REMARKS ON THE LONG TIME BEHAVIOUR

OF THE SOLUTIONS OF HAMILTON-JACOBI EQUATIONS

Gawtum Namah^{(1)}, Jean-Michel Roquejoffre^{(2)}

(1) Mathématiques Appliquées de Bordeaux, Université Bordeaux I 33405 TALENCE CEDEX, FRANCE.

(2) UFR MIG, UMR CNRS 5640, Université Paul Sabatier 118 Route de Narbonne, 31062 TOULOUSE CEDEX, FRANCE.

INTRODUCTION

Let M be a smooth compact connected N- dimensional manifold without boundary. The purpose of this note is to study the long-time behaviour of the viscosity solutions of the following Hamilton-Jacobi equation:

(1)  u_{t} = H(X, Du) − f(X),  X ∈ M.

In this direction, there are two kinds of results we are aware of. The first one is a theorem of Dafermos [5], which proves the covergence to steady solutions to nonlinear nonhomogeneous periodic conservation laws in one space dimension; the assumptions on the flux functions are contained in the assumptions (H_{i})_{1≤i≤3} to be stated below. The second series of results is due to Kruzhkov [6], Lions [7] and Barles [1], where the convergence to a steady state solution is proved. They either assume that the Hamiltonian is X-independent, or that it depends on u, but that there is some monotonicity with respect to u.

The functions H and f will satisfy the following assumptions: (H_{1}) • for every (X, p) ∈ M × (IR^{N}\{0}), there holds H(X, p) > 0; furthermore H(X, 0) ≡ 0.

(H_{2}) • The function f(X) is continuous, nonnegative and its zero set is nonempty. (H_{3}) • The hamiltonian H is Lipschitz with respect to both variables p and X, convex with respect to p. Moreover there holds

lim_{|p|→+∞} H(X, p) = +∞, uniformly in X ∈ M.

Recall that (1) subject to the assumptions (H_{i})_{1≤i≤3} and an initial condition u_{0} ∈ C(M) admits a unique viscosity solution u belonging to C([0, +∞[×M). Steady solutions of (1) come to solving

(2)  f(X) − H(X, Dϕ) = 0, X ∈ M.

Theorem 1. Problem (2) admits a viscosity solution ϕ. Moreover, for every u_{0} ∈ C(M) there exists a solution ϕ of (2) such that the solution u(t) of the Cauchy Problem (1) with initial datum u_{0} satisfies:

lim_{t→+∞} ∥u(t) − ϕ∥_{∞} = 0.

It is well-known that such a problem can admit multiple solutions; see Lions [7], where the problem is entirely solved for convex hamiltonians. The purpose of the next theorem is to show that many solutions have a nontrivial attraction domain.

Theorem 2. Denote by Z the nonempty zero set of f. Let C be an isolated connected component of Z; let u_{0} be negative outside C, and vanish on C. Then u(t) tends to a solution of (2) vanishing exactly on C.

The organization of this paper is as follows. In Section 1, basic facts are recalled; in Section 2 two proofs of Theorem 1 are presented, and Theorem 2 is proved. Finally, Section 3 provides an application to a solid combustion model.

Before going into the discussion, we would like to state that this paper owes a lot to Professor G. Barles. We wish to acknowledge his careful reading of the paper - hence pointing out to us several mistakes -, and his kind indication of the very quick proof of Theorem 1 via the relaxed semi-limits.

1. BASIC FACTS

The definition of a viscosity solution that we will use is this paper is the one given by Crandall and Lions [3], that we recall for the reader’s convenience. Consider a general equation

(3)  F (t, X, u_{t}, Du) = 0, X ∈ M;

then u is a viscosity subsolution of (3) on [0, T] × M if and only if

(4)

∀v ∈ C^{1}([0, T] × M),

((t_{0}, X_{0}) ∈]0, T] × M is a local maximum of (u − v))
⇒(F (t, X, v_{t}, u, Dv) ≤ 0 at (t_{0}, X_{0}))

Similarly, u is a supersolution of (3) if and only if

(5)

∀v ∈ C^{1}([0, T] × M),

((t_{0}, X_{0}) ∈]0, T] × M is a local minimum of (u − v))
⇒(F (t, X, v_{t}, u, Dv) ≥ 0 at (t_{0}, X_{0}))

Finally u is a viscosity solution if it is both a sub and supersolution.

