Application of the Weighted Energy Method in the Partial Fourier Space to Linearized Viscous Conservation Laws with Non-Convex Condition

As you know, the energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space Rn. Recently, the author studied half space problems in R+ = R+ × Rn−1 and developed the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable Rn−1. Then the author applied this energy method to the half space problem for linearized viscous conservation laws with convex condition and proved the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t → ∞ (see, [14]).


Introduction
As you know, the energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space R n .Recently, the author studied half space problems in R n + = R + × R n−1 and developed the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable R n−1 .Then the author applied this energy method to the half space problem for linearized viscous conservation laws with convex condition and proved the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t → ∞ (see, [14]).
In this chapter, we consider the half space problem for linearized viscous conservation laws with non-convex condition, and derive the asymptotic stability of planar stationary waves and the corresponding convergence rate.Our proof is based on the energy method in the partial Fourier space with the anti-derivative method.
In this present chapter, we are concerned with the half space problem for the viscous conservation laws: u(x,0)=u 0 (x). (1.3) is the space variable in the half space R n + = R + × R n−1 with n ≥ 2; we sometimes write as x =(x 1 , x ′ ) with x 1 ∈ R + and x ′ =(x 2 , ••• , x n ) ∈ R n−1 ; u(x, t) is the unknown function, u 0 (x) is the initial data satisfying and u b is the boundary data (assumed to be a constant) with u b < 0; is a smooth function of u ∈ R with values in R n and satisfies for u ∈ [u b ,0).Here we note that the condition (1.4) is the necessary condition for the existence of the planar stationary waves (for the detail, see Section 2.2).We emphasize that the assumption (1.4) is weaker than the convex condition for u ∈ [u b ,0] and f 1 (0)=0.Namely, we do not assume the convex condition for our problem (1.1)- (1.3).
For viscous conservation laws (1.1) with the convex condition (1.5), there are many results on the asymptotic stability of nonlinear waves.First, Il'in and Oleinik in [3] studied the stability of nonlinear waves in the one-dimensional whole space.Liu, Matsumura and Nishihara in the paper [8] discussed the stability of stationary waves in one-dimensional half space.More precisely, they proved the asymptotic stability of several kind of nonlinear waves such as rarefaction waves, stationary waves, and the superposition of stationary waves and rarefaction waves.Later, in a series of papers [5][6][7], their stability result of stationary waves in one-space dimension was generalized to the multi-dimensional case.Kawashima, Nishibata and Nishikawa [5] first considered the stability of non-degenerate planar stationary waves in two-dimensional half space and obtained the convergence rate t −1/4−α/2 in L ∞ norm by assuming that the initial perturbation is in L 2 α (R + ; L 2 (R)).Furthermore, the papers [6, 7] studied the n-dimensional problem in the L p framework.In particular, the paper [7] showed the stability of non-degenerate planar stationary waves and obtained the convergence rate t −(n/2)(1/2−1/p)−α/2 in L p norm under the assumption that the initial perturbation belongs to . Next, we refer to viscous conservation laws with non-convex condition.Liu and Nishihara in [9] and Nishikawa in [10] investigated the asymptotic stability of travelling waves in the one-dimensional and multi-dimensional whole space, respectively.On the other hand, Hashimoto and Matsumura in [1] studied the asymptotic stability of stationary waves in the one-dimensional half space.Especially, in order to relax the convex condition, Liu and Nishihara in [9] and Nishikawa in [10] employed the anti-derivative method and achieved the desired result.Moreover, Hashimoto and the author in [2] used the same method to derive the asymptotic stability of stationary waves for damped wave equations with non-convex convection term in one-dimensional half space.Inspired by these arguments, we try to relax the convex condition (1.5) and get the asymptotic stability of planar stationary wave for the multi-dimensional problem (1.1)- (1.3).Unfortunately, Nishikawa in the paper [10] considered some special situation for the nonlinear term to make a good combination of the energy method and the anti-derivative method.For the same reason, we will treat the special situation (for the detail, see Section 3).All these stability results mentioned above are obtained by employing the energy method in the physical space.On the other hand, it is useful to apply the energy method in the partial Fourier space to show sharper convergence rate.Indeed the author's paper [14] considered our problem (1.1)-(1.3)with the convex condition (1.5) and obtained the sharper convergence rate of the planar stationary waves.We shall show the result of the paper [14] in detail.
We are interested in the asymptotic stability of one-dimensional stationary solution φ(x 1 ) (called planar stationary wave) for the problem (1.1)-(1.3):φ(x 1 ) is a solution to the problem To show the stability, it is convenient to introduce the perturbation v and write the solution u in the form u(x, t)=φ(x 1 )+v(x, t).
The original problem (1.1)-(1.3) is then reduced to where Under the convex condition (1.5), the author in [14] showed the asymptotic stability of the planar stationary wave φ(x 1 ) by proving a sharp decay estimate for the perturbation v(x, t).
To this end we employed the energy method in the partial Fourier space Rn which is obtained by taking the Fourier transform with respect to the tangential variable is the Fourier variable corresponding to x ′ ∈ R n−1 .For the variable x 1 ∈ R + in the normal direction, we use L 2 space (or weighted L 2 space).As the result, for the corresponding linearized problem with f (φ + v) − f (φ) replaced by f ′ (φ)v in (1.8), we showed the following pointwise estimate with respect to where C and κ are positive constants.Here F denotes the Fourier transform with respect to x ′ ∈ R n−1 and |•| L 2 is the L 2 norm with respect to x 1 ∈ R + .This pointwise estimate (1.11) enables us to get the following sharp decay estimate: where • L 2 denotes the L 2 norm with respect to , and C is a positive constant.Furthermore, when the planar stationary wave φ(x 1 ) is non-degenerate, the author applied the weighted energy method in the partial Fourier space Rn . Namely, we used the weighted space L 2 α with respect to x 1 ∈ R + .In this case, the pointwise estimate (1.11) is improved to where where For the above results, we refer to the reeder [14] in detail.
The main purpose of this chapter is to derive the sharp decay estimate (1.11)- (1.14) for the linearized problem of (1.8)-(1.10)with non-convex condition (1.4), i.e., (1.15) with (1.9), (1.10).To overcome the difficulty occured by the non-convex condition, we make a good combination of the weighted energy method in partial Fourier space employed in [14] and the anti-derivative method employed in [2,9], and get the desired results.Once we obtain the linear stability results for the problem (1.15), (1.9), (1.10), we may apply this results to the asymptotic stability for the nonlinear problem (1.8)-(1.10).
The remainder of this chapter is organized as follows.In Section 2, we introduce function spaces and some preliminaries used in this chapter.Especially, we reformulate our problem (1.15), (1.9), (1.10) by using the anti-derivative method in Section 2.3.In the final section, we treat the half space problem for the reformulated viscous conservation laws and (1.15), (1.9), (1.10), and develop the weighted energy method in the partial Fourier space with the anti-derivative method.In this section, we derive pointwise estimates of solutions and prove the corresponding decay estimates.

