Stability of traveling waves of models for image processing with non-convex nonlinearity

We establish the existence and stability of smooth large-amplitude traveling waves to nonlinear conservation laws modeling image processing with general flux functions. We innovatively construct a weight function in the weighted energy estimates to overcome the difficulties caused by the absence of the convexity of fluxes in our model. Moreover, we prove that if the integral of the initial perturbation decays algebraically or exponentially in space, the solution converges to the traveling waves with rates in time, respectively. Furthermore, we are able to construct another new weight function to deal with the degeneracy of fluxes in establishing the stability.

This paper is concerned with existence and asymptotic stability of smooth traveling wave solutions to nonlinear conservation laws with general flux functions f arising from image processing for x ∈ R and t > 0, where u(x, t) is the gray scale of images and g(s) = 1 1 + s 2 . (2) The initial data, the original image, is given by where u − and u + are nonnegative constants. Before the development of denoising techniques based on nonlinear PDEs [9,23], the linear filtering PDEs method was introduced by Marr and Hildereth [16], and was developed by [25,8,3]. However, after denosing image by the linear PDEs methods, it is hard to recognize the original important contents such as edges in 960 TONG LI AND JEUNGEUN PARK the denoised image. To resolve such blurriness caused by the linear methods, Perona and Malik [23] introduced the nonlinear PDEs u t = ∇ · (g(|∇u|)∇u) (4) where u(·, t) : Ω → R, t > 0 and Ω ⊂ R 2 denotes an image domain. A typical g in (4) is given in (2) and it plays a role in preserving edges in the image while reducing overall noise of the image.
In [9], Kurganov et al. introduced a convection term in (4) for x ∈ R and t > 0, where g(s) is given in (2) with the initial data (3). In fact, such a convection motion has been applied to image inpainting in a region where the image data is lost and is filled within the image information surrounded the region (e.g. see [1,2,5]). Kurganov et al. [9] and Goodman et al. [4] investigated shock or jump type behavior of solutions to (5) which are typical edges in images. The existence of smooth traveling wave solutions to the Burgers-type problem (5) was studied by authors in [9,26,5]. Rigorous proofs of its stability were achieved by Wu [26] and Li and Park [15]. Wu [26] obtained the stability of the smooth traveling waves in some exponential weighted space by spectral analysis. In [15], the authors proved nonlinear stability of traveling waves for general small initial perturbations. The authors in [15] also used a weighted energy method to show that if the initial perturbation decays algebraically or exponentially as |x| → ∞, then the Cauchy problem solution approaches to the traveling wave at corresponding rates as t → ∞. It is noticed that equation (5) is equation (1) with f (u) = u 2 2 which is convex.
In the current paper, we prove existence and stability of smooth large-amplitude traveling wave solutions to (1)-(3) with non-convex fluxes f ∈ C 2 . To show the existence of smooth traveling solutions to (1)-(3), we impose the entropy condition [17] h(u) where c satisfies the Rankine-Hugoniot condition We show the stability with or without decay rates for both non-degenerate case and degenerate case In establishing the stability theorems, the absence of convexity of f causes difficulties in energy estimates. To overcome such obstacles, Matsumura and Nishihara [17] used weighted energy estimate methods for scalar viscous conservation laws. Their weight function has been widely adopted to obtain stability with non-convex f , for example, [11,13,14,19]. However the above weight function does not apply to our problem (1)-(3) with large-amplitude traveling waves due to the different dissipative term. In order to prove the stability for general fluxes f and for arbitrary wave strengths, we construct a new weight function which is piecewisely defined on bounded and unbounded domains, see (47). On the bounded domains where f changes its convexity, we set up w in (47) so that the L 2 -integral of φ does not appear in the L 2 -estimate (52). On the unbounded domains, using the entropy condition (6), we derive the desired L 2 -estimates (52). It is worth mentioning that in proving our stability results, we only need the entropy condition (6) instead of imposing additional conditions on the fluxes as in the previous related results [7,10,11].
Combining the same weight function with algebraic and exponential weights, we prove our stability with decay rates. For related previous results, see [6,12,15,17,18,19,21,22]. For the degenerate case (9), we construct a new unbounded weight function (81) whose essential feature at far fields comes from the degeneracy. Moreover, using the weight function (47), we are able to improve the previous results in [15] where f is convex. See Remarks 3 and 4.