The subsequent proposition is well-known; we recall it because of the shortness of its proof.

Proposition 3. (Lions, Papanicolaou, Varadhan [8]) Problem (2) is solvable.

PROOF. Let us consider the sequence of problems

(6)  f(X) − H(X, Dϕ) + εϕ = 0, X ∈ M.

which have unique Lipschitz solutions ϕ_{ε} in W^{1,∞}(M) satisfying

−∥f∥_{∞} ≤ εϕ_{ε} ≤ 0;

thus the sequence (Dϕ_{ε})_{ε} is bounded in L^{∞}(M). Hence the sequence (ϕ_{ε}− < ϕ_{ε} >)_{ε} is bounded in W^{1,∞}(M) due to assumption (H_{3}), thus relatively compact in C(M). This yields a real number λ ≤ 0 such that the problem

(7)  λ + f(X) − H(X, Du) = 0  X ∈ M

is solvable. This equation holds almost everywhere; therefore we have, for almost every X ∈ M: λ ≥ −f(X); as a consequence λ ≥ 0, hence λ = 0. • 2. LONG-TIME BEHAVIOUR

The strategy comes from dynamical system theory. From [4], Equation (1) defines a nonlinear semigroup in C(M), that will be denoted by S(t) in the sequel. Occasionally, the function S(t)u_{0} will be denoted by u(t) if no confusion is possible. Let us recall that the ω-limit set of an initial datum u_{0} with respect to S(t) is defined by

ω(u_{0}) = {ψ ∈ C(M) : ∃(t_{n})_{n}, lim_{n→+∞} t_{n} = +∞, lim_{n→+∞} S(t_{n})u_{0} = ψ}.

This set is non void if {S(t)u_{0}, t ≥ 0} is relatively compact in C(M). In parabolic equations, the flow is regularizing, and ω(u_{0}) is known to exist as soon as a uniform L^{∞} estimate is known. In the case of Hamilton-Jacobi equations, this is not always the case; therefore we will devote the next few lines to such compactness properties.

Proposition 4. Let u_{0} belong to W^{1,∞}(M), and u(t) = S(t)u_{0}. There exists M > 0 such that ∥u_{t}∥_{L^{∞}(IR_{+}×M)} + ∥Du∥_{L^{∞}(IR_{+}×M)} ≤ M.

PROOF. This property is known for finite time intervals [4], and we only have to prove this estimate uniformly with respect to t. Let M therefore be the Lipschitz constant of u on [0, 1] × M. For any h ∈ [0, 1] there holds u_{0} − Mh ≤ u(h, .) ≤ u_{0} + Mh; by the maximum principle and the fact that the semigroup S(t) commutes with the addition of constants there holds S(t)u_{0} − Mh ≤ S(t + h)u_{0} ≤ S(t)u_{0} + Mh, which ensures that u is Lipschitz in time with constant M. This, together with Assumption (H_{3}), implies th result. •

Let ϕ be a solution of (2). For u_{0} belonging to W^{1,∞}(M), there holds, from the maximum principle:

∥S(t)u_{0}∥_{∞} ≤ ∥u_{0}∥_{∞} + ∥ϕ∥_{∞}.

This easily implies

Proposition5. The set {S(t)u_{0}, t ≥ 0} is relatively compact in C(M).

Propositions 5 and 4 also imply that ω(u_{0}) is compact in C(M), connected and positively invariant by the flow. In fact, every element of ω(u_{0}) is Lipschitz, with a common Lipschitz constant. The last wellknown fact which will be useful to us is the L^{∞}-stability of all the trajectories, namely:

Proposition 6. For every (u_{0}, v_{0}) ∈ C(M), there holds ∥S(t)u_{0} − S(t)v_{0}∥_{∞} ≤ ∥u_{0} − v_{0}∥_{∞}.

This last proposition enables us to see that it suffices to prove Theorem 1 for a Lipschitz initial datum: if u_{0} is only continuous, one approximates it by a sequence of Lipschitz initial data, then uses the L^{∞} stability of the steady states.

Recall that Z is the zero set of f. The following lemma is the key point of our convergence proof.

Lemma 7. For every X ∈ Z, the function t ↦ S(t)u_{0}(X) is nondecreasing.

