Dept. of Math. Univ. of Oslo Pure Mathematics No. 20 ISSN 0806–2439 September 2005
CONVERGENT DIFFERENCE SCHEMES FOR THE HUNTER–SAXTON EQUATION
H. HOLDEN, K. H. KARLSEN, AND N. H. RISEBRO
Abstract. We propose and analyze several finite difference schemes for the Hunter–Saxton equation
ut+uux=1 2
Z x
0
(ux)2dx, x >0, t >0. (HS) This equation has been suggested as a simple model for nematic liquid crystals.
We prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to the derivation of several a priori estimates that yield some basic convergence results, is to prove strong convergence of the discrete spatial derivative of the numerical approximations of u, which is achieved by analyzing various renormalizations (in the sense of DiPerna and Lions) of the numerical schemes. Finally, we demonstrate through several numerical examples the proposed schemes as well as some other schemes for which we have no rigorous convergence results.
Contents
1. Introduction 1
2. Some preliminaries 5
3. The semi-discrete upwind scheme 7
4. The implicit upwind scheme 15
5. The explicit upwind scheme 20
6. The casev0∈L1∩L2 24
7. Numerical examples 37
References 41
1. Introduction
Liquid crystals are mesophases, i.e., intermediate states of matter between the liquid and the crystal phase [13]. They exhibit characteristics of fluid flow and have optical properties typically associated with crystals. Liquid crystals consist of strongly elongated molecules, typical sizes are 5−10˚A, that can be considered invariant under rotation by an angle of π. Nematic liquid crystals are commonly described by two linearly independent vector fields; one describing the fluid flow and one describing the orientation of the so-called director field that gives the orientation of the rod-like molecule. In this paper we will specialize to stationary flow, and hence focus exclusively on the dynamics of the director field, a map n: R3 →S3
Date: September 6, 2005.
2000Mathematics Subject Classification. Primary: 35D05, 65M12; Secondary: 65M06.
Key words and phrases. nonlinear hyperbolic partial differential equation, weak solutions, finite difference schemes, convergence, liquid crystals.
Partially supported by the BeMatA program of the Research Council of Norway and the European network HYKE, contract HPRN-CT-2002-00282. The research of KHK is supported by an Outstanding Young Investigators Award from the Research Council of Norway.
1
from the Euclidean space to the unit ball, see Saxton [12]. It is common to consider the Oseen–Franck expression for the internal energy (see [12], [13], [4])
W(n,∇n) =α|n×(∇ ×n)|2+β(∇ ·n)2+γ(n· ∇ ×n)2,
where α,β andγare constants. Physically,αcorrelates with “splay”;β correlates with “twist”; and γ with “bend” (see, e.g., [13] for an extensive discussion). The dynamics of nis governed by the action principle
δ δn
Z Z
(n2t−W(n,∇n))dx dt. (1.1) We here further specialize to consider planar director fields given by
n=n(x, t) = cos(ψ(x, t))ex+ sin(ψ(x, t))ey,
where ex and ey are orthonormal vectors in the x and y direction, respectively.
Inserting this into (1.1) we find the Lagrangian L=
Z Z
ψt2−c2(ψ)ψx2 dx dt with
c(ψ)2=αcos2ψ+βsin2ψ, which yields the Euler–Lagrange equation
ψtt=c(ψ)(c(ψ)ψx)x.
We now consider the equation satisfied by expansions around the constant state.
More precisely, assume [4]
ψ(x, t, ε) =ψ0+εψ1(θ, τ) +O(ε2)
withθ=x−c(ψ0)t(assumingc0(ψ0)6= 0) andτ =εt. Introduceu=c0(ψ0)ψ1and redefinexbyx= sign (c0(ψ0))θ. Then
(ut+uux)x=1
2(ux)2, u|t=0=u0, or
ut+uux= 1 2
Z x 0
(ux)2dx, u|t=0=u0, (1.2) which is the Hunter–Saxton equation [4]. By introducing
v=ux, we may write this as
vt+uvx=−1
2v2, v=ux (1.3)
or
vt+ (uv)x= 1
2v2, v=ux.
The equation possesses many intriguing properties: It is completely integrable [5];
indeed, let
L=∂x
1 uxx
∂x, A= 1
2(u∂x+∂xu).
Then
Lt= [L, A] is formally equivalent to (uxt+uuxx+1
2u2x)x= 0.
Equation (1.3) also has infinitely many conservation laws (see [5]); the first few reading
(|vx|1/2)t+ (u|vx|1/2)x= 0, (v2)t+ (uv2)x= 0,
(uv2)t−(2uvut+u2t)x= 0.
Furthermore, it is bivariational and bi-Hamiltonian (see [5]). Characteristics are given by
d
dtΦ(x, t) =u(Φ(x, t), t), Φ(x,0) =x.
We consider the half-line problem and assume u(0, t) = 0 and v(x,0) = v0. If v0≥0, then
Φ(x, t) = Z x
0
(1 +12v0(y)t)2dy, u(Φ(x, t), t) =
Z x 0
(1 +12v0(y)t)v0(y)dy, v(Φ(x, t), t) = 2v0(x)
2 +v0(x)t.
In contrast to hyperbolic conservation laws where characteristics in general will collide, the characteristics for the Hunter–Saxton equation will only focus, that is, approach the same tangent.
