¼ ðf;U;HÞ 2 F0 and we denote byL:D ! F0 the map which to anyðu;lÞ 2 D associatesðf;U;HÞ 2 F₀ as given by (2.5).

From the Lagrangian variables, we can return to Eulerian variables using the following transformation.

Definition 2.7. Given any elementXinF. Then, the pairðu;lÞdefined as follows:

u xð Þ ¼Uð Þn for any n such that x¼yð Þ;n (2.6a)

l¼y#ðHndnÞ (2.6b)

(15)

belongs to D. Here, the push-forward of a measure by a measurable functionf is the measure f# defined byf#ðBÞ ¼ðf¹ðBÞÞ for all Borel setsB.We denote by M:F ! D the map which to anyXinF associatesðu;lÞas given by (2.6).

The key properties of these transformations are (cf. [13, Prop. 2.11])

LMj_F₀¼IdF₀; ML¼IdD: (2.7) The formalism up to this point has been stationary, transforming back and forth between Eulerian and Lagrangian variables. Next, we can take into consideration the time-evolution of the solution of the HS equation.

The evolution of the HS equation in Lagrangian variables is determined by the system (cf. (2.4))

S_t :F ! F; X tð Þ ¼S_tð Þ;X₀ X_t ¼S Xð Þ; Xj_t¼0 ¼X₀: (2.8) of ordinary differential equations. Here

S Xð Þ ¼ 1 U 2H1

4C 0 0 B@

1 CA;

Next, we address the question about relabeling. We need to identify Lagrangian solutions that correspond to one and the same solution in Eulerian coordinates. Let G be the subgroup of the group of homeomorphisms onR such that

fId and f¹Id both belong to W^1;1ð ÞR ; fn1 belongs to L²ð ÞR :

By default, the HS equation is invariant under relabeling, which is given by equivalence classes

½ ¼X X~ 2 F j there exists g2G such that X¼X~ g F=G¼½ jX X2 F ;

: The key subspace of F is denotedF0 and is defined by

F₀ ¼X¼ðy;U;HÞ 2 F jyþH¼Id :

The map into the critical space F0 is taken care of by (cf. [13, Def. 2.9]) P:F ! F₀; Pð Þ ¼X XðyþHÞ¹;

with the property that PðF Þ ¼ F0. We note that the map X7!½Xfrom F0 toF=Gis a bijection. Then we have that (cf. [13, Prop. 2.12])

PS_tP¼PS_t; and hence we can define the semigroup

~S_t ¼PS_t :F₀! F₀: (2.9)

We can now provide the solution of the HS equation. Consider initial data ðu0;l₀Þ 2 D, and defineX0 ¼ ðy₀;U0;H0Þ ¼Lðu0;l₀Þ 2 F0 given by

(16)

y₀ð Þ ¼n supxjxþFð0;xÞ<n

; U₀ð Þ ¼n u₀y₀ð Þn

; H0ð Þ ¼n ny₀ð Þ;n

with Fð0;xÞ ¼l₀ðð1;xÞÞ. Next, we want to determine the solution ðuðtÞ;lðtÞÞ 2 D (we suppress the dependence in the notation on the spatial variable x when convenient) for arbitrary time t.

Define

X tð Þ ¼StX02 F; X tð Þ ¼~StX0 2 F0: (2.10) The advantage of XðtÞ is that it obeys the differential Eq. (2.8), while X(t) keeps the relation yþH¼Id for all times. From (2.9) we have that

X tð Þ ¼PX tð Þ :

We know thatXðt; nÞ ¼ ðyðt;nÞ;Uðt;nÞ;Hðt; nÞÞ 2 F is the solution of

y_tðt;nÞ ¼U t;ð nÞ; (2.11a) U_tðt;nÞ ¼1

2H t;ð nÞ1

4C; (2.11b)

