Asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws with explicit decay rates

We consider scalar conservation laws with nonlocal diffusion of Riesz-Feller type such as the fractal Burgers equation. The existence of traveling wave solutions with monotone decreasing profile has been established recently (in special cases). We show the local asymptotic stability of these traveling wave solutions in a Sobolev space setting by constructing a Lyapunov functional. Most importantly, we derive the algebraic-in-time decay of the norm of such perturbations with explicit algebraic-in-time decay rates.

Taking α = 2 and θ = 0 in (1), we formally obtain a classical viscous conservation law: The existence and asymptotic stability of traveling wave solutions of equation (5) has been studied thoroughly. A first example of equation (1) with nonlocal diffusion is with a fractional Laplacian D α 0 , 0 < α ≤ 2, which has been studied e.g. in [6,11]. For 1 < α ≤ 2, the Cauchy problem for (6) with f ∈ C ∞ (R) and essentially bounded initial data has a global-in-time mild solution which becomes smooth for positive times, see [11] and its extension to (1) in [2].
Other examples of equation (1) with nonlocal diffusion appear in viscoelasticity [27] and fluid dynamics [21]. In particular, with 0 < γ < 1 is used as a model for the far-field behavior of uni-directional viscoelastic waves [27], and derived as a model for the internal structure of hydraulic jumps in near-critical single-layer flows [21]. Moreover the nonlocal operator D 1/3 appears in Fowler's equation which models the uni-directional evolution of sand dune profiles [13]. In the theory of water waves similar models ∂ t u + ∂ x u 2 = N [u] with different (nonlocal) Fourier multiplier operators N are studied, see the book [25] and references therein.
To explain our main results, we introduce traveling wave solutions for equation (1). Traveling wave solutions (TWS) are of the form u(x, t) = u(ξ) for some profile u with ξ = x−st and (constant) wave speed s ∈ R. We are interested in TWS with profiles u connecting distinct endstates u ± such that lim x→±∞ u(x) = u ± .
Using this ansatz in equation (1) and assumption (9), we find that the wave speed s has to satisfy the Rankine-Hugoniot condition Here, an extension of Riesz-Feller operators to non-integrable functions is needed, see Appendix A. Due to translational invariance of equation (1), traveling wave solutions are only unique up to a shift. For classical viscous conservation laws (5), the profile of a TWS satisfies an ordinary differential equation In fact, TWS exist only for parameters (u − , u + ; s) satisfying (10) and u + < u − . In case of equation (7), the existence and asymptotic stability (without decay rates) of traveling wave solutions for parameters (u − , u + ; s) satisfying (10) and u + < u − has been shown [1,8]. Here, a profile satisfies a fractional differential equation The proof of existence relies on the causality of the Caputo derivative D γ , i.e. to evaluate D γ u at x the profile u on (−∞, x) is needed. In contrast, the profile for a TWS of a nonlocal conservation law (6) for 1 < α < 2 has to satisfy Thus D α 0 u(x) depends on the whole profile u. For fractal Burgers equation, i.e. equation (6) with 1 < α < 2 and Burgers flux function f (u) = u 2 , the existence of traveling wave solutions has been proven recently [7]. The idea is to approximate the operators D α 0 by convolution operators K ǫ [u] = K ǫ * u − u for suitable convolution kernels K ǫ ∈ L 1 (R). The existence of TWS for the approximate equations is known and the TWS is established as the limit of this family. It is conceivable to use this approach to prove the existence of traveling wave solutions in the general case (1) for convex flux functions f with 1 < α < 2 and |θ| ≤ 2 − α.