This paper is organized as follows. In Section 2, we show existence of smooth traveling wave solutions for (1)-(3), and describe properties of the traveling wave solutions. We also state our main results. In Section 3, we reformulate our problem in terms of perturbation quantities. Our main stability theorems are proved in Section 4 for the non-degenerate case (8) and for the degenerate case (9) in Section 5. In Section 6, we investigate algebraic and exponential decay rates for the nondegenerate case (8).
Notation. We denote C > 0 as a generic constant so each C can be a different number at different context. We write f ( We denote H k the usual Sobolev space W k,2 with the norm f k for any function f ∈ H k , k ≥ 1, We also denote L 2 w the space of measurable functions f on R which satisfy w(x)f ∈ L 2 , where w(x) ≥ 0 is a weight function. Then the space is endowed with the norm Similarly, H k w denotes the weighted Sobolev space of L 2 w -functions f whose deriva- We define Weight functions in each sections and subsections differ. Particularly, a weight function in Subsection 6.1 is of the form w(x) ∼ x α for α ≥ 0 and we denote where c is the traveling wave speed and z = x − ct is the traveling wave variable. Constants u ± and c satisfy the Rankine-Hugoniot condition (7) and the entropy condition (6). The condition (6) implies We consider a non-convex function f (u) and the case u + < u − . It follows from (6) that Substituting (12) into (1) and using the definition of h(u) in (6), we have Integrating (15) over (±∞, z) and using the Rankine-Hugoniot condition (7), one obtains Solving for U z , we have where 0 < |h(U )| ≤ 1 2 . Since the critical threshold of discontinuity of solutions of (16) occurs at |h(U )| = 1 2 , in order to have smooth traveling waves, we need a condition This is analogous to the analysis of existence of smooth traveling wave solutions in [9]. Let b > 0 be the upper bound of the strength of smooth traveling wave solutions, namely, For example, b=2 in [9,15]. Then for each u ± satisfying (19), there is a constant Under condition (19), we have a unique smooth solution of (16) and the following theorem.
Theorem 2.1. Let f (u), u ± , c satisfy the Rankine-Hugoniot condition (7) and the entropy condition (14) and (19). If there are nonnegative integers n ± such that h (u ± ) = · · · = h (n±) (u ± ) = 0 and h (n±+1) (u ± ) = 0, there exists a unique smooth traveling wave solution U (z) of (1)- (2). and the conservation law (1) implies that Hence, we arrive at Now we decompose the solution u to the Cauchy problem (1)-(3) into the traveling wave U and its perturbation as Without loss of generality, let x 0 = 0. The initial data of φ is given by Our main theorems are stated as follows.
Theorem 2.2 (Non-degenerate case (8)). Let U (z) be a smooth traveling wave solution obtained in Theorem 2.1. Let u 0 − U be integrable and φ 0 ∈ H 2 . There is a constant ε 1 > 0 such that if φ 0 2 < ε 1 , then the Cauchy problem (1)-(3) has a unique global solution u(x, t) satisfying In addition, it holds we have the same stability theorem as Theorem 2.3 by replacing z + by respectively.
In addition, we investigate algebraic and exponential decay rates for the nondegenerate case (8).
Theorem 2.4 (Algebraic decay rates for the non-degenerate case (8)). Under the same assumptions of Theorem 2.2, let φ 0 ∈ H 2 ∩ L 2 α for any α > 0, where L 2 α is defined in (11). Then the solution u(x, t) satisfies for some C > 0 and for any t ≥ 0.
Theorem 2.5 (Exponential decay rates for the non-degenerate case (8)). Under the same assumptions in Theorem 2.2, let φ 0 ∈ H 2 ∩ L 2 we the weight function w e (z) satisfying w e (z) ∼ e d|z| as z → ±∞ for some d > 0. Then there are constants θ = θ(d, U z ) > 0 and C > 0 such that for any t ≥ 0.
3. Reformulation of the problem. As discussed in Subsection 2.1, we look for a solution of (1)-(3) of the form in certain solution spaces which are different in each sections. For simplicity of notation, we omit the bar inφ z throughout this paper. Without loss of generality, we assume u + = 0 and f (u + ) = 0.