Smooth solutions of (1.3) can be expressed as the solution of a system (see [6]) u=u0(ξ) +tg(ξ) +h0(ξ),
x=ξ+tu0(ξ) +1
2t2g(ξ) +h(ξ),
where his any function withh(0) =h0(0) = 0, andg0(ξ) = 12u00(ξ)2. However, the Hunter–Saxton equation will not in general enjoy classical solutions. More precisely, ifu0isnotmonotone increasing, then
inf(ux)→ −∞ast↑t∗= 2/sup(−u00). (1.4) The concept of a weak solution is more complicated. Two different solution concepts can be found in the literature, namely that of aconservative solution and that of a dissipative solution (see Hunter and Zheng [6, 7] and Zhang and Zheng [16]). Before we recall these definitions and for later reference, let us state the problem that we intend to study in this paper, i.e., the Hunter–Saxton equation augmented with appropriate initial and boundary conditions:
vt+uvx=−1
2v2, ux=v, (x, t)∈QT, v(x,0) =v0(x), x∈R+,
u(0, t) = 0, t∈[0, T],
(1.5)
whereT >0 is a fixed final time (T =∞is possible) andQT denotes the space-time cylinderR+×(0, T), where R+= (0,∞). Sometimes we also use the notationQT
for the setR+0 ×[0, T], whereR+0 is short-hand for the half-closed interval [0,∞).
Conservative solutions of (1.5) are defined as triplets (v, u,Φ) satisfying v∈L∞((0, T);L2(R+)), u∈C(QT), Φ,Φt∈C(QT),
vt+ (uv)x=12v2, ux=v (v2)t+ (uv2)x= 0
)
in the sense of distributions onQT,
∂tΦ(x, t) =u(Φ(x, t), t), Φ0(x) =x, Z Φ(x2,t)
Φ(x1,t)
v(y, t)2dy=
Z Φ(x2,0) Φ(x1,0)
v(y,0)2dy, x1< x2.
Moreover, the function u(x, t) is zero at x = 0 as a continuous function in x for eacht∈[0, T], while the functionv(x, t) takes on the initial datav0(x) att= 0 in
the sense of C(R+, L1(R+)). Since we are not interested in conservative solutions in this paper, we refer to the papers [6, 7, 14, 15, 16] by Hunter, Zhang, and Zheng for more information about them and their properties.
However, in this paper we are going to work with dissipative solutions, so we choose to state this notion of solution explicitly in a definition. It is convenient first to define a weak solution.
Definition 1.1. A pair of functions (v, u) is a weak solution of (1.5) provided v∈L∞((0, T);L2(R+)), u∈C(QT),
vt+ (uv)x=1
2v2 andux=v in the sense of distributions onQT, Z ∞
0
v(x, t)2dx≤ Z ∞
0
v0(x)2dx for almost allt∈(0, T), u(x, t)→0 asx→0 for eacht∈[0, T],
v(x, t)→v0(x) inC([0, T];L1(R+)) ast→0+.
Definition 1.2. A pair of functions (v, u) is a dissipative solution of (1.5) provided the pair (v, u) is a weak solution of (1.5) and
v≤2
t a.e. onQT.
Dissipative solutions of the Hunter–Saxton equation are well understood, and we refer to a long series of papers by Hunter, Zhang, and Zheng [6, 7, 14, 15, 16]
for various types of results. This series culminated with the paper [16] by Zhang and Zheng, which established the existence and uniqueness of dissipative solutions for the (natural) case of pure L2 initial datav0.
Thus the Hunter–Saxton equation is well studied from a mathematical point of view. However, there has been no rigorous analysis of numerical schemes for (1.5).
For general initial data, there are no closed-form solutions to the Hunter–Saxton equation, and therefore the study of numerical schemes is important. It is the chief purpose of this paper to propose and analyze some numerical schemes of finite difference type for the Hunter–Saxton equation.
The numerical schemes that we propose are deliberately based on discretizing the non-conservative form (1.3) and not the conservative form (1.2). One might expect the latter form to be natural since it can be viewed as a perturbation of Burgers’ equation, where the perturbation takes the form of a non-local integro operator. For Burgers’ equation and other nonlinear conservation laws there exist a rich literature on several types of numerical schemes. Many of the schemes devel- oped for conservation laws are also accompanied by a sound theoretical foundation, sometimes using rather sophisticated analytical tools like, for example, compen- sated compactness. However, we have not be able to prove that any “reasonable”
finite difference scheme based on the conservation law form (1.2) converges to a dis- sipative solution. For this reason we will focus exclusively on the form (1.3), which is a linear transport equation with a quadratic right-hand side plus an additional side constraint relating the derivative of the “velocity” uto the unknownv.
Let us be a bit more precise about our achievements in this paper. In the case where the initial data v0 is a non-negative function in L1∩Lq with q > 2, we describe semi-discrete, implicit, and explicit upwind finite difference schemes, and for all these schemes we show that the corresponding approximate solutions converge to the unique dissipative solution of the Hunter–Saxton equation (1.5).
Then we consider the more complicated case where v0 does not have a definite sign and merely belongs toL1∩L2. Here we define a semi-discrete upwind scheme
and again we show that the suggested scheme converges to the unique dissipative solution of the Hunter–Saxton equation.
The fact that our numerical schemes are of upwind type means that the finite differencing of the transport part uvxis biased in the direction of incoming waves, making it possible to resolve discontinuities without excessive smearing. We stress that the use of upwind schemes is quite natural, as one would expect them to dissipate the energy and as such generate dissipative solutions in the limit as the discretization parameters tend to zero. Our analysis confirms this intuition.