The asymptotic stability of traveling wave solutions of classical viscous conservation laws (5) has been studied thoroughly. At first, historically, Il'in and Oleinik [16] proved the asymptotic stability of nonlinear waves for viscous conservation laws (5) by making use of the maximum principle for linear parabolic equations. For Burgers' equation, i.e. equation (5) with Burgers' flux function f (u) = u 2 , Nishihara [26] obtained the decay estimates toward traveling wave solutions by making use of the explicit solution formula. And, Kawashima and Matsumura [18] generalized Nishihara's time decay result to a class of viscous conservation laws. They considered weighted L 2 spaces and used a weighted energy method. Furthermore, Kawashima, Nishibata and Nishikawa [19] extended the L 2 energy method to general L p spaces. Their techniques have been applied to a model system for compressible viscous gas in [24] and a hyperbolic system with relaxation in [28].
Assuming the existence of a traveling wave solution of (1) with monotone decreasing profile, we show that asymptotic stability of a traveling wave solution in a Sobolev space setting follows from a standard Lyapunov functional argument: To investigate the stability of the traveling wave solution with profile u, we consider initial data u 0 such that u 0 − u is integrable and determine the unique shift x 0 which yields ∞ −∞ (u 0 (ξ) − u(ξ + x 0 )) dξ = 0. Moreover, we restrict the domain of initial data u 0 further such that W 0 (ξ) = ξ −∞ (u 0 (η) − u(η)) dη exists (using a suitable shifted profile u) and satisfies W 0 ∈ H 2 . (For details, we refer to [28].) More precisely, we can derive the following theorem.
Theorem 1. Suppose 1 < α ≤ 2 and θ ≤ min{α, 2 − α}. Let the flux function f ∈ C 2 (R) be convex and let u(x, t) = u(x − st) be a traveling wave solution of (1) with monotone decreasing profile u. Let u 0 be an initial datum for (1) Then there exists a positive constant δ 0 such that if W 0 H 2 ≤ δ 0 , then the Cauchy problem (1) has a unique global solution converging to the traveling wave in the sense that The proof of Theorem 1 for the general equation (1) is similar to the one of [1, Theorem 4] for the special case (7) without decay rates.
Our main result is to prove the asymptotic stability with algebraic-in-time decay rate for traveling wave solutions of (1) with monotone decreasing profiles.
Remark 2. We employ sharp interpolation inequalities in Sobolev spaces to derive (11). In this way optimal decay estimates for the asymptotic stability of viscous rarefaction waves in scalar viscous conservation laws (5) have been derived in [14].
Remark 3. We want to explain the functional setting in Theorem 2: We considered the function spaces The choice H 1 (R) ∩ W 1,1 (R) leads to the restriction α ∈ (3/2, 2) if we use an estimate of the nonlinearity like Dix [9,10] to establish the existence of solutions for the Cauchy problem. Assuming higher regularity of the initial data removes the need for this restriction: Under the assumptions of Theorem 1 with W 0 ∈ H 2 (R) ∩ W 2,1 (R), the solution constructed in Theorem 1 satisfies for t ≥ 0, where E 1 := W 0 H 2 + W 0 W 2,1 and a constant C independent of time t. Our choice W 0 ∈ W 1,∞ (R) ∩ W 1,1 (R) in Theorem 2 leads to the technical assumption f ∈ C ∞ (R), since we use a result on the existence of global-in-time solutions for the Cauchy problem with essentially bounded initial data [11,2]. The assumption f ∈ C 2 (R) in Theorem 1 could be retained by aiming for less regularity in their approach.
Unfortunately it is difficult to apply the weighted energy method in [18] to our problem (to derive the convergence rate). Instead of this method, we employ another technique which focuses on the interpolation property in Sobolev space. For example, this argument is utilized in [14].
The contents of this paper are as follows. In Section 2, we reformulate our problem and consider the well-posedness of the new one. In Section 3, we derive the asymptotic stability result by uniform energy estimates as a-priori estimates of solutions in the Sobolev space H 2 . Furthermore, our main result on the asymptotic stability with explicit algebraic decay rate in Theorem 2 is proved in Section 4, by using the energy method with an L 2 -L 1 interpolation argument. In Appendix A, we collect results on the singular integral representation of Riesz-Feller operators.