Htðt;nÞ ¼0; (2.11c)

where C¼l₀ðRÞand Xð0Þ ¼ X0. Straightforward integration yields y tð;nÞ ¼1

4 H₀ð Þn 1 2C

t²þU₀ð Þtn þy₀ð Þn ; U t;ð nÞ ¼1

2 H₀ð Þn 1 2C

tþU₀ð Þ;n H t;ð nÞ ¼H0ð Þ ¼n ny₀ð Þn:

The solutionðuðtÞ;lðtÞÞ ¼MðXðtÞÞ in Eulerian variables reads u t;ð xÞ ¼U t;ð nÞ; y t;ð nÞ ¼x;

lðt;xÞ ¼y#Hnðt;nÞdn

; with Fðt;xÞ ¼lðt;ð1;xÞÞ ¼Ð

yðt;nÞ<xð1y_0;nðnÞÞdn.

However, forXðt;nÞ ¼ ðyðt;nÞ;Uðt;nÞ;Hðt;nÞÞ 2 F0, which satisfies X tð Þ ¼X tð Þ yþH1

2 F0; we see, using (2.7), that

y t;ð nÞ ¼supxjxþF t;ð xÞ<n

¼supxjxþF tð;xÞ<y tð;nÞ þH tð;nÞ

;

where we in the second equality use that XðtÞ 2 F₀. Note that we still have ðuðtÞ;lðtÞÞ ¼MðXðtÞÞ ¼MðXðtÞÞ, and thus

(17)

u t;ð xÞ ¼U t;ð nÞ; y t;ð nÞ ¼x;

lðt;xÞ ¼y#Hnðt;nÞdn

; with Fðt;xÞ ¼lðt;ð1;xÞÞ ¼Ð

yðt;nÞ<xHnðt;nÞdn. Since XðtÞ ¼XðtÞ ðyþHÞ, we find that

y tð;nÞ ¼y t;y tð;nÞ þH tð ;nÞ

¼supxjxþF tð ;xÞ<y tð ;nÞ þH tð;nÞ

: (2.12) This is the only place in this construction where we use the quantity X(t).

Define now

vðt;gÞ ¼supxjlðt;ð1;xÞÞ<g

¼supxjF tð;xÞ<g : We claim that

vðt;gÞ ¼y t; l t;ð gÞ

; where we have introducedlðt; Þ:½0;C !R by

l tð;gÞ ¼supnjH tð ;nÞ<g

: (2.13)

Note that sinceHt ¼0, we have that

l t;ð gÞ ¼lð0;gÞ and ltðt;gÞ ¼0:

Recall that for each timet,we have (cf. (2.12))

y t;ð nÞ ¼supxjxþF t;ð xÞ<y t;ð nÞ þH t;ð nÞ

; which implies that

y t;ð nÞ þF t; y t;ð nÞ

y t;ð nÞ þH t;ð nÞ y t;ð nÞ þF t; y t;ð nÞþ : Subtractingyðt;nÞ in the above inequality, we end up with

F t; y t;ð nÞ

H t;ð nÞ F t; y t;ð nÞþ

for all n2R:

Comparing the last equation and (2.13), we have F t; y t; l t;ð gÞ

H t; l t;ð gÞ

¼gF t;y t; l t;ð gÞ

þ

: Sinceyðt; Þis surjective and non-decreasing, we end up with

vðt;gÞ ¼supxjF t;ð xÞ<g

¼y t; l t;ð gÞ : Introduce the new function

Uðt;gÞ ¼U t ;l tð;gÞ :

We are now ready to derive the system of ordinary differential equations for vðt;gÞ andUðt;gÞ. Therefore recall that

H t; l t;ð gÞ

¼H0;l t;ð gÞ

¼Hð0;lð0;gÞÞ ¼g for all g2½0;C; sinceHð0;nÞis continuous. Direct calculations yield

(18)

v_tðt;gÞ ¼ d

dty t;ð lð0;gÞÞ ¼y_tðt;lð0;gÞÞ ¼U t;ð lð0;gÞÞ ¼U t; l t;ð gÞ

¼ Uðt;gÞ;

U_tðt;gÞ ¼ d

dtU t;ð lð0;gÞÞ ¼U_tðt;lð0;gÞÞ ¼1

2H t;ð lð0;gÞÞ1 4C

¼1 2g1

4C:

