Abstract
We construct a time-asymptotic expansion with pointwise remainder estimates for solutions to 1D compressible Navier–Stokes equations. The leading-order term is the well-known diffusion wave and the higher-order terms are a newly introduced family of waves which we call higher-order diffusion waves. In particular, these provide an accurate description of the power-law asymptotics of the solution around the origin \(x=0\), where the diffusion wave decays exponentially. The expansion is valid locally and also globally in the \(L^p({\mathbb {R}})\)-norm for all \(1\le p\le \infty \). The proof is based on pointwise estimates of Green’s function.
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1 Introduction
The equations
describe the motion of a 1D viscous compressible flow. Here v(x, t) is the specific volume (the reciprocal of the density \(\rho \)) and u(x, t) is the flow velocity; t is the time and x is the Lagrangian mass coordinate related to the Eulerian coordinate X by \(x=\int _{X_0(t)}^{X}\rho (X',t)\, dX'\), where \(X_0(t)\) is the trajectory of a particle moving with the fluid and initially placed at \(X_0(0)=0\). The system above models barotropic flow, that is, the pressure p(v) does not depend on the temperature. We assume that \(p'(v)<0\) and \(p''(v)\ne 0\) for \(v>0\) and that the viscous coefficient \(\nu \) is a positive constant. The system is often called the p-system and is a typical example of quasilinear hyperbolic–parabolic viscous conservation laws.
The purpose of this paper is to construct a time-asymptotic expansion of the solution to (1) together with pointwise estimates for the remainder. We shall consider solutions close to the steady state \((v_S,u_S)\equiv (1,0)\). To study the long-time asymptotics of such solutions, it is convenient to consider
instead of (v, u). Here \(c=\sqrt{-p'(1)}\) is the speed of sound for the state \((v_S,u_S)\).
It is well-known that \(u_i\) has the diffusion wave \(\theta _i\) as its asymptotic profile. Here \(\theta _i\) (\(i=1,2\)) is the self-similar solution to the convective viscous Burgers equation
where \(\lambda _i=(-1)^{i-1}c\), \(M_i=\int _{-\infty }^{\infty }u_i(x,0)\, dx\), and \(\delta (x)\) is the Dirac delta function. An explicit formula for \(\theta _i\) is available through the use of Cole–Hopf transformation:
The diffusion wave \(\theta _i\) describes the leading-order asymptotics in the \(L^p({\mathbb {R}})\)-norm. In fact, we have the following optimal decay estimates [14]:
The key to proving the \(L^p\)-decay estimates above—especially for \(p=1\)—is the pointwise estimates for Green’s function of the linearization of (1) around \((v_S,u_S)\). These, in fact, allow us to obtain pointwise estimates for the solution itself [10]:
The \(L^p\)-decay estimates are obtained by integrating this.
Pointwise estimates (5) allow us to deduce not just global \(L^p\)-estimates but also local ones. In particular, we have \(|(u_i-\theta _i)(x,t)|\lesssim t^{-3/4}\) for \(x=\lambda _i t+O(1)\). Since \(\theta _i(x,t)\lesssim t^{-1/2}\) for \(x=\lambda _i t+O(1)\), the diffusion wave \(\theta _i\) also describes the leading-order asymptotics locally around the characteristic line \(x=\lambda _i t\). However, the situation is different around the origin \(x=0\). As can be seen from (4), the diffusion wave \(\theta _i\) decays exponentially fast around the origin \(x=0\) but (5) implies \(|(u_i-\theta _i)(x,t)|\lesssim t^{-3/2}\) for \(x=O(1)\). Thus the diffusion wave \(\theta _i\) provides almost no information about the long-time asymptotics around \(x=0\); we need new waves to capture the asymptotic behavior there.
In [12], van Baalen, Popović, and Wayne constructed a time-asymptotic expansion of \(u_i\) in an \(L^2\)-framework. The leading-order term of the expansion is the diffusion wave \(\theta _i\) but the first higher-order term beyond \(\theta _i\) turns out to be a wave decaying algebraically as \(t^{-3/2}\) around the origin. It is then natural to expect that this new wave captures the leading-order asymptotics of the flow around \(x=0\). However, the decay estimate for the remainder of the expansion is given in the \(H^1({\mathbb {R}})\)-norm. This implies only a far from optimal decay estimate around \(x=0\). For this reason, we cannot conclude that the higher-order term describes the leading-order asymptotics of the flow around \(x=0\).