Regarding the convergence analysis, we derive several a priori estimates in Lebesgue and Sobolev spaces, which yield some basic convergence results. An interesting mathematical issue is that we need to prove that the spatial derivative of the nu- merical solutions, i.e.,v∆x= (u∆x)x, which only is weakly compact a priori, in fact converges strongly. Strong convergence of v∆x is necessary if we want to recover the Hunter–Saxton equation when we take the limit in the finite difference schemes.
Strong convergence ofv∆xis obtained by analyzing various renormalizations (in the sense of DiPerna and Lions [2, 10, 11]) of the numerical schemes and correspond- ing defect measures. In addition, to preventv2∆x from exhibiting concentrations as
∆x→0, we need to derive higher (thanL2) integrability estimates for v∆x. Our arguments can be viewed as discrete counterparts of those employed by Zhang and Zheng [14, 15, 16] to prove existence of a dissipative solution.
The organization of this paper goes as follows: In Section 2 we introduce some (finite difference) notation and recall a few well known mathematical results useful for the subsequent analysis. In Section 3 we present and analyze the semi-discrete scheme. The particular form of the scheme and the analysis rely on the assumption that the initial data are nonnegative and belongs toL1∩Lqfor someq >2. Sections 4 and 5 are devoted to similar analyses for implicit and explcit upwind schemes. In Section 6 we extend our analysis to the case of initial data in L1∩L2. Finally, in Section 7 we present several numerical examples, which demonstrate the proposed numerical schemes as well as some other schemes which do not have a theoretical foundation.
2. Some preliminaries
We start by introducing some notation needed to define the finite difference schemes. Throughout this paper we reserve ∆x and ∆t to denote two small pos- itive numbers that represent the spatial and temporal discretization parameters, respectively, of the numerical schemes. Given ∆x,∆t >0, letD± denote the dis- crete forward and backward differences, respectively, in the spatial direction, i.e.,
D±g(x) =± 1
∆x g(x±∆x)−g(x)
for any functiong: R→Radmitting pointvalues. Similarly, we letD±t denote the forward and backward differences, respectively, in the time direction, i.e.,
Dt±h(x, t) =± 1
∆t h(x, t±∆t)−h(x, t) for any functionh:QT →Radmitting pointvalues.
Forj∈N0=N∪ {0}we setxj =j∆x, and forn= 0,1, . . . , N, whereN∆t=T for some fixed time horizonT >0, we settn=n∆t.
For any function g = g(x) admitting pointvalues we write gj = g(xj), and similarly for any functionh=h(x, t) admitting pointvalues we writehnj =h(xj, tn).
Furthermore, let us introduce the spatial and temporal grid cells Ij= [xj−1/2, xj+1/2), Ijn =Ij×[tn, tn+1),
wherexj±1/2=xj±∆x/2. Thus in this notation,D±gj=±(gj±1−gj)/∆x. Also, a discrete Leibniz rule holds
D±(gjhj) =gjD±hj+hj±1D±gj. (2.1) If we extend a sequence {gj}j∈
N0 to a piecewise constant function defined on R+0
(actually on [−∆x/2,∞)) by
g∆x(x) = X
j∈N0
gj1Ij(x), (2.2)
where 1A denotes the characteristic function of the setA, viz.
1A(x) =
(1, forx∈A, 0, forx /∈A, then clearly
kg∆xkLp(R+)=
∆xX
j∈N0
|gj|p1/p . Letf be aC2function. By using a Taylor expansion we find
f(b) =f(a) +f0(a)(b−a) +1
2f00(ξ)(b−a)2 (2.3) for some ξ betweena andb. Let {vj}j∈N
0 be a given sequence. Using the Taylor expansion (2.3) on the sequence{f(vj)}j∈
N0 we obtain D±f(vj) =f0(vj)D±vj±∆x
2 f00(ξj±)(D±vj)2 (2.4) for someξ±j betweenvj±1 andvj. We will make frequent use of (2.4), which states that a discrete chain rule holds up to an error term of order ∆x(D±vj)2.
In this paper we will exploit some well-known results related to weak convergence and convex functions. For the convenience of the reader we collect these results in a lemma (for proofs, see, for example, [3]).
Lemma 2.1. LetO be a bounded open subset ofRM, withM ≥1.
Let {vn}n≥1 be a sequence of measurable functions onO for which sup
n≥1
Z
O
Φ(|vn(y)|)dy <∞,
for some continuous functionΦ : [0,∞)→[0,∞). Then there exists a subsequence (which is not relabeled) such that
g(vn)* g(v)in L1(O) for all continuous functions g:R→Rsatisfying
lim
|v|→∞
|g(v)|
Φ(|v|) = 0.
Let g:R → (−∞,∞] be a lower semicontinuous convex function and {vn}n≥1 a sequence of measurable functions on O, for which
vn * v inL1(O),g(vn)∈L1(O)for each n,g(vn)* g(v)inL1(O).
Then
g(v)≤g(v)a.e. onO.
Moreover, g(v)∈L1(O)and Z
O
g(v)(y)dy≤lim inf
n→∞
Z
O
g(vn)(y)dy.
If, in addition, g is strictly convex on an open interval(a, b)⊂Rand g(v) =g(v)a.e. onO,
then, passing to a subsequence if necessary,
vn(y)→v(y)for a.e.y∈ {y∈O|v(y)∈(a, b)}.
