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

Yoshihiro Ueda *Faculty of Maritime Sciences, Kobe University Japan*

#### **1. Introduction**

18 Will-be-set-by-IN-TECH

248 Fourier Transform Applications

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Wellesley, MA.

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***<sup>n</sup>* <sup>+</sup> = **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>***n*−<sup>1</sup> 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:

$$
\Delta u\_t - \Delta u + \nabla \cdot f(u) = 0,\tag{1.1}
$$

$$
\mu(0, \mathfrak{x}', t) = \mathfrak{u}\_{\mathfrak{b}'} \tag{1.2}
$$

$$
\mu(\mathbf{x}, \mathbf{0}) = \mu\_0(\mathbf{x}). \tag{1.3}
$$

Here *<sup>x</sup>* = (*x*1, ··· , *xn*) is the space variable in the half space **<sup>R</sup>***<sup>n</sup>* <sup>+</sup> <sup>=</sup> **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>***n*−<sup>1</sup> with *<sup>n</sup>* <sup>≥</sup> 2; we sometimes write as *x* = (*x*1, *x*� ) with *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> and *<sup>x</sup>*� = (*x*2, ··· , *xn*) <sup>∈</sup> **<sup>R</sup>***n*−1; *<sup>u</sup>*(*x*, *<sup>t</sup>*) is the unknown function, *u*0(*x*) is the initial data satisfying

$$
\mu\_0(\mathfrak{x}) \to 0 \quad \text{as} \quad \mathfrak{x}\_1 \to \infty,
$$

and *ub* is the boundary data (assumed to be a constant) with *ub* < 0; *f*(*u*)=(*f*1(*u*), ··· , *fn*(*u*)) is a smooth function of *<sup>u</sup>* <sup>∈</sup> **<sup>R</sup>** with values in **<sup>R</sup>***<sup>n</sup>* and satisfies

$$f\_1(0) = 0, \qquad f\_1(u) > f\_1(0) \,(=0) \tag{1.4}$$

for *u* ∈ [*ub*, 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

To show the stability, it is convenient to introduce the perturbation *v* and write the solution *u*

<sup>251</sup> Application of the Weighted Energy Method in the Partial Fourier Space to Linearized Viscous Conservation Laws with Non-Convex Condition

*u*(*x*, *t*) = *φ*(*x*1) + *v*(*x*, *t*).

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*).

which is obtained by taking the Fourier transform with respect to the tangential variable *x*� =

**<sup>R</sup>***n*−1. For the variable *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> in the normal direction, we use *<sup>L</sup>*<sup>2</sup> space (or weighted *<sup>L</sup>*<sup>2</sup> space). As the result, for the corresponding linearized problem with *f*(*φ* + *v*) − *f*(*φ*) replaced

(*φ*)*<sup>v</sup>* in (1.8), we showed the following pointwise estimate with respect to *<sup>ξ</sup>* <sup>∈</sup> **<sup>R</sup>***n*−<sup>1</sup>

where *C* and *κ* are positive constants. Here F denotes the Fourier transform with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−<sup>1</sup> and |·|*L*<sup>2</sup> is the *<sup>L</sup>*<sup>2</sup> norm with respect to *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**+. This pointwise estimate (1.11)

Furthermore, when the planar stationary wave *φ*(*x*1) is non-degenerate, the author applied

2*t*

*v*(0, *x*�

To this end we employed the energy method in the partial Fourier space **R**ˆ *<sup>n</sup>*


*vt* − Δ*v* + ∇ · (*f*(*φ* + *v*) − *f*(*φ*)) = 0, (1.8)

, *t*) = 0, (1.9)

*<sup>ξ</sup>* is the Fourier variable corresponding to *x*� ∈

�*v*(*t*)�L<sup>2</sup> <sup>≤</sup> *Ct*−(*n*−1)/4�*v*0�*L*<sup>2</sup>(*L*<sup>1</sup>), (1.12)

*<sup>α</sup>* with respect to *x*<sup>1</sup> ∈ **R**+. In this case, the pointwise estimate (1.11) is