To overcome this issue, we construct a time-asymptotic expansion of \(u_i\) with pointwise estimates for the remainder. The leading-order term is the diffusion wave \(\theta _i\) and the higher-order terms are higher-order diffusion waves \(\xi _{i;n}\) (\(n\ge 1\)) defined in the next section. It turns out that \(|\xi _{i;n}(x,t)|\lesssim t^{-(2-1/2^n)}\) for \(x=O(1)\) as \(t\rightarrow \infty \). Setting \(n=1\), we see that \(|\xi _{i;1}(x,t)|\lesssim t^{-3/2}\) for \(x=O(1)\). The pointwise estimates for the remainder imply \(|(u_i-\xi _{i;1})(x,t)|\lesssim t^{-7/4}\) for \(x=O(1)\), thus it is rigorously proved that \(\xi _{i;1}\) describes the leading-order asymptotics of \(u_i\) for \(x=O(1)\). In addition, thanks to the pointwise estimates, our asymptotic expansion is valid not only in the \(L^2({\mathbb {R}})\)-norm but also in the \(L^1({\mathbb {R}})\)-norm.
The proof is based on pointwise estimates of Green’s function, and the basic strategy follows that of [10]. The most non-trivial part of the proof is perhaps the definition of the higher-order diffusion waves \(\xi _{i;n}\) (\(n\ge 1\)); see (7). Although the differential equation defining \(\xi _{i;n}\) does not seem to have a simple solution formula such as (4),Footnote 1 we use its structure (by the help of Lemma A.1) to analyze cancellation effects which are crucial in nonlinear estimates; see the proof of Lemma 3.6.
Before concluding the introduction, we briefly comment on related works. Diffusion wave approximations and pointwise estimates of solutions has been extensively studied for hyperbolic–parabolic systems [10], hyperbolic–elliptic systems [3], hyperbolic balance laws [13, 15], the Boltzmann equation [8], and so on. In these works, nonlinear diffusion waves similar to \(\theta _i\) were constructed and pointwise estimates of solutions were obtained. However, to the best of our knowledge, time-asymptotic expansions with pointwise estimates have not been obtained previously. We mention that the author already analyzed the second-order term \(\xi _{i;1}\) in connection with a fluid–structure interaction problem in [7]; the complete asymptotic expansion, however, was not given. We also comment that for multidimensional incompressible Navier–Stokes equations, time-asymptotic expansions were studied for example in [1, 2]. Because the nonlinearity is weaker compared to the 1D case, nonlinear waves similar to \(\xi _{i;n}\) do not appear in these works.
In the next section, we state our main results. These are proven in Sect. 3.
2 Main Results
To state our main results (Theorem 2.1) we start by defining and discussing the properties of the higher-order diffusion waves \(\xi _{i;n}\) (\(n\ge 1\)) mentioned in the introduction.
2.1 Higher-Order Diffusion Waves
Let (v, u) be the solution to (1). Then define \(u_i\) by (2) and set
Let \(\xi _{i;0}=\theta _i/2\) with \(\theta _i\) defined by (3). We then define the higher-order diffusion waves \(\xi _{i;n}\) (\(n\ge 1\)) inductively by the equations
Here \(\lambda _i=(-1)^{i-1}c\) and \(i'=3-i\), that is, \(1'=2\) and \(2'=1\). We remind the reader that \(c=\sqrt{-p'(1)}>0\).