Occasionally we will use the following standard interpolation inequality.
Lemma 2.2. Let O be an open subset of RM, with M ≥ 1. Let 1 ≤p0 < pθ <
p1≤ ∞,θ∈(0,1), and
1 pθ
= θ p0
+1−θ p1
. Then, for any v∈Lp0(O)∩Lp1(O),
kvkLpθ(O)≤ kvkθLp0(O)kvk1−θLp1(O)≤ kvkLp0(O)+kvkLp1(O).
Finally, let us recall the definition of a standard mollifier, which will be used several times in this paper. Let ω(x) be a smooth non-negative function with support inside [−1,1],ω(−x) =ω(x), andR
ω dx= 1. Then a standard mollifier ωε=ωε(x),ε >0, is defined by
ωε(x) =1 εωx
ε
, x∈R. 3. The semi-discrete upwind scheme
In this section we present and analyze the semi-discrete scheme, relying on the notation introduced in Section 2.
For the analysis in this section we assume that the initial function satisfies v0≥0 andv0∈L1(R+)∩Lq(R+) for someq >2. (3.1) By interpolation the functionv0belongs toLp(R+) for any 1≤p≤q. The general case where v0 belongs merely to L2 and may change sign is more involved and will be treated in Section 6. The L1 requirement is a natural replacement of the compact support condition on v0 used by Zhang and Zheng [14, 15, 16].
Let vj0
j∈N0 be sequence of discrete initial data chosen such that v∆x0 (x) = X
j∈N0
vj01Ij(x) (3.2)
converges to the initial datav0inL2(R+) as ∆x→0. We make the approximation such thatv0j ≥0 and vj0= 0 for j > J∆x:= 1/(∆x2). It is not hard to construct such a sequence. For example, we may take
vj0= 1
∆x Z
Ij
v0(x)dx, j = 1,2, . . . , J∆x,
and set v00 = v01 and v0j = 0 for all j ≥ J∆x. For t ≥ 0, let {(vj(t), uj(t))}j∈
N0
satisfy the finite system of ordinary differential equations
˙
vj+ujD−vj=−1
2(vj)2, j∈[0, J∆x], vj = 0, j > J∆x, D+uj =vj, j∈N0, u0(t) = 0,
vj|t=0=v0j, j∈[0, J∆x],
(3.3)
where ˙vj denotes differentiation ofvj with respect tot. Whenever it is convenient we also extend vj anduj to be zero forj <0. Observe that it follows from (3.3) that
uj(t) = ∆x
j−1
X
i=0
vi(t) forj∈N. Using the discrete Leibniz rule (2.1), we have
D−(ujvj) =ujD−vj+vj−1D−uj =ujD−vj+ (vj−1)2, and hence we may write the scheme (3.3) in conservative form
˙
vj+D−(ujvj) =1
2(vj)2+ (vj−1)2−(vj)2= 1
2(vj)2−∆x D−(vj)2. (3.4) For positive ∆x, equation (3.3) is a finite dimensional system of ordinary differential equations, which has a C1 solution at least until some blowup time. Below (see Lemma 3.2) we shall show that blowup does not happen. For the convergence analysis, we need to introduce the two pointwise defined functions
v∆x(x, t) = X
j∈N0
vj(t)1Ij(x) and u∆x(x, t) = Z x
0
v∆x(y, t)dy, (3.5) which are piecewise constant and piecewise linear and continuous, respectively.
Before we continue we need to establish that the numerical solution (u∆x, v∆x) remains nonnegative if it initially started so. We also prove that v∆x is bounded from above, independently of ∆x, as soon as t > 0. This latter estimate is a consequence of an Ole˘ınik type (one-sided Lipschitz) estimate for u∆x. Besides ensuring uniqueness of the dissipative solution, the Ole˘ınik type estimate is not used directly in the convergence proof in this and the next two sections. It will however play an important role in the convergence proof in Section 6, where we allowv0(and thus the solution) to change sign. We emphasize that for the arguments in this and the next two sections it is important that the functionsu∆x, v∆xare nonnegative.
Lemma 3.1. Fort >0 andj∈N0 we have 0≤vj(t)≤ 2
t. (3.6)
Proof. We have thatv0(0) =v00≥0. Since
˙ v0=−1
2(v0)2,
it trivially follows that v0(t) ≥ 0 for all t. Let t0 ≥ 0 and k > 0 be such that vk(t0) = 0, andvj(t0)≥0 for allj < k. ThenD−vk(t0)≤0 anduk(t0)≥0, and hence
˙
vk(t0) =−ukD−vk(t0)≥0.
Hence vj(t)≥0 anduj(t)≥0 for allj andt.
Set
¯k(t) = sup{k|vk(t)≥vj(t) for all j}
and ¯v∆x(t) =v¯k(t)(t). Since ¯v∆x(t) is the maximum of a finite number of continu- ously differentiable functions, it is continuous and differentiable almost everywhere.
At every differentiable point, we know that d
dt¯v∆x(t)≤ −1
2¯v∆x(t)2,
since if ¯k >0, then D−vk¯(t)≥0, while if ¯k= 0 the above inequality is an equality.
Now the comparison principle for ordinary differential equations yields the last
inequality of the lemma.
Let f: R→Rbe a twice continuously differentiable function. Multiplying the scheme (3.3) byf0(vj) and using the discrete chain rule (2.4), we find that
d
dtf(vj) +ujD−f(vj) +∆x
2 ujf00(ξj)(D−vj)2=−1
2f0(vj)(vj)2. (3.7) This is our main tool for proving the next lemma, which collects some uniform a priori estimates satisfied by the numerical approximations.