Notations and function spaces
Let us consider functions defined in the half space x j denote the standard gradient and Laplacian with respect to x =( x 1 ,...,x n ), respectively.The symbol where . For a nonnegative integer s, we denote by , where ∂ k x ′ denotes the totality of all the k-th order derivatives with respect to x ′ ∈ R n−1 .Also, we denote by L p (R + ) the L p space with respect to . Now we introduce function spaces over the half space The norm is denoted by • L q (L p ) .When q = p, we simply write as We sometimes use By the definition (2.1) of the Fourier transform, we see that sup with C =(2π) −(n−1)/2 .Also, it follows from the Plancherel theorem that Let T > 0 and let X be a Banach space defined on the half space R n + .Then C([0, T]; X) denotes the space of continuous functions of t ∈ [0, T] with values in X.
In this paper, positive constants will be denoted by C or c.
(ii) Degenerate case where f ′ 1 (0)=0: In this case the problem (1.6)-(1.7)admits a unique smooth solution φ(x 1 ) if and only if u b < 0. The solution verifies φ x 1 > 0 and where q is the degeneracy exponent of f 1 and C is a positive constant.
In this chapter we only treat the stationary solutions φ(x 1 ) with φ x 1 > 0 and discuss their stability; however, we must get the similar stability result of the monotone decreasing stationary solutions by using the same argument introduced in this chapter.(We refer the reader to [2].)

Reformulated problem
In this subsection we reformulate our problem by the anti-derivative method.To this end we introduce a new function z(x, t) as Here, we assume the integrability of v(x, t) over R + .This transformation is motivated by the argument in Liu-Nishihara [9].By using (2.4), we can reformulate (1.8)-(1.10) in terms of z(x, t) as ) where Once we obtain the solution for the problem (2.5)-(2.7), the differentiation v = z x 1 is the solution for (1.8)-(1.10).Namely, we will apply the weighted energy method in the partial Fourier space and try ot derive the global solution in time to the reformulated problem (2.5)-(2.7).We will discuss this reformulated problem in Section 3 to prove our main theorems.

Weight function
We introduce the weight function employed in the weighted energy method.Our weight function is defined as where A is a positive constant determined in Lemma 2.2.This weight function is very important to derive a priori estimate in the latter section.For this weight function, we obtain the following lemma.
Then there exists a positive constant δ such that if A ≥ δ, then w(u) satisfies the following conditions: Here, C and c are some positive constants which independent of x 1 .
The detail of the proof is omitted here.For the details, we refer the reader to [2].