Although we do not have a simple explicit formula for \(\xi _{i;n}\) like (4), we can still understand its asymptotic behavior quite well. To explain this, we introduce
and
Then we have the following decay estimates for \(\xi _{i;n}\) (we postpone the proof until we later prove a finer version in Lemma 3.1):
Proposition 2.1
Let \(n\ge 1\) and \(\varepsilon =\max (M_1,M_2)\). For \(k\ge 0\), if \(\varepsilon \) is sufficiently small, we have
for some positive constant \(C_{n,k}\). In particular, when \(|x|\le K\) for some fixed \(K>0\), we have
for some \(C_{n,K}>0\). Moreover, for any \(1\le p\le \infty \), there exists \(C_{n,p}>0\) such that
We can also prove more detailed estimates if we focus on x with \((-1)^{i-1}x\ge 0\). Let
and
We then have the following asymptotic formula. This is obtained from Lemma 3.2 proved in the next section.
Proposition 2.2
Let \(n\ge 1\) and \(\varepsilon =\max (M_1,M_2)\). For any \(K>0\), if \(\varepsilon \) is sufficiently small, there exist \(A_{i;n}\), \(B_{i;n}\), and \(C_n>0\) such that
for x with \(-K\le (-1)^{i-1}x\). The constants \(A_{i;n}\) and \(B_{i;n}\) are determined from \((M_1,M_2)\) defined by (6).
Remark 2.1
The function \(f_{i;n}\) appears in [12, Section 4]. It is shown that \(f_{i;n}(z)\) decays exponentially as \((-1)^{i-1}z\rightarrow \infty \) but decays algebraically as \(f_{i;n}(z)\sim z^{-\alpha _{n-1}}\) in the limit \((-1)^{i-1}z\rightarrow -\infty \). In particular, if \(|x|\le K\) for some fixed \(K>0\), we have
Remark 2.2
With some additional effort, we can show that, for \(-K\le (-1)^{i-1}x\), the higher-order diffusion waves \(\xi _{i;n}\) (\(n\ge 1\)) are asymptotically equivalent to the higher-order terms of the asymptotic expansion constructed in [12].
2.2 Time-Asymptotic Expansion with Pointwise Remainder Estimates
Let \(\varvec{u}_0=(v_0-1,u_0)\) and denote its anti-derivatives by \(\varvec{u}_{0}^{\pm }\), that is,
Our main theorem is the following:
Theorem 2.1
For \(\varvec{u}_0=(v_0-1,u_0)\in H^6({\mathbb {R}})\times H^6({\mathbb {R}})\), let (v, u) be the solution to (1). Define \(u_i\), \(\theta _i\), and \(\xi _{i;n}\) by (2), (3), and (7), respectively. Set
where \(i'=3-i\) and \(\gamma _i=(-1)^i \nu /(4c)\). Then for \(n\ge 1\), there exist positive constants \(\delta _n\) and \(C_n\) such that if
the solution (v, u) satisfies the pointwise estimates
for all \(x\in {\mathbb {R}}\) and \(t\ge 0\). Here \(\Psi _{i;n}\) is defined by (8).
As a corollary, we obtain the following \(L^p\)-decay estimates. Combining this with Proposition 2.1, it follows that \(u_i \sim \theta _i+\sum _{n=1}^{\infty }\xi _{i;n}\) is a time-asymptotic expansion in the \(L^p({\mathbb {R}})\)-norm for all \(1\le p\le \infty \).
Corollary 2.1
Under the assumptions of Theorem 2.1, we have the optimal \(L^p\)-decay estimate
Proof
The same bound for \(u_i-\theta _i-\sum _{k=1}^{n}u_{i;k}\) easily follows from Theorem 2.1. We can replace \(u_{i;k}\) by \(\xi _{i;k}\) thanks to (4) and Proposition 2.1. \(\quad \square \)
We also obtain the following local-in-space decay estimates:
Corollary 2.2
Under the assumptions of Theorem 2.1, when \(|x|\le K\) for some fixed \(K>0\), we have
Moreover, there exist constants \(\{ A_{i;k} \}_{k=1}^{n}\) determined from \((M_1,M_2)\) such that
Here \(M_i\) and \(f_{i;k}\) are defined by (6) and (11), respectively.
Proof
Again, the same bound for \(u_i-\theta _i-\sum _{k=1}^{n}u_{i;k}\) easily follows from Theorem 2.1. We can then replace \(u_{i;k}\) by \(\xi _{i;k}\) thanks to (4) and Proposition 2.1. The second inequality follows from Proposition 2.2. \(\quad \square \)
By Corollary 2.2, and also Remark 2.1, we now have a detailed picture of the power-law asymptotics of the solution around \(x=0\) where the diffusion waves decay exponentially.