<sup>−</sup>(*n*−1)/4�*v*0�*L*<sup>2</sup>

*<sup>α</sup>* norm with respect to *x*<sup>1</sup> ∈ **R**+. Consequently, we had the decay

−*κ*|*ξ*| 2*t* ) <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>*


<sup>+</sup> <sup>=</sup> **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>***n*−<sup>1</sup>

*α*

*<sup>x</sup>*�). For the above results, we refer to the reeder

(*φ*)*v*) = 0 (1.15)


<sup>+</sup>, �·�*L*<sup>2</sup>(*L*<sup>1</sup>) is the norm in

*<sup>ξ</sup>* . Namely, we used

, (1.13)

*α*(*L*<sup>1</sup>), (1.14)

<sup>+</sup> <sup>=</sup> **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>***n*−<sup>1</sup>

*<sup>ξ</sup>* :

*ξ*

*v*(*x*, 0) = *v*0(*x*), (1.10)

in the form

by *f* �

*L*2(**R**+; *L*<sup>2</sup>

improved to

where |·|*L*<sup>2</sup>

where �·�*L*<sup>2</sup>

[14] in detail.

estimate

*<sup>x</sup>*� <sup>∩</sup> *<sup>L</sup>*<sup>1</sup>

the weighted space *L*<sup>2</sup>

*<sup>α</sup>* denotes the *<sup>L</sup>*<sup>2</sup>

*<sup>α</sup>*(*L*<sup>1</sup>) is the norm in *<sup>L</sup>*<sup>2</sup>

The original problem (1.1)–(1.3) is then reduced to

(*x*2, ··· , *xn*) <sup>∈</sup> **<sup>R</sup>***n*−1; *<sup>ξ</sup>* = (*ξ*2, ··· , *<sup>ξ</sup>n*) <sup>∈</sup> **<sup>R</sup>***n*−<sup>1</sup>

enables us to get the following sharp decay estimate:

where �·�L<sup>2</sup> denotes the *<sup>L</sup>*<sup>2</sup> norm with respect to *<sup>x</sup>* = (*x*1, *<sup>x</sup>*�

*<sup>x</sup>*�), and *C* is a positive constant.

the weighted energy method in the partial Fourier space **R**ˆ *<sup>n</sup>*


�*v*(*t*)�L<sup>2</sup> <sup>≤</sup> *<sup>C</sup>*(<sup>1</sup> <sup>+</sup> *<sup>t</sup>*)−*α*/2*<sup>t</sup>*

*<sup>α</sup>*(**R**+; *<sup>L</sup>*<sup>2</sup>

linearized problem of (1.8)–(1.10) with non-convex condition (1.4), i.e.,

*<sup>x</sup>*� <sup>∩</sup> *<sup>L</sup>*<sup>1</sup>

*vt* − Δ*v* + ∇ · (*f* �

The main purpose of this chapter is to derive the sharp decay estimate (1.11)–(1.14) for the

where *v*0(*x*) = *u*0(*x*) − *φ*(*x*1); notice that *v*0(*x*) → 0 as *x*<sup>1</sup> → ∞.

the assumption (1.4) is weaker than the convex condition

$$f\_1''(u) > 0\tag{1.5}$$

for *u* ∈ [*ub*, 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 [5–7], 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* <sup>−</sup>1/4−*α*/2 in *L*<sup>∞</sup> norm by assuming that the initial perturbation is in *L*<sup>2</sup> *<sup>α</sup>*(**R**+; *<sup>L</sup>*2(**R**)). Furthermore, the papers [6, 7] studied the *n*-dimensional problem in the *L<sup>p</sup>* framework. In particular, the paper [7] showed the stability of non-degenerate planar stationary waves and obtained the convergence rate *t* <sup>−</sup>(*n*/2)(1/2−1/*p*)−*α*/2 in *L<sup>p</sup>* norm under the assumption that the initial perturbation belongs to *L*2 *<sup>α</sup>*(**R**+; *<sup>L</sup>*<sup>2</sup> *<sup>x</sup>*�).