Integrating (36) with respect to z over (z, ∞), we have the reformulated problem

TONG LI AND JEUNGEUN PARK
By (16), equation (37) is rewritten as and Here, F = O(φ 2 z ) and G = O(φ 2 zz ) provided that φ 2 z and φ 2 zz are small. We then state the following theorems. (8)). Under the assumptions of Theorem 2.2, there exists a constant δ 1 > 0 such that if φ 0 2 < δ 1 , the problem (39)-(38) has a unique global solution satisfying for some C > 0, and constants a > 0, R 1 < 0 and R 2 > 0 are determined in Section 4. Consequently, Theorem 3.2 (Degenerate case (9)). Under the assumptions of Theorem 2.3, there exists a constant δ w > 0 such that if φ 0 2 < δ w , the problem (39)-(38) has a unique global solution satisfying for some C > 0, where a > 0 and R < 0 are determined in Section 5, and n + is given in (82). Consequently, 4. The non-degenerate case -proof of Theorem 2.2. In this section, we prove Theorem 2.2 by proving Theorem 3.1. To achieve our stability result, we introduce a new weight function for the non-degenerate case (8).
For a small constant 0 < a < m, where m is defined in (23), there are constants R 1 < 0 and R 2 > 0 satisfying and With such constants a, R 1 , and R 2 , we now define our new weight function w(z) (47) From Proposition 1, one can easily check that w(z) is positive and uniformly bounded for all R, namely, for some M > 0. One can also check that w(z) belongs to C 2 (R).
In previous works [7,10,11], authors imposed conditions for the fluxes to be almost convex. In this paper, piecewisely defined w in (47) enables us to deal with non-convexity of fluxes. Due to our construction of w, the L 2 -integral of φ on the bounded domain does not appear in the L 2 -estimate of φ in (52). Now by manipulating the size of a > 0 in (47) and using the entropy condition (6), we complete the L 2 -estimates for φ. It is worthwhile to mention that we only need the entropy condition (6) instead of imposing additional conditions on the fluxes. Now we define the solution space as the Sobolev embedding theorem gives us for some C > 0. The global existence of φ in Theorem 3.1 is a consequence of the following two propositions, the local existence and the a priori estimates, by the continuation arguments.
The local existence can be shown in a standard way as in [20], so we omit the proof. In this section, we intend to establish the a priori estimates.
for some small 0 < a < m, where m is defined in (23) and R 1 < 0 and R 2 > 0 are defined in (45).
Proof. Multiplying (39) by 2wφ, we have where w is defined in (47). After integrating (53) with respect to z over R, we have Now we estimate the second term on the left hand side of (54) by considering two cases: z ∈ R\(R 1 , R 2 ) and z ∈ (R 1 , R 2 ), where R 1 and R 2 are defined in (45).
From the definition of w(z) in (47) and equation (25), direct calculation gives us and Combining (55) and (56), we obtain Differentiating (57) with respect to z, and substituting (25) and (26), it follows By (22) U z < 0 and by (46) Uz a + 1 > 0 for z ∈ R\(R 1 , R 2 ), we obtain from (58) that In order to establish L 2 -estimate (52), we show the following inequality for any z ∈ R\(R 1 , R 2 ) with some choice of a. To prove the inequality (60) with some choice of a, we find an upper bound of the left hand side of (60). Indeed, by (23) and (24), the first term on the left hand side of (60) is bounded by 6m 1−m 4 . For the second term on the left hand side of (60), noting that h (U ) = f (U ) − c from (6), h (U ) = f (U ), and there is an M > 0 such that |f (U )| ≤ M for any z ∈ R. Moreover, the non-degeneracy condition (8) yields that h (U ) = f (U ) − c is nonzero for any z ∈ R\(R 1 , R 2 ). Hence, the second term is bounded by Therefore, the left hand side of (60) is bounded by Now if we let a = m 2 , there is an α > 0 such that |f (U ) − c| ≥ α > 0 for any z ∈ R\(R 1 , R 2 ), where R 1 and R 2 are defined in (45) with a . Plugging |f (U ) − c| ≥ α into (61), we are able to find an a 0 > 0 such that For any a satisfying 0 < a ≤ min{a 0 , a },
. By the definition of w(z) in (47) and equation (25), it holds On the other hand, by (47), we obtain Thus, we have Hence, we arrive at Combining the estimates (64) and (68), and using (23), (24), (48), one deduces that for some C > 0. Consequently, we complete the proof of (52) by integrating (69) with respect to t.