Lemma 3.2. Suppose (3.1)holds. Then for anyt >0 we have
kv∆x(·, t)kLp(R+)≤ kv∆x(·,0)kLp(R+)≤C, p∈[2, q]. (3.8) Furthermore, there holds
kv∆xkLq+1(QT)≤ 2
q−2kv∆x(·,0)kLq(R+)≤C.
For any t >0 there holds
kv∆x(·, t)kL1(R+)≤ kv∆x(·,0)kL1(R+)+t
2kv∆x(·,0)kL2(R+)≤C(t).
For any t >0 there holds
ku∆x(·, t)kL∞(R+)≤ kv∆x(·,0)kL1(R+)+ t
2kv∆x(·,0)kL2(R+)≤C(t).
Remark 3.3. The first estimate (3.8) also states that the approximate solutions remain inside some ball in “RJ∆x”, and thus do not blow up. Therefore the solution of the system of ordinary differential equations (3.3) exists for allt >0.
Proof. Choosing f(v) =vpin (3.7), we obtain d
dt(vj)p+ujD−(vj)p+p(p−1)
2 ujξp−2j (D−vj)2∆x=−p
2(vj)p+1 (3.9) with ξ = {ξj}j∈N0 being a sequence of nonnegative numbers. Multiplying (3.9) with ∆xand summing overj, yields (using that uj and vj are non-negative) the fundamental identity
d
dtkvp∆x(·, t)kL1(R+)+p(p−1)
2 (∆x)2X
j
ujξjp−2(D−vj)2
=−∆xX
j
ujD−(vj)p+p 2
v∆xp+1(·, t) L1(
R+)
= ∆xX
j
(D+uj)(vj)p−p 2
vp+1∆x (·, t) L1(
R+)
= 1−p
2
v∆xp+1(·, t) L1(
R+).
(3.10)
Integrating (3.10) from 0 tot we end up with kv∆xp (·, t)kL1(
R+)+ (∆x)2p(p−1) 2
Z t 0
X
j
ujξjp−2(D−vj)2 ds
= 1−p
2 v∆xp+1
L1(Q
T)+kvp∆x(·,0)kL1(R+)≤C,
(3.11)
for some constantCindependent of ∆x. As the second term of the left-hand side is nonnegative and the first term on the right-hand side is nonpositive, (3.11) implies that the first and second claims of the lemma hold.
Next, we setp= 1 in (3.10) and (3.8) withp= 2 to obtain d
dtkv∆x(·, t)kL1(R+)= 1
2kv∆x(·, t)k2L2(R+)≤ 1
2kv∆x(·,0)k2L2(R+),
which proves the third claim. The fourth claim follows from the third one, since
|u∆x(x, t)| ≤ kv∆x(·, t)kL1(R+)
≤ kv∆x(·,0)kL1(R+)+t
2kv∆x(·,0)k2L2(R+).
Using the estimates above we can prove some useful convergence results.
Lemma 3.4. Supposev0 satisfies the conditions in (3.1). Extracting subsequences if necessary, we have the following basic convergence results as ∆x→0:
u∆x→uuniformly in[0, R]×[0, T] for eachR >0and pointwise in QT
and the limitubelongs toW1,q+1(QT); (3.12)
v∆x=∂xu∆x* ∂xu=:v inLq+1(QT) andv∆x=∂xu∆x
* ∂? xu=:v inL∞((0, T);L1(R+)∩Lq(R+)); (3.13) (v∆x)2* winLq+12 (QT)
and(v∆x)2* w? in L∞((0, T);L1(R+)∩Lq2(R+)); (3.14) u∆xv∆x* uv inLq+1(QT)
andu∆xv∆x
* uv? inL∞((0, T);L1(R+)∩Lq(R+)). (3.15) Proof. The second part of Lemma 3.2 shows that ∂xu∆x = v∆x is bounded in Lq+1(QT) independently of ∆x. Next, we bound ∂tu∆x. Recalling that u−1 = v−1= 0, we find that
d
dtuj = ∆x
j−1
X
i=0
˙ vi
= ∆x
j−1
X
i=0
−D−(uivi) +1
2(vi)2−∆xD−v2i
=−uj−1vj−1−∆xv2j−1+∆x 2
j−1
X
i=0
v2i. Thus, using (3.8), we find
d dtuj
≤ ku∆xkL∞(QT)|vj−1|+kv∆x(·,0)kL2(R+).
Fix anyR >0 and letJ be an integer such thatJ∆x≤R. Then it follows that
∆x
J
X
j=0
d dtuj
q+1
≤C1+kv∆xkq+1Lq+1(QT)≤C2,
where C1 and C2 depend on R but are independent of ∆x. Consequently, u∆x is uniformly bounded inW1,q+1([0, R]×[0, T]), a space which is compactly embedded into the H¨older spaceC0,`([0, R]×[0, T]), where`= 1−2/(q+ 1). In other words, there exists a continuous function u:QT →Rsuch that the following convergence holds, extracting a subsequence if necessary,
u∆x→uuniformly on [0, R]×[0, T] and pointwise inQT as ∆x→0.
Now the claim (3.12) follows from this and a standard diagonal argument on a sequenceR`→ ∞.