Thus we established rigorously the linear system

v_tðt;gÞ ¼ Uðt;gÞ; (2.14a) U_tðt;gÞ ¼1

2g1

4C (2.14b)

of ordinary differential equations, with solution vðt;gÞ ¼ 1

4g1 8C

t²þ Uð0;gÞtþvð0;gÞ;

Uðt;gÞ ¼ 1 2g1

4C

tþ Uð0;gÞ:

Example 2.8. Recall Example 2.2. Let us compute the corresponding quantities in the case withu₀ ¼0 andl₀¼d₀. Here we find that

y₀ð Þ ¼n n; for n0; 0; for n2ð0;1Þ;

n1; for n1;

8<

: U0ð Þ ¼n 0;

H₀ð Þ ¼n 0; for n0;

n; for n2ð0;1Þ;

1; for n1:

8<

: The solution ofX ¼StX0 (cf. (2.10)) reads

y t;ð nÞ ¼

nt²

8; for n0; t²

4nt²

8; for n2ð0;1Þ;

t²

8þn1; for n1; 8>

>>

><

>>

:

U tð;nÞ ¼ t

4; for n0;

t 2nt

4; for n2ð0;1Þ;

t

4; for n1;

8>

>>

<

>>

>:

(19)

H t;ð nÞ ¼ 0; for n0;

n; for n2ð0;1Þ;

1; for n1:

8<

:

If we compute the corresponding quantitiesXðtÞ ¼XðtÞ ðyþHÞ ¹, we find

y t;ð nÞ ¼

n; for n t²

8; t²

t²þ4n t²

2t²þ8; for n2 t² 8;t²

8þ1

;

n1; for nt²

8þ1;

8>

>>

<

>>

>:

U tð ;nÞ ¼ t

4; for n t²

8; 2t

t²þ4 nþt² 8

t

4; for n2 t² 8;t²

8þ1

; t

4; for nt²

8þ1;

8>

>>

<

>>

>:

H t;ð nÞ ¼

0; for n t²

8; 4

t²þ4 nþt² 8

; for n2 t²

8;t² 8þ1

;

1; for nt²

8þ1;

8>

>>

<

>>

>:

with the property thatyþH¼Id. Finally, we find l t;ð gÞ ¼ 1; for g¼0;

g; for g2ð0;1; (

vðt;gÞ ¼y t; l t;ð gÞ

¼t² 4 g1

2

; g2ð0;1;

Uðt;gÞ ¼U t;l tð ;gÞ

¼ t 2 g1

2

; g2ð0;1; as expected.

Theorem 2.9. Let u02E2 and l0 be a non-negative, finite Radon measure with C¼l₀ðRÞ. Let ðuðtÞ;lðtÞÞ denote the conservative solution of the HS equation.

Define

v₀ð Þ ¼g supxjl₀ðð1;xÞÞ<g

; U₀ð Þ ¼g u₀ðv₀ð ÞgÞ:

(20)

If limg!0v₀ðgÞ and limg!Cv₀ðgÞ are finite, we define

vðt;gÞ ¼ C

8t²þtU₀ð Þ þ0 v₀ð Þ þ0 g; if g<0;

t²

4 gC

2

þtU₀ð Þ þg v₀ð Þ;g if g2ð0;C; C

8t²þtU₀ð Þ þC v₀ð Þ þC gC; if g>C;

8>

>>

<

>>

>:

Uðt;gÞ ¼ C

4tþ U₀ð Þ0 ; if g<0;

t

2 gC

2

þ U₀ð Þ;g if g2ð0;C; C

4tþ U₀ð ÞC ; if gC:

8>

>>

<

>>

>: Then we have

x;t;u t;ð xÞ

¼ðvðt;gÞ;t;Uðt;gÞÞ 2R³jt2½0;1Þ; g2R : Iflimg!0v₀ðgÞ ¼ limg!Cv₀ðgÞ ¼ 1,we define

vðt;gÞ ¼t²

4 gC

2

þtU₀ð Þ þg v₀ð Þ;g if g2ð0;CÞ;

Uðt;gÞ ¼t

2 gC

2

þ U0ð Þ;g if g2ð0;CÞ:

Then we have

x;t;u t;ð xÞ

¼ðvðt;gÞ;t;Uðt;gÞÞ 2R³jt2½0;1Þ;g2ð0;CÞ : Similar results hold if one of limg!0v₀ðgÞ and limg!Cv₀ðgÞ is finite.

