Time global stability of theLax-Friedrichs scheme
Kohei SOGA(Waseda University)
第５回日独流体数学国際研究集会

2012.6.14

Introduction
Let c, h(c) ∈ R be given constants and consider
ut + H(x, t, c + u)x = 0 in T × (0, T ],
0(x) ∈ L∞(T) on T := R/Z,(CL)u(x,0)=u∫
u0dx = 0, k u0 kL∞ ≤ r.



(HJ)

Tvt + H(x, t, c + vx) = h(c) in T × (0, T ],v(x, 0) = v 0(x) ∈ Lip(T) on T, k vx0 kL∞ ≤ r.

If u0 = vx0, then∃/ u ∈ C 0((0, T ]; L∞): entropy sol.,∃/ v ∈ Lip(T × [0, T ]): viscosity sol. s.t. vx = u.
(CL) is well-studied with the variational structures of (HJ).

Consider discretization of (CL) and (HJ) by the L-F scheme:{kDtuk+1+DH(x,t,c+uxm+2km+2) = 0m+1(CL)∆k , u0 = u0 (x ).
ukm±2N = vmm∆ m
k+1k Dtvm+ H(xm, tk , c + Dxvm+1) = h(c)(HJ)∆kk00 vm+1±2N = vm+1, vm+1 = v∆(xm+1).
k+1Dtvm:=kk
k+1 vm−1 +vm+1vm −2∆tklvm+1−vm−1
k, Dxvm+1 :=(Lax-Friedrichs scheme).
2∆xk, if u0 = vx0.
(CL)∆ and (HJ)∆ are equivalent: ukm = Dxvm+1

We want to show under hyperbolic scaling 0 < λ0 ≤ λ = ∆t/∆x < λ1•

Stability of (CL)∆ for k → ∞ with ﬁxed ∆ = (∆x, ∆t),•

Error estimate between ukm and u for ∆ → 0,•

Asymptotic behavior of (CL)∆ for k → ∞ with ﬁxed ∆.Known Results on (CL)∆(i) The case of H(x, t, p) = f (p)Huge literature ontime global stability,L1-convergence,L1-error estimates,long time behavior.
(ii) The case of H(x, t, p) = f (p) + F (x, t)[T] Takeno (’01): stability within [0, 1] for existence of periodic sol.
[NS] Nishida-S (’12): time global stability, L1-convergence, long timebehavior.
(iii) The case of H(x, t, p)[O] Oleinik (’57): stability within a restricted time interval, L1convergence.
[S] S (submitted): stability within arbitrary time intervals, pointwise
convergence, approximation of characteristic curves.
• All, except [S], are based on L1-framework.
• [S] is based on stochastic and variational methods
with the theory of viscosity sol. of Hamilton-Jacobi eq.

Main Results
Suppose that the C 2-ﬂux function H(x, t, p) : T2 × R → R satisﬁes
H(x, t, p)= ∞, |Lx| ≤ α(1 + |L|) (L := H ∗).
Hpp > 0,lim|p||p|→∞u∆: the step function given by ukm,kv∆: the linear interpolation of vm+1.
Thm. ∃ λ1 > 0 such that if 0 < λ0 ≤ λ := ∆t/∆x < λ1, thefollowing hold:
1. The L-F scheme (CL)∆ is globally stable: For all k ≥ 0 and m−1 (CFL-condition).
<λ|Hp(xm, tk , c + ukm)| ≤ λ−1
1

2. For each t > 0, ∃ α(t) > 0 independent of ∆x, ∆t and initial
1
4datas s.t. k u∆(·, t) − u(·, t) kL1(T)≤ α(t)∆x .
3. For each c, ∃! ūc∆: periodic diﬀerence entropy sol., which absorbs any other diﬀerence entropy sol. exponentially as t → ∞.