Notation. Before closing this section, we give some notations used in this paper. We define the Fourier transform for v ∈ S in the Schwartz space S aŝ and the inverse Fourier transform as The Fourier transform and its inverse are linear operators and F and F −1 will denote also their respective extensions to L 2 (R). For 1 ≤ p ≤ ∞, we denote by L p = L p (R) the usual Lebesgue space over R with norm · L p , and W s,p = W s,p (R) the usual Sobolev space over R with norm · W s,p . Using the ) denotes the space of ℓ-times continuously differentiable functions (respectively with bounded derivatives) on the interval I with values in the Banach space X.
The constants in our estimates may change their value from line to line.

Reformulation for the problem
In the special case (7), the existence and asymptotic stability of traveling wave solutions u(x, t) = u(x−st) with monotone decreasing profile u has been proven without rates of decay [1,8]. However, assuming in the general case (1) the existence of a traveling wave solution u(x, t) = u(x − st) with monotone decreasing profile u, then the proof of asymptotic stability generalizes with obvious modifications: To prove the asymptotic stability of a traveling wave solution u of (1), one can follow the standard approach called the anti-derivative method introduced in [18] for viscous conservation laws. It is convenient to cast (1) in a moving coordinate frame (x, t) → (ξ, t), such that and u is a stationary solution of (12). The Cauchy problem for (12) with initial datum u 0 governs the evolution of u 0 . If its solution u is considered as a perturbation of the traveling wave solution u, then this perturbation U (ξ, t) := u(ξ, t) − u(ξ) satisfies the Cauchy problem where U 0 (ξ) := u 0 (ξ) − u(ξ). To obtain the desired result, we try to construct the L 2 -energy estimate for U by employing the energy method. However, because of the decreasing property of traveling wave solutions, it is hard to construct the L 2 -energy estimate. To overcome this difficulty, we apply the anti-derivative method. Precisely, we introduce the new function W (ξ, t) which satisfies ∂ ξ W = U . Then we can formally rewrite (13) as If a global-in-time solution of (14) with W 0 (ξ) = ξ −∞ U 0 (η) dη is sufficiently smooth, then its derivative ∂ ξ W satisfies Cauchy problem (13). Therefore, we try to construct a global-in-time solution of (14), instead of (13). For this purpose, we discuss the well-posedness of problem (14) in this section.
The well-posedness of the Cauchy problem for (14), will follow from a contraction argument. Assuming f (u) = u 2 and α > 3/2 allows to estimate the nonlinearity in the fashion of Dix [9,10] implying the well-posedness in H 1 . For general flux functions and α ∈ (1, 2], we have to require more regularity of the initial data, e.g. W 0 ∈ H 2 .
To prove Proposition 1, we first present some properties of the fundamental solution of Due to the properties of G α θ , it is easy to show that D α θ generates a semigroup. Lemma 2. For 1 < α ≤ 2, |θ| ≤ min{α, 2 − α} = 2 − α, the Riesz-Feller operator D α θ generates a strongly continuous, convolution semigroup for all 1 ≤ p < ∞ and some C p > 0.
Proof. Due to (G3) and Young's inequality for convolutions, is a semigroup, since S t+s = S t S s for all s, t ≥ 0 holds due to (G4) and S 0 := Id. Strong continuity of (S t ) t≥0 follows from a standard result about convolutions [22, p.64] and (G2). The dispersion property can be proved using (G5) and Young's inequality [22, p.98-99].
Proof. For arbitrary constants where the integral is bounded by 4 φ 2 H σ . Thus, the Dominated Convergence Theorem allows to pass to the limit under the integral sign, which completes the proof.