We can now introduce the new Lipschitz metric. Define d u ₁ð Þt ;l₁ð Þt

;u₂ð Þt ;l₂ð Þt

¼ kU1ð Þ Ut ₂ð Þkt _L1ð½0;CÞþ kv₁ð Þ t v₂ð Þkt _L1ð½0;CÞ; which implies that

d u ₁ð Þt ;l₁ð Þt

¼ kU₁ð Þ Ut ₂ð Þkt _L1ð½0;CÞþ kv₁ð Þ t v₂ð Þkt _L1ð½0;CÞ

ð1þCtÞkU1ð Þ U0 2ð Þk0 _L¹_ð_½_0;C_Þþ kv₁ð Þ 0 v₂ð Þk0 _L1ð½0;CÞ

ð1þCtÞd u ₁ð Þ0;l₁ð Þ0

;u₂ð Þ0 ;l₂ð Þ0

:

(21)

A drawback of the above construction is the fact that we are only able to compare solutions ðu1;l₁Þ and ðu2;l₂Þ with the same energy, viz. l₁ðRÞ ¼l₂ðRÞ ¼C. The rest of this section is therefore devoted to overcoming this limitation.

A closer look at the system (2.14) of ordinary differential equations reveals that we can rescale vðt;gÞ andUðt;gÞin the following way. Let

^

vðt;gÞ ¼vðt;CgÞ;

U^ðt;gÞ ¼ Uðt;CgÞ:

Then ^vðt; Þ:½0;1 !R; U ðt^ ; Þ:½0;1 !R for allt and

^

v_tðt;gÞ ¼U^ðt;gÞ;

U^tðt;gÞ ¼1

2C g1 2

: Direct computations then yield

kU^1ðt;Þ U^2ðt;Þk_L¹_ð_½_0;1_Þ kU^1ð0; Þ U^2ð0; Þk_L¹_ð_½_0;1_Þþ1

2tjC₁C2j kg1

2k_L¹_ð_½_0;1_Þ kU^₁ð0; Þ U^₂ð0; Þk_L¹_ð_½_0;1_Þþ1

4tjC₁C₂j

and

k^v₁ðt;Þ ^v₂ðt; Þk_L1ð½0;1Þ k^v₁ð0;Þ ^v₂ð0;Þk_L1ð½0;1ÞþtkU^₁ð0;Þ U^₂ð0;Þk_L¹_ð_½_0;1_Þ þ1

8t²jC₁C2j: Thus introducing the redefined distance by

d uðð ₁;l₁Þ;ðu₂;l₂ÞÞ ¼ kv₁ðC₁Þ v₂ðC₂Þk_L1ð½0;1Þ

þ ku₁ðv₁ðC₁ÞÞ u₂ðv₂ðC₂ÞÞk_L1ð½0;1Þ

þ jC₁C₂j; we end up with

d u ₁ð Þt ;l₁ð Þt

1þtþ1 8t²

d u 1ð Þ0 ;l₁ð Þ0

;u2ð Þ0 ;l₂ð Þ0

:

In particular, dððu₁;l₁Þ;ðu₂;l₂ÞÞ ¼0 immediately implies that C₁¼l₁ðRÞ ¼ l₂ðRÞ ¼C2, which then implies v₁ðt;xÞ ¼v₂ðt;xÞ and thusu1ðt;xÞ ¼u2ðt;xÞ.