Remark 2.3
The term \(\gamma _{i'}\partial _x \theta _i\) and \(\gamma _{i'}\partial _x \xi _{i;n}\) are both neglected in the two corollaries above. These are negligible in the \(L^p({\mathbb {R}})\)-norm and locally around \(x=0\) but are important in the neighborhood of the other characteristic line \(x=-\lambda _i t\). For this reason, these terms are required in the statement of Theorem 2.1.
Remark 2.4
The rather strong \(H^6\)-regularity is required to invoke pointwise estimates of \(\partial _x(u_i-\theta _i)\) provided by [10, Theorem 2.6 and Remark 2.8]. The proof involves energy estimates up to the \(H^6({\mathbb {R}})\)-norm. These also imply a unique global-in-time existence theorem in appropriate Sobolev spaces. Off course, global-in-time existence of solutions can be proved with much lower regularity [4, 9], but proving detailed pointwise estimates for such data seems to be difficult at this point.
Remark 2.5
We add a comment on taking the limit \(n\rightarrow \infty \) in Theorem 2.1 and also on a possible route to expand the solution to even higher order. A careful examination of the proof shows that the constant \(C_n\) in Theorem 2.1 grows as \(2^n\). So we cannot simply take the limit. However, as pointed out in [12, p. 1955], it might be possible to take the limit by adding a logarithmic weight:
Here, \(C_{\infty }\) is a constant independent of n and \(\Psi _{i;\infty }(x,t)=\lim _{n\rightarrow \infty }\Psi _{i;n}(x,t)\). Since \(\Vert \Psi _{i;\infty }(\cdot ,t) \Vert _{L^{\infty }}\lesssim t^{-1}\), to study an asymptotic expansion beyond the order \(O(t^{-1}\log t)\), it seems that we need to identify waves describing this order. Such waves are identified for example in [5] for generalized Burgers equations. Analogous results for hyperbolic–parabolic systems are, as far as I know, not known. If such waves are identified, we might be able to expand the solution beyond the order \(O(t^{-1}\log t)\). And drawing an analogy between the heat equation, terms beyond this order should also depend on higher-order moments \(\int _{-\infty }^{\infty }x^k u_i(x,0)\, dx\) and not just on \(M_i=\int _{-\infty }^{\infty }u_i(x,0)\, dx\).
3 Proof
The following function appears frequently in the subsequent part of the paper:
Here \(\lambda \in {\mathbb {R}}\) and \(\alpha ,\mu >0\). Note that
for some positive constants \(A_0\) and \(B_0\). In what follows, the symbols C and \(\nu ^*\) denote sufficiently large constants.
3.1 Pointwise Estimates of the Higher-Order Diffusion Waves
We start with the proofs of Propositions 2.1 and 2.2.