Proof of Proposition 1. Using the fundamental solution G α θ of the linear evolution equation ∂ t u = D α θ u, the mild formulation of (14) reads where To employ a fix point argument, we consider the mapping G[W ] defined by on the Banach space Then we show that G is a contraction mapping on a closed convex subset S R of X, where S R := {W ∈ X; W X ≤ R} for some parameter R > 0 which will be determined later. Due to a Sobolev embedding, where we used Lemma 3 and the identity Similarly, we can calculate that Combining the above estimates, we obtain Therefore, letting T = min{1, (2C * (R)) −α/(α−1) }, we deduce where C * (R) := C 0 (R) + C 1 (R) + C 2 (R). On the other hand, letting V ≡ 0 in (19), we get where we used (16) with ℓ = r. Therefore, choosing R = 2M , we obtain G[W ] X ≤ 2M . Finally we discuss the continuity of G[W ] in time t. It follows from the continuity at time 0 and the semigroup property (G4) of G α θ . Due to Lemma 4, for W 0 ∈ H σ (R) with σ ≥ 0, the convergence lim tց0 G α θ (·, t) * W 0 = W 0 in H σ holds. Moreover, for t ∈ [0, T ] and s ≥ 0 the identity holds, where the last integral converges to zero for s → 0. Thus, for t 1 , t 2 ∈ [0, T ] with t 1 < t 2 (without loss of generality), we have Therefore, by the fact that W 0 ∈ H 2 , W ∈ X and Lemma 4, we find that the right hand side of (20) tends to zero in H 2 as t 1 → t 2 . Hence, we deduce the continuity of G[W ] in t and that G[W ] ∈ S 2M for W ∈ S 2M . Consequently, we conclude that there exist T = T (M ) such that G is a contraction mapping of S 2M . This means that the mapping G admits a unique fixed point W in S 2M , such that W = G[W ]. Hence the proof of Proposition 1 is complete.

Asymptotic stability of traveling waves
In this section, we consider the asymptotic stability of traveling wave solutions with monotone decreasing profile in (1). To this end we derive the existence of global-in-time solutions for evolution equation (14) and that these perturbations decay. Precisely we prove the following theorem.
for some positive constant C and for all t ≥ 0. Furthermore, the solution W (ξ, t) converges to zero in the sense that We note that the third integral of the left hand side in (21) is non-negative, since the flux function f ∈ C 2 is convex such that f ′′ ≥ 0 and the profile u is monotone decreasing, i.e. u ′ ≤ 0. For the solution W constructed in Theorem 3, it is easy to check that ∂ ξ W satisfies Cauchy problem (13). Consequently we obtain Theorem 1. Global existence will be the consequence of the existence of a Lyapunov functional, which also allows to deduce the asymptotic stability of traveling waves, see also [1,Theorem 4] for the special case θ = 2 − α.
Lemma 5. Suppose that the same assumptions as in Theorem 1 hold. Let W be a solution to (14) satisfying W ∈ C([0, T ]; H 2 ) for some T > 0. Then there exists some positive constant δ 1 independent of T such that if sup 0≤t≤T W (t) H 2 ≤ δ 1 , the a-priori estimate expressed in (21) holds for t ∈ [0, T ].
Proof. We rewrite the first equation of (14), and test it with W , Integrating with respect to ξ ∈ R, we obtain where L is a positive non-decreasing function. Due to a Sobolev embedding and the assumption on W , we deduce W (t) W 1,∞ ≤ W (t) H 2 ≤ δ 1 for all t ∈ [0, T ]. Thus the energy estimate becomes for some positive constant C δ1 depending on δ 1 . Note that we keep W L ∞ for further reference.
Then, integrating (29) with respect to t, we have Thus, by employing (26), and (27) with v = ∂ ξ U and σ = α/2, we arrive at for some positive constant C. Finally, using the fact that E(t) ≤ δ 2 1 C, we arrive at the desired a-priori estimate.
Proof of Theorem 3. The existence of global-in-time solutions to the initial value problem (14) can be obtained by the continuation argument based on a local existence result in Proposition 1 combined with the a-priori estimate in Lemma 5. Because the argument is standard, we may omit the details here. In the rest of this proof, we prove only the asymptotic stability result (22).