PROOF. The result is intuitively obvious, but we are unfortunately dealing with only Lipschitz solutions, which will force us to use the following argument. We may identify a small neighbourhood of X with an open subset U of IR^{N}; let us indeed do so and let ε be small enough so that the cube K_{ε}(X) := X + [−ε, ε]^{N} lies in U. Let h > 0 be given; equation (1) holds in the L^{∞} sense; therefore we may integrate it over [t, t + h] × K_{ε}(X). There holds

∫_{[t,t+h]×K_{ε}(X)} u_{t}(s, Y) dsdY = ∫_{K_{ε}(X)}(u(t + h, Y) − u(t, Y)) dY.

As a consequence we have:

\begin{aligned}\frac{1}{ε^{N}} ∫_{K_{ε}(X)}(u(t + h, Y) − u(t, Y)) dY & ≥ −\frac{h}{ε^{N}} ∫_{K_{ε}(X)} f(Y) dY \\ & ≥ −εh∥f∥_{Lip} \end{aligned}

Letting ε → 0 and using the continuity of u yields u(t + h, X) ≥ u(t, X), hence the proposition. •

The first consequence of this lemma is that the restriction to Z of the function S(t)u_{0} converges uniformly - due to the fact that it is uniformly Lipschitz in space, thanks to Proposition 4 and Dini’s Theorem - towards a Lipschitz function ϕ(X). Hence every element ψ ∈ ω(u_{0}) satisfies ψ = ϕ on Z. This remark leads us to the

PROOF OF THEOREM 1. Because ω(u_{0}) is positively invariant by the flow, the restriction of u = S(t)ψ to Z is constant in time. Let Ω denote the set M\Z. For every ψ ∈ ω(u_{0}), let us set u(t) = S(t)ψ and

(8)  u^{ε}(t, X) = u(\frac{t}{ε}, X).

The function u^{ε} is then a solution of

εu_{t}^{ε} = H(X, Du^{ε}) − f(X) (Ω)

As in [2], let us introduce the semi-limits

(9)

\overline{u}(t, X) := lim sup^{∗}u^{ε} = lim sup_{ε→0, s→t, Y →X} u^{ε}(s, Y)
\underline{u}(t, X) := lim inf_{∗}u^{ε} = lim inf_{ε→0, s→t, Y →X} u^{ε}(s, Y)

which are respectively sub- and super- solution to f(X)−H(X, Du) = 0 on Ω - [2], Chap. 4, Theorem 4.1. The Lipschitz estimates on u, and the fact that u(t, .) converges uniformly on Z, imply

(10)  \overline{u} = \underline{u} = ϕ on Z.

The strong uniqueness - [2], Chap. 4, Theorem 4.2 - for the Dirichlet problem

(11)  \begin{aligned}f(X) − H(X, Du) & = 0, X ∈ Ω \\ u & = ϕ, X ∈ ∂Ω \end{aligned}

implies \overline{u} = \underline{u} everywhere, and implies the convergence of S(t)ψ to ϕ. •

The core of the above proof is a strong uniqueness result for discontinuous solutions. Below is a longer alternative proof, whose merit is to point out where uniqueness comes into play.

ALTERNATIVE PROOF TO THEOREM 1. Because ω(u_{0}) is positively invariant by the flow, the restriction of u = S(t)ψ to Z is constant in time. Let us now perform a Kruzhkov transform; we set v = e^{u}; the function v is a viscosity solution of

(12)  v_{t} + H̃(X, v, Dv) + f(X)v = 0,

with H̃(X, v, p) = −vH(X, \frac{p}{v}). It is to be noticed that, in the range of u

that we are considering, the hamiltonian H̃ is nonsingular, and thus can be suitably modified in the vicinity of v = 0. Hence the Cauchy problem for (12) is well-posed, and generates a semigroup that we are still going to denote by S(t).

Let us consider a metric on M, which induces a distance that we call d. For every ε > 0, let Z_{ε} be an ε-neighbourhood of Z: Z_{ε} = {X ∈ M, d(X, Z) < ε}, and let us set

(14)  δ_{ε} = min_{X∉Z_{ε}} f(X);

there holds δ_{ε} > 0. For K > 0, the function

(15)  v̄(t, X) = v(t, X) + K(ε + e^{−δ_{ε}t})

is a super-solution to (12) on M\Z̄_{ε}. Indeed, let w(t, X) be a C^{1} test function, and let (t_{0}, X_{0}) be a minimum point of v̄−w; this implies that, if we set w̄ = w−K(ε+e^{−δ_{ε}t}), then (t_{0}, X_{0}) is a minimum point of v−w̄. This implies

(16) w_{t} +H̃(X_{0}, v, Dw)+f(X_{0})v +Kδ_{ε}e^{−δ_{ε}t} ≥ 0 at (t, X) = (t_{0}, X_{0}).