Proposition 2.1 follows from the following finer version:
Lemma 3.1
Let \(n\ge 1\) and \(\varepsilon =\max (M_1,M_2)\). If \(\varepsilon \) is sufficiently small, we have
for any integer \(k\ge 0\). In particular, we have
Proof
We assume \(t\ge 4\) in what follows (the lemma is otherwise easier to prove). The lemma is trivial for \(n=0\) if we set \(\xi _{i;0}=\theta _i/2\) and \(\xi _{i';-1}=0\). So it suffices to prove the lemma for n assuming that it holds for \(n-1\ge 0\). In what follows, we only prove the case of \(i=1\) and \(k=0\) since the other cases are similar. Note first that, by (7) and Duhamel’s principle, we have \(\xi _{i;n}(x,t)=\zeta _{1;n}(x,t)+\eta _{1;n}(x,t)\), where
and
We first consider \(\zeta _{1;n}(x,t)\). Set \(I(x,t)=-\sqrt{2\pi \nu }\zeta _{1;n}(x,t)\) and \(f=\theta _2 \xi _{2;n-1}\). By Lemma A.1, we have
where
and
Here \(L_2=\partial _t-c\partial _x-(\nu /2)\partial _{x}^{2}\). By the induction hypothesis, we have
By Lemmas A.2 and A.3, we obtain
Next, note that (3) and (7) imply
Then by Lemma A.4, we obtain
We have thus proved that
We next consider \(\eta _{1;n}(x,t)\). Note that it is the solution to
This variable coefficient equation can be solved by an iteration scheme. Let \(\eta _{1;n}^{(1)}\) be the solution to
and \(\eta _{1;n}^{(k)}\) (\(k\ge 2\)) be the solution to
Then we can write \(\eta _{1;n}\) as
We now give bounds for \(\eta _{1;n}^{(k)}\) (\(k\ge 1\)) inductively. Note first that
and that (18) implies
for some positive constants \(A_1\) and \(\nu '\). Then by [10, Lemma 3.2], we obtain
for some \(M>0\). This means that the inequality
holds for \(l=1\). We then show that (21) holds for \(l=k+1\) assuming that it holds for \(l=k\). By the induction hypothesis and (14), we have
Applying [10, Lemma 3.2] again, this time to the integral representation
we obtain
Therefore, (21) holds for any \(l\ge 1\), and by taking \(\varepsilon \) sufficiently small, we get
Combining this with (18), we obtain (15). \(\quad \square \)
Remark 3.1
The proof above can be modified to show that
holds for all \(m\ge n\) with the smallness of \(\varepsilon \) depending only on n and k.
We next prove Proposition 2.2. (The proof is rather lengthy and may be skipped; the rest of the paper can be read independently.) Define \(\zeta _{i;n}\) and \(\eta _{i;n}\) by (16) and (17), respectively. Then Proposition 2.2 is a direct consequence of the following lemma.
Lemma 3.2
Let \(n\ge 1\) and \(\varepsilon =\max (M_1,M_2)\). Fix \(k\ge 0\). For any \(K>0\), if \(\varepsilon \) is sufficiently small, there exist \(A_{i;n}\), \(B_{i;n}\), and \(C_{n,k}>0\) such that
and
when \(-K\le (-1)^{i-1}x\). Here g and \(f_{i;n}\) are defined by (9) and (11), respectively.
Proof
The lemma is proved by induction in n.
We first consider the case of \(n=1\). Let \(i=1\) and \(k=0\) (the other cases are similar). We start with the proof of (22). Note that (16) implies
In addition, by (4) and (10), we have
Hence we may write
where
and
Noting that \(\lim _{x\rightarrow \infty }r(x,t)=0\), we can show that
where \(L_2=\partial _t-c\partial _x-(\nu /2)\partial _{x}^{2}\). We then have
Concerning the second term, similar calculations leading to the bound of \(\zeta _{1;n}(x,t)\) in Lemma 3.1 imply
for \(-K\le x\). For the first term, note that a simple change of variable yields
Therefore,
We then set
and show that \(|I(x,t)|\le C\Theta _2(x,t;c,\nu ^*)\) for \(-K\le x\). We first consider Case (i) \(|x-c(t+1)|\le (t+1)^{1/2}\). In this case, we simply have
We next consider Case (ii) \(-K\le x\le c(t+1)-(t+1)^{1/2}\). The integral over \((-1,0)\) is easy to handle. For \(s\in (t,\infty )\) on the other hand, when t is large (the case when t is not large is easier), we have
Hence
We end the analysis of \(\zeta _{1;1}\) by considering Case (iii) \(x\ge c(t+1)+(t+1)^{1/2}\). When \(s>-1\), we have
From these, it follows that \(|I(x,t)|\le C\Theta _2(x,t;c,\nu ^*)\) as in Case (ii). These prove (22) for \(n=1\) by setting