Asymptotic stability with convergence rates
In the final section, we apply our weighted energy method in the partial Fourier space to the linearized problem.We consider the linearized problem corresponding to the half space problem (2.5)-(2.7).Namely, we consider (2.5) with g j = 0 for j = 1, ••• , n.For this linearized equation, we treat the special situation that Then our initial value problem of the linearized equation is written as together with (2.6) and (2.7).Taking the Fourier transform with respect to x ′ ∈ R n−1 for the linearized problem (3.1), (2.6), (2.7), we obtain . This is the formulation of our linearized problem in the partial Fourier space Rn Here we note that v = z x 1 .By applying the weighted energy method to the above problems, we obtain the pointwise estimate of solutions.

Energy method
We apply the energy method to the problems (3.2) and (3.3) formulated in the partial Fourier space and derive pointwise estimates of solutions to (3.2).We use L 2 space for the variable x 1 ∈ R + in the normal direction.The result is given as follows.
Proof of Theorem 3.1.Throughout this proof, we use the weighted L 2 norm: , where w is the weight function defined by (2.8).For this weighted norm, by using Lemma 2.2, we see the following properties.
for the degenerate case : f ′ (0)=0. (3.10) We prove (i) and (ii) in Theorem 3.1 in parallel.We first derive (3.4).We multiply (3.2) 1 by w(φ) z and take the real part, obtaining where and w is a weight function defined by (2.8).By virtue of (2.9) 2 in Lemma 2.2, we have where c is a some positive constant.Therefore, integrating (3.11) in with a positive constant c 1 , where Here, by virtue of (2.9) 1 , the last term of the left-hand side of (3.13) is positive.We multiply (3.13) by e κ|ξ| 2 t (κ > 0) to get where C is a positive constant.
Then this yields the desired estimate: where γ ≥ 0, and C is a positive constant.These will be used in the next subsection.
, where C is a positive constant.Here we used (2.2) and the simple inequality with a constant C. By applying the same argument to the pointwise estimates (3.5) and (3.6) we can derive the decay estimate (3.8) and (3.9), respectively.Thus this completes the proof.

Weighted energy method
In the last subsection, we restrict to the non-degenerate case f ′ 1 (0) < 0 and apply the weighted energy method to the problems (3.2) and (3.3).This yields sharp pointwise estimates of solutions to (3.2).We use the weighted space L 2 α (R + ) (α ≥ 0) for x 1 ∈ R + in the normal direction and this gives the additional decay (1 + t) −α/2 .The result is stated as follows.

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The proof of Corollary 3.4 is completely same as the proof of Corollary 3.2 and omitted here.
Proof of Theorem 3.3.We use the weighted energy method to the problems (3.2) and (3.3) formulated in the partial Fourier space.Our computation is similar to the one used in [4,10,12,14] and divide into four steps.
Step 1. First, we show the following space-time weighted energy inequality: for γ ≥ 0 and 0 ≤ β ≤ α, where and C and κ are positive constants.Notice that D0 coincides with D1 in (3.14).
To prove (3.31), we use the equality (3.11).Notice that, by virtue of (2.9) 1 in Lemma 2.2, we have with positive constants c 2 and C. Now we multiply (3.11) by We integrate this equality over x 1 ∈ R + and use (3.12) and (3.32), obtaining for 0 ≤ β ≤ α, where we define and C is a positive constant.Here, by virtue of (2.9) 1 , the last term of the left-hand side of (3.33) is positive.We now observe that for any ǫ > 0, where C ǫ is a constant depending on ǫ.We apply this inequality to the term on the right-hand side of (3.33) by taking a = ẑx 1 .Noting that Dβ ≥|ẑ , we choose ǫ > 0so small that αCǫ ≤ c 1 .This yields As in (3.15), we choose κ > 0 such that κ ≤ c 3 .Then we multiply the resulting inequality by (1 + t) γ (γ ≥ 0) and integrate over [0, t].This yields Step 2. Next we show the following estimate for α ≥ 0: for each integer l with 0 ≤ l ≤ [α], where C and κ are positive constants.Note that if α ≥ 0is an integer, then (3.36) with l = α gives the desired estimate (3.28).

.37)
Then we prove (3.36) for l = j.To this end, we put γ = j and β = α − j in (3.31).This gives where we used (3.37) in the last estimate.This shows that (3.36) holds true also for l = j and therefore the proof of (3.36) is complete.
Step 3. Next, when α > 0 is not an integer, we show that for γ > α, where D1 is defined in (3.12), and C and κ are positive constants.Notice that (3.38) gives the desired estimate (3.28) even if α > 0 is not an integer.
To prove (3.38), we recall the inequality (3.25) which is the same as (3.31) with β = 0. We need to estimate the second term on the right-hand side of (3.25).This can be done by applying the technique due to Nishikawa in [10].When α > 0 is not an integer, we have from (3.36) with l =[α] that Step www.intechopen.com