Remark 2.
By choosing the constant a > 0 in (45) satisfying (63), the difficulty in dealing with the non-convexity of f is overcome, see (60). It is worth mentioning that the existence of such an a is guaranteed only when f (u ± ) = c. Since (60) no longer holds for the degenerate case (9) and (21), we need to reconstruct the weight function w(z) in Section 5.
Next, we derive estimates of the first order derivative of φ.
Proof. Differentiating (39) with respect to z, multiplying the result by 2φ, and integrating the resulting equation with respect to z over R, we have Integrating (71) with respect to t, using (23) and (24), and combining the L 2estimates (52), we obtain the desired H 1 -estimates (70).
We shall estimate the second order derivative of φ.
for all 0 ≤ t ≤ T.
Proof. Differentiating (39) with respect to z twice, multiplying the result by 2φ zz , and integrating the resulting equation with respect to z over R, we obtain Integrating (73) with respect to t, using (23) and (24), and combining the L 2estimates (52) and H 1 -estimates (70), the proof of (72) is completed.
In conclusion, from the L 2 -estimates (52), H 1 -estimates (70) and H 2 -estimates (72), there is a constant C > 0 such that for all 0 ≤ t ≤ T, where R 1 , R 2 and a are given in (45) and (60). Now we shall estimate the last term on the right hand side of (74), which is a collection of the cubic terms. From (40) and (41), one deduces Indeed, all the cubic terms in (74) are bounded by CN (t) t 0 φ z (·, τ ) 2 2 dτ for some C > 0, where N (t) is defined in (50). For example, t 0 R G zz φ zz dzdτ is estimated by using integration by parts, the Cauchy-Schwarz inequality and (75) as follows for some C > 0. Therefore, substituting the estimated results of the cubic terms into (74), we derive that for some C > 0. Taking δ 2 > 0 in Proposition 3 such that 4Cδ 2 < 1, from (77) we conclude that for any 0 ≤ t ≤ T , where T > 0 is from Proposition 3, for some C > 0 as desired in Proposition 3. Therefore, the proof of (42) in Theorem 3.1 is completed by the continuation arguments based on the local existence in Proposition 2 and the a priori estimates in Proposition 3. It remains to prove (43) in Theorem 3.1. From (42), we derive φ z (·, t) 2 1 → 0 as t → ∞, and it follows for all z ∈ R that We finally complete the proof of Theorem 3.1. Consequently, Theorem 2.2 is proved.

Remark 3.
In the case f (u) = u 2 2 , authors in [9,26,5] proved that there exist smooth traveling wave solutions for 0 < |u − − u + | < 2 and solutions become discontinuous at |u − − u + | = 2. Stability of smooth traveling waves were shown when 0 < |u − − u + | ≤ 1.8 in [15]. Applying the method developed in this paper to [15], we can improve the stability results in [15] 5. The degenerate case -proof of Theorem 2.3. We prove Theorem 2.3 by proving Theorem 3.2 in this section. We assume that f (u + ) = c < f (u − ) and n + satisfies (21). For simplicity of notation, we replace n + by n throughout this section.
We show global existence of φ to the problem (39)-(38) by using the weighted energy estimates. However, due to the degeneracy condition f (u + ) = c, the weight function w(z) in (47), which is introduced for the non-degenerate case (8), does not work for z in a neighborhood of ∞. Thus, to overcome such a difficulty, we need to construct a new weight functionw(z).
On the other hand, since c < f (u − ) which is nondegenerate, we can still use similar arguments in defining the weight function w(z) in (47) when z is bounded above.
For a small 0 < a < m, where m is defined in (23) and a is to be determined in Lemma 5.1, there is an R < 0 such that |U z (R)| = a and 0 < |U z (z)| < a, z < R.
With a and R satisfying (80), we introduce a new weight functionw(z) as follows.