Example 2.10. Recall Example 2.2. Consider initial datau0¼0 and l_0;i¼a_id₀ yielding solutionsðu₀ðtÞ;l_iðtÞÞ. Here we find

dððu₀;l_0;1Þ;ðu₀;l_0;2ÞÞ ¼dðð0;a₁d₀Þ;ð0;a₂d₀ÞÞ ¼ ja₁a₂j:

(22)

Thus

d u 1ð Þt ;l₁ð Þt

;u2ð Þt ;l₂ð Þt

1þtþ1 8t²

ja1a2j:

Theorem 2.11. Consider u_0;j and l_0;j as in Theorem 2.9 for j¼ 1, 2 with C_j¼l_0;jðRÞ.

Assume in addition that ð₀

1F_0;jð Þdxx þ ð₁

0

C_jF_j;0ð Þx

dx<1; j¼1;2: (2.15) Define the metric

d u ₁ð Þ;t l₁ð Þt

¼ kU1ðt;C1Þ U2ðt;C2Þk_L¹_ð_½_0;1_Þ

þ kv₁ðt;C₁Þ v₂ðt;C₂Þk_L1ð½0;1Þþ jC₁C₂j:

Then we have d u ₁ð Þ;t l₁ð Þt

1þtþ1 8t²

d uðð _0;1;l_0;1Þ;ðu_0;2;l_0;2ÞÞ:

Proof. It is left to show that v_jðtÞ 2L¹ð½0;C_jÞ for all t positive and j¼1, 2. For the remainder of this proof, fixj and drop it in the notation.

Note that (2.15) is equivalent tovð0;gÞ 2L¹ð½0;CÞ. Indeed, denote byg₁the point at which vð0;gÞ changes from negative to positive, then by definition

vð0;gÞ ¼supxjFð0;xÞ<g

; and thus

kvk_L1ð½0;CÞ ¼ ð_C

0

jv gð Þ jdg¼ ðg₁

0

v gð Þdgþ ð_C

g₁

v gð Þdg

ð₀

1Fð0;xÞdxþ ð1

0

CF xð Þ

ð Þdx:

It remains to show thatvðtÞ remains integrable, i.e.,vðtÞ 2L¹ð½0;CÞfor alltpositive.

Translating the condition (2.15), we find ðn₁

1

Hy_nð Þdnn þ ð₁

n₁

CH

ð Þy_nð Þdnn <1; (2.16)

where n1 is chosen such that yðn₁Þ ¼0. Note that it does not matter if there exists a single point or a whole interval such that yðnÞ ¼0, since in the latter case y_nðnÞ ¼0.

Denote by nðtÞ the time-dependent function such thatyðt;nðtÞÞ ¼0 for all t, which is not unique. Then the first term can be rewritten as

(23)

ð_n_{ð Þ}_t

1

H t;ð nÞy_nðt;nÞdn¼ ð_n_{ð Þ}_t

1

Hð0;nÞy_nðt;nÞdn

¼ ðnð Þt

1

Hð0;nÞ y_nð0;nÞ þtUnð0;nÞ þ1

4t²Hnð0;nÞ

dn;

ðnð Þt

1ð1þtÞHy_nð0;nÞdnþ ðnð Þt

1 tþ1 4t²

HHnð0;nÞdn

¼ð1þtÞ ðnð Þ0

1

Hy_nð0;nÞdnþð1þtÞ ðnð Þt

nð Þ0

Hy_nð0;nÞdnþ 1 2tþ1

8t²

H²ð0;nð Þt Þ

ð1þtÞ ðnð Þ0

1

Hy_nð0;nÞdnþð1þtÞ ðnð Þt

nð Þ0

Hy_nð0;nÞdnþ 1 2tþ1

8t²

C²;

where we used (2.11) and that U²_nðt;nÞ ¼Hny_nðt;nÞ. The term on the right-hand side will be finite if we can show that the second integral on the right-hand side is finite.