Idea of Proof.
1. Use variational techniques, not usual L1-framework,
2. Convert the equation into the Hamilton-Jacobi type with thetheory of viscosity solutions,
3. Use Variational characterization of (CL), (HJ), (CL)∆, (HJ)∆:
Let Lc := L(x, t, ξ) − cξ.
[∫ t](1) v(x, t) =inf
Lc(γ(s), s, γ 0(s))ds+v 0(γ(0)) +h(c)t,γ∈AC,γ(t)=x0(2) If (x, t): regular point of v (i.e. ∃ vx(x, t)) and γ ∗: minimizer,∫ tu(x, t) =Lcx(γ ∗(s), s, γ ∗0(s))ds + u0(γ ∗(0)),0(3) Stochastic variational problems for v∆, u∆ ([S]).
4. Necessary boundedness is derived from calculus of variations,
5. Due to periodic setting, iteration of time-1 analysis yields timeglobal properties.

Stability• Due to calculus of variations, velocity of every minimizing curveγ ∗ for v(·, t) is uniformly bounded: For each t > 0|γ ∗0(t)| ≤ β1(t).

(β1(t) is indep. of r, k vx0 kL∞ ≤ r)

• Since u(x, t) = vx(x, t) = Lcξ (x, t, γ ∗0(t)) a.e. x ∈ T, u is uniformly
bounded: For each t > 0k u(·, t) kL∞ =k vx(·, t) kL∞ ≤ β2(t).
• u∆(·, 1) → u(·, 1) (∆ → 0) in L1(T) is uniform w.r.t. initial datas
bounded by r (k u0 kL∞ ≤ r).
•∃ E(t) > 0 indep. of initial datas bounded by r s.t.
u∆(x, t) − u∆(y, t)≤ E(t).
x−y•(entropy condition)

By [S], stability holds in [0, 1] for all initial datas bounded by r.

Therefore, ∃ δ > 0 s.t. if ∆x, ∆t ≤ δk u∆(·, 1) kL∞ ≤ β2(1) + 1.
Otherwise the entropy condition is violated. Take r ≥ β2(1) + 1.

Error Estimate• Due to stochastic and variational approach [S]√k v∆(·, t) − v(·, t) kC 0 ≤ α1(t) ∆x.
• Due to the entropy condition, u∆(·, t), u(·, t) is B.V. In Particularfor w(·) := u∆(·, t) − u(·, t)∑|w(xj ) − w(xj−1)| ≤ 4E(t).
j• Let 0 = x0 < x1 < x2 < · · · < xn = 1 be s.t. w(·) ≥ 0 on
I2j = [x2j , x2j+1] and < 0 on I2j+1. Deﬁne{1}1−˜˜J := j | max{|I2j |, |I2j+1|} ≤ ∆x 4 , J := {j} \ J. (]J ≤ ∆x 4 )Then∑k u∆(·, t) − u(·, t) kL1(T) ≤max w(x)|I2j | − min w(x)|I2j+1|x∈I2jx∈I2j+1∑∫∫

j∈J+j∈J˜≤

I2jw(x)dx −w(x)dx
I2j+1

√11
˜4E(t)∆x 4 + ]J · 4α1(t) ∆x ≤ α(t)∆x 4 .

Long Time Behavior
0∞0• Initial datas belong to L∞r (T) := {u ∈ L (T) | k u kL∞ ≤ r}.
•Take r ≥ β2(t) + 1. Then the time-1 map of (CL)∆ is∞(T) 3 u0 7→ u (·, 1) ∈ L∞(T).
φ1(·;c):L∆rr∆∆

N into itself and a ﬁxed
• φ1
isessentiallyamapfrom
B(0)⊂Rr∆point ū0
∆ exists.

• ūc∆(·, t) := φt∆(ū0∆; c) is a periodic diﬀerence entropy sol.
•φt∆(·; c) is strictly contractive.

Therefore ūc∆ is unique w.r.t. c and absorbs any other sol.