The claims (3.13) and (3.14) are consequences of the uniform Lq+1 bound on v∆x, while (3.15) holds thanks to (3.12) and (3.13).
Remark 3.5. By the weak lower semicontinuity property of norms, the limitsu, v inherit the a priori bounds in Lemma 3.2, that is, Lemma 3.2 holds withu∆x, v∆x
replaced byu, v, respectively.
We are going to prove strong convergence of{v∆x}∆x>0by analyzing a particular renormalization (in the sense of DiPerna–Lions) of the numerical scheme and its limit. As mentioned before, strong convergence is needed if we want to prove that the weak limitv solves the Hunter–Saxton equation.
Lemma 3.6. The limit triplet(v, u, w)from Lemma 3.4 satisfies vt+ (uv)x= 1
2w, ux=v (3.16)
in the sense of distributions onQT, and v∈C([0, T];Lp(R+)), lim
t→0kv(·, t)−v0kLp(R+)= 0, (3.17) for any p∈[1, q]. Moreover,
wt+ (uw)x≤0 (3.18)
in the sense of distributions onQT and limt→0
Z ∞ 0
w(x, t)−v0(x)2
dx= 0. (3.19)
Proof. Set
ϕj(t) = 1
∆x Z
Ij
ϕ(x, t)dx,
where ϕis a non-negative test function, that is, 0≤ ϕ∈Cc∞(QT). We multiply (3.4) with ∆xϕj, integrate from 0 toT and sum overj, obtaining
− Z T
0
∆xX
j∈N0
vjϕ0jdt− Z T
0
∆xX
j∈N0
ujvjD+ϕjdt
= Z T
0
∆xX
j∈N0
1
2(vj)2ϕjdt+ Z T
0
∆xX
j∈N0
(vj)2∆xD+ϕjdt
| {z }
E1
, (3.20)
after a partial integration in t and a partial summation inj. Due to the choice of ϕj, we can rewrite this as
− Z Z
QT
v∆xϕt+u∆xv∆xϕx+1 2v2∆xϕ
dtdx
=E1+ Z T
0
X
j∈N0
∆xujvjD+ϕj− Z
Ij
v∆xu∆xϕxdx
dt
| {z }
E2
.
To establish (3.16), we must show that lim∆x→0(E1+E2) = 0. Observe E1≤∆xkϕxkL∞(QT)
Z T 0
Z ∞ 0
v2∆x(x, t)dx dt
≤∆xkϕxkL∞(QT)
Z T 0
Z ∞ 0
v2∆x(x,0)dx dt
= ∆xkϕxkL∞(QT)Tkv∆x(·,0)kL2(R+),
and thusE1 vanishes with ∆x. RegardingE2 we have that the integrand equals X
j∈N0
vj
Z
Ij
ujD+ϕj−u∆xϕx
| {z }
A
dx.
We split the integrandAabove by writing
A= (uj−u∆x)D+ϕj+u∆x(D+ϕj−ϕx). Forx∈Ij we have
uj(t)−u∆x(x, t) = (xj−1/2−x)vj, and
D+ϕj(t)−ϕx(x, t) = 1
∆x Z
Ij
ϕ(y+ ∆x, t)−ϕ(y, t)
∆x −ϕx(x, t)
dy
= 1
(∆x)2 Z
Ij
Z y+∆x y
ϕx(z, t)−ϕx(x, t) dzdy
= 1
(∆x)2 Z
Ij
Z y+∆x y
Z z x
ϕxx(w, t)dw dz dy.
Therefore
|D+ϕj(t)−ϕx(x, t)| ≤ kϕxxkL∞(QT)∆x.
Collecting this we find that
|E2| ≤ Z T
0
X
j∈N0
vj Z
Ij
h
xj−1/2−x
vjkϕxkL∞(QT)+ ∆xkϕxxkL∞(QT)
i dx dt
≤ 1
2kϕxkL∞(QT)+kϕxxkL∞(QT)
∆x Z T
0
X
j∈N0
vj∆x dt
≤∆x 1
2kϕxkL∞(QT)+kϕxxkL∞(QT)
Tku∆xkL∞(QT).
From this we see that alsoE2 vanishes when ∆xbecomes small, and the first part of (3.16) holds. The second part of (3.16) follows from (3.13).
To prove the time continuity/initial data statements (3.17) we can apply stan- dard renormalization arguments, see for example [14, 15].
To prove that (3.18) holds, we recall that we have a scheme for (vj)2 using (3.9) with p= 2:
d
dt(vj)2+ujD−(vj)2+uj(D−vj)2∆x=−(vj)3. Using the Leibniz identity (2.1), we can rewrite this as
d
dt(vj)2+D− uj(vj)2
+uj(D−vj)2∆x=−∆xD−(vj)3. (3.21) The third term above is certainly non-negative, so after multiplying with ∆x ϕj, summing overj and integrating over t, we find that
− Z Z
QT
v∆x2 ϕt+u∆xv∆x2 ϕx dx dt
≤∆xkϕxkL∞(QT)
Z Z
QT
v3∆xdx+ Z T
0
X
j∈N0
∆xuj(vj)2D+ϕj− Z
Ij
u∆xv2∆xϕxdx
dt
| {z }
E3
.