Therefore observe that ð_n_{ð Þ}_t

nð Þ0

Hy_nð0;nÞdn¼

ð_y_ð_0;n_{ð Þ}_t_Þ

yð0;nð Þ0ÞF xð Þdx¼

0

F xð Þdx¼

y t;nð ð ÞtÞ F xð Þdx Cjy t;ð nð Þt Þyð0;nð Þt Þ j C tjUð0;nð Þt Þ j þt²

8C

C tku₀k_L1ð ÞR þt² 8C

<1:

Similar considerations yield that the second integral in (2.16) remains finite as

time evolves. w

Remark 2.12. Observe that the distance introduced in Theorem 2.11 gives at most a quadratic growth in time, while the distance in [13] has at most an exponential growth in time.

We make a comparison with the more complicated Camassa–Holm equation in the next remark.

Remark 2.13. Consider an interval½n₁;n₂ such thatU0ðnÞ ¼U0ðn₁Þand Hðn₁Þ ¼HðnÞ for all n2 ½n₁;n₂. This property will remain true for all later times. In particular, this means that these intervals do not show up in our metric, and the functionvðt;gÞ always has a constant jump at the corresponding point g. This is in big contrast to the Camassa–Holm equation where jumps in vðt;gÞ may be created and then subsequently disappear immediately again. Thus the construction for the Camassa–Holm equation is much more involved than the HS construction.

This is illustrated in the next examples.

(24)

Example 2.14. Given the initial dataðu0;l₀Þ ¼ ð0;d0þ2d1Þ, direct calculations yield

u tð;xÞ ¼ 3

4t; x 3

8t²; 2

tx; 3

8t²x 1 8t²; 1

4t; 1

8t²x11 8t²; 2

tðx1Þ; 11

8t²x1þ3 8t²; 3

4t; 1þ3 8t²x;

8>

>>

<

>>

>:

F t;ð xÞ ¼

0; x 3

8t²; 3

2þ 4

t²x; 3

8t² x 1 8t²;

1; 1

8t² x11 8t²; 3

2þ 4

t²ðx1Þ; 11

8t² x1þ3 8t²;

3; 1þ3

8t² x:

8>

>>

<

>>

>:

Calculating the pseudo inverse vðt;gÞ and Uðt;gÞ ¼uðt;vðt;gÞÞ for each t, then yields

vðt;gÞ ¼

1; g¼0;

t²

4 g3

2

; 0<g1;

1þt²

4 g3

2

; 1<g3;

8>

>>

><

>>

:

Uðt;gÞ ¼t 2 n3

2

; for all g2½0;3:

Here two observations are important. Note that Uðt;gÞ is continuous and differenti- able with respect to g, whilevðt;gÞ on the other hand has at each timeta discontinuity atg¼1 (and of course atg¼0). In particular, one has

g!1lim vðt;gÞ ¼ 1

8t² and lim

g!1þvðt;gÞ ¼11 8t²:

Thus the jump in function value remains unchanged even if the limit from the left and the right are time dependent.

In order to understand the behavior of lðt;gÞ, let us have a look at the solution in Lagrangian coordinates, which is given by

(25)

y t;ð nÞ ¼

n3

8t²; n0;

1 4 n3

2

t²; 0n1; n11

8t²; 1n2;

1þ1 4 n5

2

t²; 2n4;

n3þ3

8t²; 4n; 8>

>>

><

>>

:

U t;ð nÞ ¼ 3

4t; n0;

1 2 n3

2

t; 0n1;

1

4t; 1n2;

1 2 n5

2

t; 2n4;

3

4t; 4n;

8>

>>

><

>>

:

H t;ð nÞ ¼

0; n0;

n; 0n1;

1; 1n2;

n1; 2n4;

3; 4n:

8>

>>

<

>>

>:

Hence direct computations yield that

l t;ð gÞ ¼lð0;gÞ ¼

1; g¼0;

g; 0<g1;

gþ1; 1<g3:

8>

<

>:

Note that lð0;gÞ is non-constant on the intervals where both Uð0;gÞ and vð0;gÞ are constant.

Acknowledgments

This research was done while the authors were at Institut Mittag-Leffler, Stockholm.

Disclosure statement

No potential conflict of interest was reported by the authors.