One can easily check thatw(z) ∈ C 2 (R). Moreover, there is a significant difference betweenw(z) in (81) and w(z) defined for the non-degeneracy in (47). While w(z) is bounded (48),w(z) is unbounded and the rate is determined by the degeneracy of f at far fieldsw see (28). Now we define the solution space as with 0 < T ≤ ∞. By the Sobolev embedding theorem, if we set for some C > 0. We then prove Theorem 3.2 by establishing the following a priori estimates.
for all 0 ≤ t ≤ T and for some small 0 < a < m, where m is defined in (23) and R < 0 is defined in (80).
Proof. Multiplying (39) by 2wφ, and integrating the resulting equation with respect to z over R, we obtain wherew is defined in (81). In order to estimate the second term of (87), due to the structure ofw in (81), we consider two cases: z ≤ R and z > R.
For the non-degenerate part z ≤ R, we choose a small a > 0 in a similar way in obtaining (60) in Lemma 4.1. Similar to the derivation of (64) gives us for some C > 0. For the degenerate part z > R, from (65)-(67), it holds From (88) and (89), the second term of (87) is estimated as for some C > 0. Substituting (90) into (87), one has for some C > 0. Integrating (91) with respect to t, the proof of (86) is completed.
Moreover the following estimates are proved. We omit details.
for some C > 0 and any 0 ≤ t ≤ T.
for any 0 ≤ t ≤ T.
Now, in order to derive the desired a priori estimates for (44), we shall combine all the results from the weighted L 2 -estimates (86) and H 1 -and H 2 -estimates (92), (93). We then conclude that there is a C > 0 such that for all 0 ≤ t ≤ T, where a > 0 is determined in Lemma 5.1 and R < 0 is defined in (80). The third term on the right hand side of (94) can be estimated by the following inequalities for some C > 0, where Nw(t) is defined in (84). In fact, by (82), z 2n φ z φ zzz , and the Cauchy-Schwarz inequality, the second term of (96) is further estimated as for some C > 0. It is noticed that the last term on the right hand side of (94) is estimated by CNw(t) t 0 φ z (·, τ ) 2 2 dτ for some C > 0. If Nw(0) < δ w for some δ w > 0 small enough, the same arguments in obtaining the a priori estimates (78) from the L 2 -estimates (52), H 1 -estimates (70) and H 2estimates (72) yield ≤CN 2 w (0) for some C > 0 and all 0 ≤ t ≤ T. By (82) we finally obtain the desired estimates (44). 6. Decay rates for the non-degenerate case. In this section, we investigate algebraic and exponential rates of convergence for the non-degenerate case (8).
6.1. Algebraic decay rates. In this subsection, we prove that the perturbation converges algebraically in time if the initial perturbation decays algebraically in space. Kawashima and Matsumura [6] and Matsumura and Nishihara [17] introduced the rates of asymptotic speed based on the weighted energy estimates. We also refer to [15,19,21,22] for the improved techniques.
For α > 0, we define the solution space as then the Sobolev embedding theorem yields for some C > 0, where z α is defined in (10). Theorem 2.4 is a consequence of the following theorem.
the problem (39)-(38) has a unique global solution φ ∈ X α (0, ∞) satisfying and for any ε > 0 for some C > 0 and for all 0 ≤ t ≤ T . Theorem 6.1 is shown by the continuation arguments based on the local existence and the a priori estimates in a similar framework in proving Theorem 3.1. Hence, we devote to establish the a priori estimates in this subsection.
First, we derive L 2 −estimates for φ on |z| > r for some r > 0 in the following lemma.
Proof. With w defined in (47), multiplying (39) by 2(1+t) γ z β wφ, and integrating the resulting equation with respect to z over R, we have d dt R (1 + t) γ z β wφ 2 dz + R (1 + t) γ I 1 (z, t) + I 2 (z, t) dz + 2 where To estimate the second term on the left hand side of (104), we shall estimate R I 1 (z, t)dz and R I 2 (z, t)dz.
Integrating the resulting inequality (114) with respect to t, we finally obtain the desired (103).
Different from the L 2 -estimates without weights (52), the weighted L 2 -estimates (103) are not enough to conclude our algebraic decay rate results. Indeed, we need to obtain the weighted L 2 -estimate for φ on (−r, r) to be added to (103) so that we can yield the desired estimate (130) as in [6,17].