To this end, we prepare the following interpolation inequality.

Convergence rate toward traveling waves
We consider the convergence rate of the solution toward the corresponding traveling waves. Kawashima, Nishibata and Nishikawa [19] proposed an L p energy method to study the asymptotic stability and the associated convergence rates of planar viscous rarefaction waves of multi-dimensional viscous conservation laws. When the authors obtain the convergence estimate, they derived the L 1 estimate by using the energy method associated with the sign function. This approach is useful. It is however difficult to apply this method because of a Riesz-Feller operator. To overcome this difficulty, we employ not only the energy method but also the representation of the mild solution. Precisely, our purpose in this section is to derive the following theorem.
Theorem 4. Suppose that the same assumptions as in Theorem 1 and f ∈ C ∞ (R) hold. Then the Cauchy problem (14) with W 0 ∈ W 1,1 (R) ∩ W 1,∞ (R) has a unique global solution W (ξ, t) satisfying with estimates (37) and (38). Moreover, there exists a positive constant δ 1 for t ≥ 0, where E 1 := W 0 H 1 + W 0 W 1,1 and C is a certain positive constant independent of t.
The proof of the existence of global-in-time solutions is based on results for the Cauchy problem (1) with fractional Laplacian [11] and its extension to the Cauchy problem (1) with Riesz-Feller operators [2]. There the assumption f ∈ C ∞ (R) is made to simplify the presentation. The method is applicable also in case of f ∈ C k (R), k ≥ 2, but yields a lower regularity for the unique solution u.
Lemma 6. Suppose that f ∈ C ∞ (R) and W 0 ∈ W 1,1 (R) ∩ W 1,∞ (R). Then Cauchy problem (14) has a unique mild solution W ∈ C([0, T ]; for 0 ≤ t ≤ T , where L is a positive non-decreasing function. Moreover, for any positive time ∞)) and it is a classical solution of the first equation of (14).
Proof. We use again U = ∂ ξ W and analyze the Cauchy problem (13) with initial datum U 0 := ∂ ξ W 0 ∈ L 1 (R) ∩ L ∞ (R) first. We recall U = u − u where u and u solve equation (12), and u is a monotone decreasing function satisfying lim ξ→±∞ u(ξ) = u ± . Thus, u 0 := U 0 + u is essentially bounded. Due to [11,Theorem 1] and its extension to equations with Riesz-Feller operators in [2], the Cauchy problem for (12) with initial datum u 0 ∈ L ∞ (R) has a (unique) solution which satisfies u(t) L ∞ (R) ≤ u 0 L ∞ (R) for all t ≥ 0; in fact, the solution u takes values between the essential lower and upper bounds of u 0 . Therefore, U (t) = u(t) − u ∈ L ∞ (R ξ ) for all t ≥ 0 and estimate (34) follows.
Due to [11,Remark 1.2] and its extension to equations with Riesz-Feller operators, equation (12) supports an L 1 contraction principle: If u 0 , v 0 ∈ L ∞ (R) satisfy u 0 −v 0 ∈ L 1 (R), then the associated solutions u and v of the Cauchy problem for (12) Then, we are left to prove that W (t) ∈ L 1 (R ξ ) for all t ≥ 0 and the stated continuity in time. Considering the mild formulation (17), we obtain the estimate for t ≥ 0 by using the local Lipschitz continuity of f and the previous estimates on U = ∂ ξ W ; again, L is a positive non-decreasing function. Moreover, for any positive time t 0 > 0, U ∈ C ∞ b (R×(t 0 , ∞)) and U = ∂ ξ W satisfies the first equation of (13) in the classical sense, see [11,1]. Due to integrability of U , also W is a global-in-time solution of (14), and W ∈ C ∞ b (R × (t 0 , ∞)) is a classical solution of the first equation of (14) for all t ≥ t 0 > 0.