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

$$-\phi\_{\mathbf{x}\_1\mathbf{x}\_1} + f\_1(\boldsymbol{\phi})\_{\mathbf{x}\_1} = \mathbf{0},\tag{1.6}$$

$$
\phi(0) = \mathfrak{u}\_{b'} \qquad \phi(\mathfrak{x}\_1) \to 0 \quad \text{as} \quad \mathfrak{x}\_1 \to \infty. \tag{1.7}
$$

To show the stability, it is convenient to introduce the perturbation *v* and write the solution *u* in the form

$$
\mu(\mathbf{x}, t) = \phi(\mathbf{x}\_1) + \upsilon(\mathbf{x}, t).
$$

The original problem (1.1)–(1.3) is then reduced to

2 Will-be-set-by-IN-TECH

for *u* ∈ [*ub*, 0] and *f*1(0) = 0. Namely, we do not assume the convex condition for our problem

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 [5–7], 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

studied the *n*-dimensional problem in the *L<sup>p</sup>* framework. In particular, the paper [7] showed the stability of non-degenerate planar stationary waves and obtained the convergence rate

<sup>−</sup>(*n*/2)(1/2−1/*p*)−*α*/2 in *L<sup>p</sup>* 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

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

−*φx*<sup>1</sup> *<sup>x</sup>*<sup>1</sup> + *f*1(*φ*)*x*<sup>1</sup> = 0, (1.6)

*φ*(0) = *ub*, *φ*(*x*1) → 0 as *x*<sup>1</sup> → ∞. (1.7)

<sup>1</sup> (*u*) > 0 (1.5)

*<sup>α</sup>*(**R**+; *<sup>L</sup>*2(**R**)). Furthermore, the papers [6, 7]

<sup>−</sup>1/4−*α*/2 in *L*<sup>∞</sup> norm by

*f* ��

the assumption (1.4) is weaker than the convex condition

in two-dimensional half space and obtained the convergence rate *t*

assuming that the initial perturbation is in *L*<sup>2</sup>

situation (for the detail, see Section 3).

(1.1)–(1.3).

*t*

*L*2 *<sup>α</sup>*(**R**+; *<sup>L</sup>*<sup>2</sup>

*<sup>x</sup>*�).

$$v\_t - \Delta v + \nabla \cdot (f(\phi + v) - f(\phi)) = 0,\tag{1.8}$$

$$v(0, \mathbf{x}', t) = 0,\tag{1.9}$$

$$v(\mathbf{x},0) = v\_0(\mathbf{x}),\tag{1.10}$$

where *v*0(*x*) = *u*0(*x*) − *φ*(*x*1); notice that *v*0(*x*) → 0 as *x*<sup>1</sup> → ∞.

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 **R**ˆ *<sup>n</sup>* <sup>+</sup> <sup>=</sup> **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>***n*−<sup>1</sup> *ξ* which is obtained by taking the Fourier transform with respect to the tangential variable *x*� = (*x*2, ··· , *xn*) <sup>∈</sup> **<sup>R</sup>***n*−1; *<sup>ξ</sup>* = (*ξ*2, ··· , *<sup>ξ</sup>n*) <sup>∈</sup> **<sup>R</sup>***n*−<sup>1</sup> *<sup>ξ</sup>* is the Fourier variable corresponding to *x*� ∈ **<sup>R</sup>***n*−1. For the variable *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> in the normal direction, we use *<sup>L</sup>*<sup>2</sup> space (or weighted *<sup>L</sup>*<sup>2</sup> space). As the result, for the corresponding linearized problem with *f*(*φ* + *v*) − *f*(*φ*) replaced by *f* � (*φ*)*<sup>v</sup>* in (1.8), we showed the following pointwise estimate with respect to *<sup>ξ</sup>* <sup>∈</sup> **<sup>R</sup>***n*−<sup>1</sup> *<sup>ξ</sup>* :

$$|\mathcal{F}v(\cdot,\xi,t)|\_{L^2} \le \mathcal{C}e^{-\kappa|\xi|^2t}|\mathcal{F}v\_0(\cdot,\xi)|\_{L^2} \tag{1.11}$$