Sincev∆x∈L3(QT) with anL3norm that is independent of ∆x, the first term on the right-hand side vanishes with ∆x. The second termE3 is similar to E2, the only difference being that we havev∆x2 instead ofv∆x. Hence we can boundE3 as
|E3| ≤ 1
2kϕxkL∞(QT)+kϕxxkL∞(QT)
∆x Z T
0
kv∆x(·, t)k2L2(R+)dt
≤∆x 1
2kϕxkL∞(QT)+kϕxxkL∞(QT)
Tkv∆x(·,0)k2L2(R+). Consequently, lim∆x→0E3= 0 and (3.18) holds.
Finally, let us prove (3.19), which also follows from standard arguments. Thanks to (3.13), we have that
v2(x, t)≤w(x, t) for a.e. in (x, t)∈QT, (3.22) so that by the energy estimate (the first part of Lemma 3.2 with p= 2, cf. also Remark 3.5) we obtain
t→0lim Z ∞
0
v(x, t)2dx≤lim inf
t→0
Z ∞ 0
w(x, t)dx
≤lim sup
t→0
Z ∞ 0
w(x, t)dx≤ Z ∞
0
v0(x)2dx.
On the other hand, (3.17) yields
t→0lim Z ∞
0
v(x, t)2dx= Z ∞
0
v0(x)2dx,
which finishes the proof of (3.19).
We state and prove the next lemma in a form that is slightly more general than what we actually need in this section to conclude that the sequence {v∆x}∆x>0 is strongly convergent.
Lemma 3.7. Supposeuis bounded and continuous inQT with u(0, t) = 0fort∈ [0, T],v∈L∞((0, T);L2(R+))∩L3(QT),v≥0a.e. inQT,w∈L∞((0, T);L1(R+))∩
L32(QT), andw≥v2 a.e. inQT. Assume that
t→0lim Z ∞
0
w−v2
(·, t)dx= 0 (3.23)
and that the triplet(v, u, w)satisfies the system vt+ (uv)x= 1
2w, (3.24)
wt+ (uw)x≤0, (3.25)
ux=v (3.26)
in the sense of distributions onQT. Then
w=v2 a.e. in QT.
Proof. The proof is a standard exercise in the theory of renormalized solutions, so we include it only for the sake of completeness. Set vε = v ? ωε, wε = w ? ωε, where ωε is a standard mollifier acting on the spatial variable. Then according to the DiPerna–Lions folklore lemma [2], as well as (3.24) and (3.26),vεsolves
vtε+uvxε=1
2wε−(vε)2+rε, (3.27) where rε=uvεx−(uvx)? ωε+ (vε)2−v2? ωε and
rε→0 inLp(QT) for anyp∈[1,3/2].
Multiplying this equation by vεwe get (vε)2
2
t
+u (vε)2
2
x
= 1
2wεvε−(vε)3+rεvε,
or, thanks to (3.26), (vε)2
2
t
+
u(vε)2 2
x
=1
2wεvε−(vε)3+v(vε)2
2 +rεvε. Sending ε↓0, we obtain
v2
t+ uv2
x=w v−v3= w−v2 v≥0,
where we have used (3.22) to derive the last inequality. Comparing this inequality with (3.25), keeping in mind that w≥v2a.e. inQT, we find
w−v2
t+ u w−v2
x≤0
in the sense of distributions onQT. In particular, this implies that Z T
0
Z ∞ 0
w−v2
(x, t)∂tψ dx dt≥0
for any non-negativeψ∈Cc∞((0, T)). Hence, for any two Lebesgue pointst1, t2∈ (0, T),t1< t2, of theL1 function
(0, T)3t7→
Z ∞ 0
w−v2
(x, t)dx, we obtain
Z ∞ 0
w−v2
(x, t2)dx≤ Z ∞
0
w−v2
(x, t1)dx,
and combining this with (3.23) we have proved the lemma.
We summarize our findings in the following main theorem.
Theorem 3.8. Letv0be a function satisfying (3.1). Define the semi-discrete finite difference approximation (v∆x, u∆x) for ∆x positive using (3.5), (3.2), and (3.3).
Then {(v∆x, u∆x)}∆x>0 converges to a dissipative solution (v, u) of (1.5) in the sense of Definition 1.2. More precisely, as ∆x→0
ku∆x−ukL∞(QT)→0, kv∆x−vkLp(QT)→0 for anyp∈[1, q+ 1). (3.28) Proof. In view of Lemmas 3.6, 3.7, and 2.1 we conclude that w = v2 a.e. in QT
and that there exists a subsequence of {v∆x}∆x>0 that converges tov a.e. in QT, where vis the (weak) limit from Lemma 3.4. Moreover, Lemma 3.4 implies that
v∈L∞((0, T);Lp(R+))∩Lq+1(QT)∩C([0, T];Lp(R+)), p∈[1, q], which clearly proves the second part of (3.28). The first part follows from (3.12).
The fact that the limit (v, u) solves the Hunter–Saxton equation (1.5) in the sense of distributions (i.e., the second requirement in Definition 1.2) follows from Lemma 3.6 and the identification w=v2a.e. inQT.
The remaining requirements in Definition 1.2 are straightforward consequences of the subsequent strong convergence of{v∆x}∆x>0and Lemmas 3.1–3.6.
For any given sequence we have proved that we can find a subsequence ∆xj →0 for which all statements hold. However, Zhang and Zheng [16] have proved that the Hunter–Saxton equation has a unique global dissipative solution. Hence the limit exists for all subsequences, which concludes the proof of the theorem.
Remark 3.9. In addition to the properties stated in Theorem 3.8, the proof also shows that the limitsu, v possess the following properties:
u∈W1.q+1(QT),
v∈L∞((0, T);Lp(R+))∩Lq+1(QT), p∈[1, q], v∈C([0, T];Lp(R+)), p∈[1, q].