On the other hand, we note that we have, because t > 0 ↦ tH(X, \frac{p}{t}) is

nonincreasing: H̃(X, v̄, p) ≥ H̃(X, v, p); therefore we have

w_{t} + H̃(X_{0}, v̄, Dw) + f(X_{0})v̄
≥w_{t} + H̃(X_{0}, v, Dw) + f(X_{0})v + Kf(X_{0})(ε + e^{−δ_{ε}t})
≥w_{t} + H̃(X_{0}, v, Dw) + f(X_{0})v + Kδ_{ε}e^{−δ_{ε}t}
≥0

hence v̄ is a super-solution. Let us now notice that there exists a constant K > 0 such that, on ∂Z_{ε} there holds, for any couple (t, s) ∈ IR_{+}^{2};

(17)  ∥S(t)ψ − S(s)ψ∥_{∞} ≤ Kε.

Let h ∈ [0, 1] be fixed. By choosing K > 0 large enough, taking into account (17) and the compactness estimates there holds: v(h, X) ≤ v̄(0, X) on M\Z̄_{ε}, and v(t + h, X) ≤ v̄(t, X) on ∂Z_{ε}. The maximum principle implies then: v(t + h, X) ≤ v̄(t, X). Let us now pass to the limit as t → +∞; for any ψ̃ ∈ ω(e^{ψ}) there holds: S(h)ψ̃ ≤ ψ̃ + ε on M\Z̄_{ε}; this inequality being valid for every ε > 0 we get S(h)ψ̃ ≤ ψ̃ on M\Z̄. But this inequality is also trivially valid on Z; hence it holds on all M. Hence the function S(t)ψ̃ converges to a steady state, which belongs to ω(ψ̃).

Coming back to the original unknown u = Logv, we have just proved the existence of an equilibrium state in ω(Logψ̃). However, ω(Logψ̃) ⊂ ω(ψ) ⊂ ω(u_{0}). Therefore there exists an equilibrium state in ω(u_{0}), and we end the proof by applying Proposition 6. •

Let us remark that, in spite of the simplicity of its proof, Theorem 1 is interesting due to the fact that we may encounter many steady solutions. A surprising fact - if we think in terms of viscous approximations - is that they are all L^{∞}-stable, due to Proposition 6. This raises the question of the correct definition of stability for the steady states of Hamilton-Jacobi equations, i.e. according to which criterion some solutions will be more stable than others.

PROOF OF THEOREM 2. The idea is to construct a super-solution to (2), larger than u_{0}, and whose zero set will force the unique element of ω(u_{0}) to vanish exactly on C. Due to Lemma 7, S(t)u_{0} will vanish exactly on C.

Let us first choose α > 0 such that d(C, Z\C) ≥ 2α. For every β > 0, let Z_{β} be a β-neighbourhood of Z\C; with this notation let (f_{ε})_{ε>0}, defined for ε small enough, be a sequence of continuous functions, such that

• 0 ≤ f_{ε} ≤ f,

• f_{ε} = ε on Z_{3α/2}\Z_{α},

• the zero set of f_{ε} is exactly Z_{α/2} ∪ (M\Z_{2α}).

Let us now choose a sequence (δ_{ε})_{ε} of positive numbers such that the following Dirichlet problem is solvable - see [7], Chapter 5.

\begin{aligned}f_{ε}(X) − H(X, Dϕ) & = 0 on Z_{2α}\Z_{α/2} \\ ϕ & = 0 on M\Z_{2α} \\ ϕ & = −δ_{ε} on Z_{α/2} \end{aligned}

Let ϕ_{ε} be its unique solution. Let us remark that Assumption (H_{1}) tells us that, if we call ϕ̄_{ε} the extension of ϕ_{ε} to M defined by its Dirichlet boundary values, then it is a solution to f_{ε} − H(X, Dϕ) = 0 on M. Indeed, it is differentiable at every point of the zero set of f_{ε}, and its gradient is 0.