where *C* and *κ* are positive constants. Here F denotes the Fourier transform with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−<sup>1</sup> and |·|*L*<sup>2</sup> is the *<sup>L</sup>*<sup>2</sup> norm with respect to *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**+. This pointwise estimate (1.11) enables us to get the following sharp decay estimate:

$$\|\boldsymbol{\upsilon}(t)\|\_{\mathcal{L}^2} \le \mathsf{C}t^{-(n-1)/4} \|\boldsymbol{\upsilon}\_0\|\_{L^2(L^1)'} \tag{1.12}$$

where �·�L<sup>2</sup> denotes the *<sup>L</sup>*<sup>2</sup> norm with respect to *<sup>x</sup>* = (*x*1, *<sup>x</sup>*� ) <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* <sup>+</sup>, �·�*L*<sup>2</sup>(*L*<sup>1</sup>) is the norm in *L*2(**R**+; *L*<sup>2</sup> *<sup>x</sup>*� <sup>∩</sup> *<sup>L</sup>*<sup>1</sup> *<sup>x</sup>*�), 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 **R**ˆ *<sup>n</sup>* <sup>+</sup> <sup>=</sup> **<sup>R</sup>**<sup>+</sup> <sup>×</sup> **<sup>R</sup>***n*−<sup>1</sup> *<sup>ξ</sup>* . Namely, we used the weighted space *L*<sup>2</sup> *<sup>α</sup>* with respect to *x*<sup>1</sup> ∈ **R**+. In this case, the pointwise estimate (1.11) is improved to

$$\|\mathcal{F}v(\cdot,\xi,t)\|\_{L^2} \le \mathcal{C}(1+t)^{-\mathfrak{a}/2}e^{-\kappa|\xi|^2t}|\mathcal{F}v\_0(\cdot,\xi)|\_{L^2\_{\mathfrak{a}'}} \tag{1.13}$$

where |·|*L*<sup>2</sup> *<sup>α</sup>* denotes the *<sup>L</sup>*<sup>2</sup> *<sup>α</sup>* norm with respect to *x*<sup>1</sup> ∈ **R**+. Consequently, we had the decay estimate

$$\|\|v(t)\|\|\_{\mathcal{L}^2} \le \mathcal{C}(1+t)^{-a/2} t^{-(n-1)/4} \|\|v\_0\|\|\_{L^2\_a(L^1)'} \tag{1.14}$$

where �·�*L*<sup>2</sup> *<sup>α</sup>*(*L*<sup>1</sup>) is the norm in *<sup>L</sup>*<sup>2</sup> *<sup>α</sup>*(**R**+; *<sup>L</sup>*<sup>2</sup> *<sup>x</sup>*� <sup>∩</sup> *<sup>L</sup>*<sup>1</sup> *<sup>x</sup>*�). 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.,

$$\nabla v\_t - \Delta v + \nabla \cdot (f'(\phi)v) = 0 \tag{1.15}$$

of *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> with values in *<sup>L</sup><sup>p</sup>*

When *q* = *p*, we simply write as

The space *<sup>H</sup>s*(*Lp*) = *<sup>H</sup>s*(**R**+; *<sup>L</sup><sup>p</sup>*

the space of *L*<sup>2</sup>

We sometimes use

*L*2(*Hs*) = *L*2(**R**+; *H<sup>s</sup>*

**2.2 Stationary solution**

*the following existence result:* (i) *Non-degenerate case where f* �

to [2, 8, 11–13].

is denoted by

*<sup>x</sup>*� with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1. The norm is denoted by �·�*Lq* (*Lp* ).

*<sup>x</sup>*�) consists of *<sup>H</sup><sup>s</sup>* functions of *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> with values in *<sup>L</sup><sup>p</sup>*

*Lp x*� *dx*<sup>1</sup> 1/2 .

*α*(*L*<sup>2</sup>).