Moreover, since v0 ∈ Lq(R+) with q > 2, which implies v ∈ Lq+1(QT) and in particularv∈L3([0, R]×[0, T]) for anyR >0, the dissipative solution constructed in Theorem 3.8 is energy conservative, that is, for anyt >0
Z ∞ 0
v(x, t)2dx= Z ∞
0
v0(x)2dx.
Formally this is obtained by multiplying the equation forv byv, which gives
∂t
v2 2
+∂x
uv2
2
= 0,
from which the claim follows. To make this argument rigorous one appeals to the DiPerna–Lions folklore lemma [2] and the local L3 estimate onv.
4. The implicit upwind scheme
In this section we show how to extend the convergence analysis from the previous section to an implicit upwind difference scheme, where we still work under the initial data assumption (3.1). Since many of the arguments are very similar, we have attempted to make this section brief.
Refering to Section 2 for the notation, the implicit finite difference solution vnj, unj
|j∈N0, n= 0,1,2, . . . , N is defined by
Dt+vnj +un+1j D−vjn+1=−1
2 vjn+12
, D+un+1j =vn+1j , (4.1) for 0≤j≤J∆xand n= 0, . . . , N−1, where we have setvnj = 0 forj > J∆xand set v−1 = 0. The final step N is chosen such that N∆t =T. The initial values vj0
j∈N0 are defined as in Section 3 and boundary values as specified asun0 = 0 for n= 0,1, . . . , N. Based on
(vnj, unj) we define the functions v∆x and u∆x as in Section 3 by
v∆x(x, t) = X
j∈N0
n=0,...,N
vjn1Ijn, and u∆x(x, t) = Z x
0
v∆x(y, t)dy. (4.2)
As for the semi-implicit scheme, we can derive a conservative form of (4.1):
Dt+vjn+D− un+1j vjn+1
=1
2(vn+1j )2−∆xD−(vn+1j )2. (4.3) We can solve (4.1) “upwards from left to right”, by rewriting it as
un+10 = 0, (4.4)
un+1j =un+1j−1 + ∆xvj−1n+1, 0< j≤J∆x, 0≤n vn+1j = 1
∆t q
1 +λun+1j 2
+ 2∆t vjn+λun+1j vj−1n+1
− 1 +λun+1j
(4.5) where (the constant) λ = ∆t/∆x. We have chosen the plus sign in front of the square root, since otherwise vjn+1 would be negative.
Lemma 4.1. Assume that the initial approximations are chosen so that
∆t→0lim ∆tmax
j
vj0 = 0.
Then for n∈Nandj∈N0 we have
0≤vnj ≤ 2
tnK∆t, (4.6)
where tn=n∆tand{K∆t}is a bounded sequence such that lim∆t→0K∆t= 1.
Proof. From (4.4) and (4.5) it is straightforward to see that if v0j ≥0, then also vnj ≥0 andunj ≥0. In order to show the upper bound, note first that if vn+1j ≥ vn+1j−1, thenun+1j D−vjn+1≥0, and hence, using (4.1),
vn+1j ≤vnj −∆t
2 vn+1j 2
, or vjn+1≤ 1
∆t
hq1 + 2∆tvjn−1i . Set ¯vn = maxj
vjn . Sincevn+1j ≥vj−1n+1 ifvjn+1= ¯vn+1, we deduce that
¯
vn+1=vn+1j ≤ 1
∆t
hp1 + 2∆t¯vn−1i
= 2¯vn
√1 + 2∆t¯vn+ 1. Thus in particular we see that ¯vn≤¯v0, and we can use this to deduce that
1
∆t(¯vn+1−v¯n)≤ 1
∆t
2¯vn
√1 + 2∆t¯vn+ 1 −¯vn
= v¯n
∆t
1−√
1 + 2∆t¯vn 1 +√
1 + 2∆t¯vn
=− 2¯vn2 (1 +√
1 + 2∆t¯vn)2
≤ −1 2¯vn2
2 1 +√
1 + 2∆t¯v0
2
=−1
2¯vn2K∆t. (4.7)
Applying (2.4) withf(v) = 1/vwe find Dt+ 1
¯ vn
−∆t
ξ3n Dt+¯vn2
≥1 2K∆t. Multiplying by ∆tand summing over the time variable yields
1
¯ vn+1
≥ 1
¯ v0
+tnK∆t 2 +P, where
P = ∆t2
n
X
j=0
1
ξj3 D+tv¯j2
≥0.
Rearranging we finally get
¯
vn+1≤ 2¯v0
¯
v0tnK∆t+ 2 + 2¯v0P ≤ 2 K∆ttn. Using that
∆t¯v0→0 as ∆t→0,
i.e., K∆t→1, we conclude the proof.
Similarly to (3.7), if we multiply the scheme (4.1) byf0(vjn+1) we get D+tf vnj
+un+1j D−f vjn+1
(4.8) +1
2
h∆tf00(ηnj) Dt+vjn2
+ ∆xf00(ξjn+1) D−vjn+12i
=−1
2f0 vn+1j
vjn+12 , where ηjn is betweenvjn+1 andvjn, and ξjn betweenvjn andvj−1n . With this we can show the following result.
Lemma 4.2. Lemma 3.2 holds also forv∆x andu∆x defined by (4.2)and (4.1).