Next we prove the following a-priori estimate obtained by Lemma 6.
for some positive non-decreasing function L. Integrating this inequality with respect to time and using (26), the estimates (33)-(34) as well as the smallness of W 0 W 1,1 , we arrive at (37).
To estimate W (t) W 1,1 , we recall that U (t) L 1 ≤ U 0 L 1 for all t ∈ [0, T ], due to estimate (32) in Lemma 6. Next we show that for t ∈ [0, T ]. We will use estimate (31) for small times t ≤ 1, and derive (39) for large times t ≥ 1: We multiply the first equation of (14) by s δ (W ) = (ρ δ * sgn)(W ) and obtain where h(v) := f (v) − sv is a convex function. We integrate equation (40) over R × [t 0 , t] and derive The first integral satisfies, due to Fubini's theorem and the strong convergence of S δ in L 1 , as δ → 0. Next, we prove that the integral on the right-hand side of (41) is non-positive, due to Proposition 3. We estimate the second term on the left-hand side of (41) as follows. Using the fact that |s δ (W )| ≤ 1 and h(u since the function S δ is non-negative with S δ (0) = 0, h ∈ C 2 (R) is a convex function, and u is a monotone decreasing traveling wave profile. Therefore, employing the previous estimates and taking the limit δ → 0 in equation (41) yields for t ≥ t 0 > 0 and some positive constant C; here we used (37) and (26). The estimate (44) is valid for an arbitrary positive constant t 0 . Thus we can estimate from (44) and (35) that for t ≥ 1. Eventually, combining this estimate and (35) again, we arrive at the desired estimate (39).
Proof of Theorem 4. The existence of the global solution follows from Lemma 6 and the a-priori estimates in Lemma 7. We derive just the decay estimate (30). To this end, we first introduce the following Nash inequality: v for σ > 0 and v ∈ L 1 (R) ∩ H σ (R), where C σ is a positive constant which depends on σ. Following the proof of Lemma 5, we deduce again estimate (28). Multiplying this inequality with (1 + τ ) β for β ∈ R and integrating over τ ∈ [0, t], we obtain We compute via Nash's inequality (45) with σ = α/2 and Young's inequality that for all ǫ > 0 and some positive constant C ǫ . Thus we get . Therefore, employing this estimate and (38), we obtain For this inequality, we take β and ǫ which satisfy for some positive constant c. Finally, using (26), the estimates (33)-(34) and the smallness of W 0 W 1,1 , we arrive at (1 + t) β−1/α and the desired estimate (30).

A Riesz-Feller operators
To study the existence of traveling wave solutions with smooth profiles, we need the singular integral representation of Riesz-Feller operators D α θ .
The singular integral representation (46) for Riesz-Feller operators D α θ is well-defined for C 2 b functions such that D α θ C 2 b (R) ⊂ C b (R).
Proposition 3. The integral representation (46) of D α θ with 1 < α < 2 and |θ| ≤ min{α, 2 − α} is well-defined for functions v ∈ C 2 b (R) with for some positive constant M and the positive constants c 1 and c 2 in Proposition 2. Moreover, if v ∈ C 2 b (R) is a function such that the limits lim x→±∞ v(x) exist, then R D α θ v(x) dx = 0.
Proof. The first statement follows by direct estimates on the extension of Riesz-Feller operators in (46), see [3,Proposition 2.4]. To prove the second statement, we consider the two summands in (46) separately, starting with dξ for any v ∈ C 2 b (R). Like before, we rewrite the integral where exchanging integration and taking derivatives is possible, since in each step the integrands are absolutely integrable uniformly with respect to x. Moreover, and the primitive satisfies where exchanging integration and taking limits is possible, since in each step the integrands are absolutely integrable and lim x→±∞ v(x + θξ) − v(x) = 0 due to the assumptions on v.