When ε > 0 is small enough we have ϕ̄_{ε} ≥ u_{0}. Indeed, the assumptions on f_{ε} imply that the sequence (δ_{ε})_{ε} converges to 0; therefore the sequence (ϕ̄_{ε})_{ε} converges uniformly to 0. Because ϕ̄_{ε} vanishes on (M\Z_{2α})∪Z_{α/2}, it is enough to check that it is larger than u_{0} over Z_{2α}\Z_{α/2}. However, u_{0} has a negative maximum in this region; hence ϕ̄_{ε} ≥ u_{0} as soon as ε is small enough.

Let us choose once and for all such an ε > 0. The function ϕ̄_{ε} is also a super-solution to (2), therefore S(t)ϕ̄_{ε} is time-nonincreasing. This implies that S(t)u_{0} remains below a time-independent negative constant on Z\C, which proves Theorem 2. •

Remark 1. Strictly speaking, we had no need of introducing the ω-limit set of u_{0} in the course of the first proof of Theorem 1. The second proof is much more easily readable once this set is introduced.

Remark 2. The minimal smoothness assumptions on H(X, p) seem to be: H continuous in X and Lipschitz continuous in p.

3. AN APPLICATION TO SOLID COMBUSTION THE-ORY

Our work was motivated by a problem from combustion theory, namely the propagation of a flame front in a solid medium with periodic striations - see [9] and the references therein. In fact, a model of propagation without curvature effects is governed by the Hamilton-Jacobi equation

(3.1)  u_{t} = R(X)\sqrt{1 + |∇u|^{2}}, X ∈ IT^{N}

where IT^{N} is the unit N-dimensinal torus. The unknown function u(t, X) then represents the position of the front and the function R is the local combustion rate of the solid. Equation (3.1) just asserts that the normal velocity of the front is a function of its sole position. It is expected - and numerically computed, see [9] - that it will behave like a travelling wave, i.e. a function of the form

u(t, X) = λt + ϕ(X),

where ϕ solves the steady problem

(3.2)  λ − R(X)\sqrt{1 + |∇ϕ|^{2}} = 0, X ∈ IT^{N}

Let us notice that Problem (3.1) may be reduced to (1). Indeed if we replace u by u − λt and set

\begin{aligned}H(X, p) & = R(X)(\sqrt{1 + |p|^{2}} − 1) \\ λ & = ∥R∥_{∞} \\ f(X) & = ∥R∥_{∞} − R(X) \end{aligned}

then u verifies (1) and it is easy to see that H and f defined as above satisfy the hypotheses (H_{1}) − (H_{3}). Therefore Theorem 1 applies and, for every u_{0} ∈ C(IT^{N}), the solution u(t, X) to the Cauchy Problem for (3.1) behaves as

u(t, X) = λt + ϕ(X) + o(1)

as t → +∞, where ϕ is a solution of (3.2).

REFERENCES

[1] - G. Barles, Asymptotic behavior of viscosity solutions of first order Hamilton Jacobi equations, Richerche di Matematica, 34 (1985), pp. 227-260.

[2] - G. Barles, Solutions de viscosité pour les équations de Hamilton-Jacobi, Mathématiques et Applications, 17, Springer-Verlag.

[3] - M.G. Crandall, P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. , Vol. 277, no. 1, 1983.

    [4] - M.G. Crandall, L.C. Evans, P.-L. Lions, Some proper-

ties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. , Vol. 282, 1984.

[5] - C.M. Dafermos, Trend to steady state in a conservation law with spatial inhomogeneity, Quarterly Appl. Math., Vol. 45 (1987), pp. 313-319.

[6] - S.N. Kruzhkov, Generalized solutions of nonlinear first order equations with several independent variables, Math. USSR Sbornik vol.1 n0. 1, 1967.

[7] - P.-L. Lions, Generalized solutions of Hamilton-Jacobi Equations, Research Notes in Mathematics, Pitman.

    [8] - P.-L. Lions, G. Papanicolaou, S.R.S. Varadhan, Hom-

ogenization of Hamilton-Jacobi equations, preprint.

[9] - G. Namah, Asymptotic behaviour of the solution of a Hamilton-Jacobi equation, Asymptotic Analysis, Vol. 12, 1996.

[10] - G. Namah, J.-M. Roquejoffre, Convergence to periodic fronts in a class of parabolic equations, Nonlinear Diff. Eq. and Applications, 4 (1997), pp. 521-536.