*<sup>α</sup>* = �·�*L*<sup>2</sup>

*<sup>x</sup>*�) denotes the space of *<sup>L</sup>*<sup>2</sup> functions of *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> with values in *<sup>H</sup><sup>s</sup>*

<sup>2</sup>*k*|*v*ˆ(·, *<sup>ξ</sup>*)<sup>|</sup>

We review the results on the stationary problem (1.6)–(1.7). For the details, we refer the reader

*unique smooth solution φ*(*x*1) *with φx*<sup>1</sup> > 0 *(resp. φx*<sup>1</sup> < 0*), provided that ub* < 0 *(resp.* 0 < *ub and*

<sup>|</sup>*φ*(*x*1)| ≤ *Ce*−*cx*<sup>1</sup> , *<sup>x</sup>*<sup>1</sup> <sup>&</sup>gt; 0,

2 *L*2 *α dξ* 1/2

<sup>1</sup>(0) < 0*: In this case the stationary problem* (1.6)*–*(1.7) *admits a*

*<sup>α</sup>*(*Lp*) = *<sup>L</sup>*<sup>2</sup>

*<sup>x</sup>*� with respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1. The norm

*<sup>α</sup>*(**R**+; *<sup>L</sup><sup>p</sup>*

. (2.3)

<sup>+</sup>. Then *C*([0, *T*]; *X*) denotes

<sup>1</sup>(0) ≤ 0*, we have*

<sup>1</sup>(0) ≤ 0 *is necessary for the existence of*

*<sup>x</sup>*� with

*<sup>x</sup>*� with

*<sup>x</sup>*�) denotes

<sup>L</sup>*<sup>p</sup>* <sup>=</sup> *<sup>L</sup>p*(*Lp*), �·�L*<sup>p</sup>* <sup>=</sup> �·�*Lp* (*Lp* ).

<sup>253</sup> Application of the Weighted Energy Method in the Partial Fourier Space to Linearized Viscous Conservation Laws with Non-Convex Condition

(<sup>1</sup> <sup>+</sup> *<sup>x</sup>*1)*α*�*v*(*x*1, ·)�<sup>2</sup>

respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1. The norm is denoted by �·�*Hs*(*Lp* ). Also, *<sup>L</sup>*<sup>2</sup>

*<sup>α</sup>*(*Lp* ) =

L2 *<sup>α</sup>* = *<sup>L</sup>*<sup>2</sup>

 <sup>∞</sup> 0

By the definition (2.1) of the Fourier transform, we see that

�*∂k <sup>x</sup>*� *<sup>v</sup>*�L<sup>2</sup> *<sup>α</sup>* = **R***n*−<sup>1</sup> *ξ* |*ξ*|

**Proposition 2.1** ([8])**.** *Assume the condition* (1.4)*. Then f* �

*f*1(*u*) < *f*1(0) *for* 0 < *u* < *ub). The solution verifies*

�*v*�*L*<sup>2</sup>

respect to *<sup>x</sup>*� <sup>∈</sup> **<sup>R</sup>***n*−1, whose norm is given by

= *<sup>s</sup>* ∑ *k*=0

�*v*�*L*<sup>2</sup>(*Hs*) =

*<sup>α</sup>* functions of *<sup>x</sup>*<sup>1</sup> <sup>∈</sup> **<sup>R</sup>**<sup>+</sup> with values in *<sup>L</sup><sup>p</sup>*

 <sup>∞</sup> 0

�*v*(*x*1, ·)�<sup>2</sup>

 ∞ 0 �*∂k*

> sup *<sup>ξ</sup>*∈**R***n*−<sup>1</sup> *ξ*

with *C* = (2*π*)−(*n*−1)/2. Also, it follows from the Plancherel theorem that

*solutions to the stationary problem* (1.6)*–*(1.7)*. Conversely, under the condition f* �

Let *T* > 0 and let *X* be a Banach space defined on the half space **R***<sup>n</sup>*

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*.

*<sup>α</sup>*(*L*2), �·�L<sup>2</sup>

*Hs x*� *dx*<sup>1</sup> 1/2

*<sup>x</sup>*� *<sup>v</sup>*(*x*1, ·)�<sup>2</sup>

*L*2 *x*� *dx*<sup>1</sup> 1/2 = *<sup>s</sup>* ∑ *k*=0 �*∂k <sup>x</sup>*� *<sup>v</sup>*�<sup>2</sup> L2 1/2 .

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.