We next prove (23) by using the series representation (19). We first consider
The bound (22) for \(\zeta _{1;1}(x,t)\) implies
This holds for all \(x\in {\mathbb {R}}\) since \(\theta _1(x,t)\) decays exponentially for \(x\le -K\). Plugging this into (20) and arguing similarly to the analysis of \(\zeta _{1;1}\), we get
where
Similar analysis for \(\eta _{1;1}^{(k)}(x,t)\) (\(k\ge 2\)) shows that
Taking the sum \(\sum _{k=1}^{\infty }\), it follows that (23) holds for \(n=1\) with
We next prove the lemma for n assuming that it holds for \(n-1\). Let \(i=1\) and \(k=0\) (the other cases are similar). The induction hypothesis and Lemma 3.1 imply
for all \(x\in {\mathbb {R}}\) (not just for \(x\le K\)). We also have \(\partial _x h(x,t)=O(\varepsilon ^{n+1})\Theta _{\alpha _{n-1}+2}(x,t;-c,\nu ^*)\). Using these, we can show that
where \(L_2=\partial _t-c\partial _x-(\nu /2)\partial _{x}^{2}\). Then similar calculations leading to the bound of \(\zeta _{1;1}(x,t)\) above imply (22) with
where
The bound (23) for \(\eta _{1;n}(x,t)\) is proved in a way similar to that for \(n=1\). This ends the proof of the lemma. \(\quad \square \)
For the proof of Theorem 2.1, it is convenient to unify \((\xi _{i;n})_{n=1}^{\infty }\) into a single function
Taking the infinite sum of (7), we see that \((\Xi _1,\Xi _2)\) is the solution to the system
Then Lemma 3.1 and Remark 3.1 imply the following:
Lemma 3.3
Let
Here \(\xi _{i;0}=\theta _i/2\). Then for \(n\ge 0\), if \(\varepsilon =\max (M_1,M_2)\) is sufficiently small, we have
for any integer \(k\ge 0\). In particular, we have
3.2 Proof of Theorem 2.1
Let us explain the strategy to prove Theorem 2.1. Note first that by Lemma 3.3, it suffices to prove the following:
Theorem 3.1
For \(\varvec{u}_0=(v_0-1,u_0)\in H^6({\mathbb {R}})\times H^6({\mathbb {R}})\), let (v, u) be the solution to (1). Define \(u_i\), \(\theta _i\), and \(\Xi _i\) by (2), (3), and (25), respectively. Then for \(n\ge 1\), there exist positive constants \(\delta _n\) and \(C_n\) such that if \(\delta \le \delta _n\), where \(\delta \) is defined by (12), the solution (v, u) satisfies the pointwise estimates
for all \(x\in {\mathbb {R}}\) and \(t\ge 0\). Here \(\gamma _i=(-1)^i \nu /(4c)\) and \(i'=3-i\).
To prove Theorem 3.1, we set
and define P(t) by
Our goal is then to prove the inequality
From this inequality, taking \(\delta \) sufficiently small, we can conclude that \(P(t)\le C\delta \) for all \(t\ge 0\) by a standard argument (see Sect. 3.2.4).
Remark 3.2
For the argument above to work, we first need to show that P(t) is finite. This can be proved, for example, by examining the iterative scheme in [6, Section 2.1] for the construction of the local-in-time solution to (1). The key step of the scheme consists of solving a variable coefficient parabolic equation, and by the Levi parametrix method, we can prove a gaussian upper bound for the fundamental solution. This bound allows us to control the spatial decay of each approximate solution, and by taking the limit, we can check that P(t) is finite at least for a short period of time. By the calculations below, it follows that (28) and hence \(P(t)\le C\delta \) hold for this short duration. Then a standard continuity argument shows that \(P(t)\le C\delta \) actually holds for all \(t\ge 0\).
The proof of (28) is based on pointwise estimates of Greens’ function [10, 11] which we shall explain in the next section. We also give an integral formulation of (1). In the remaining sections, we prove bounds for the terms appearing in the integral equations which yield (28).
3.2.1 Pointwise Estimates of Green’s Function and Integral Equations
Our equations (1) can be written in the form
with
The matrix A has right and left eigenvectors \(r_i\) and \(l_i\) (\(i=1,2\)), corresponding to the eigenvalue \(\lambda _i=(-1)^{i-1}c\), given by
We note that (2) can be written as \(u_i=l_i(v-1 \, u)^{T}\).
We define Green’s function \(G=G(x,t)\in {\mathbb {R}}^{2\times 2}\) for the linearization of (29) as the solution to
where \(\delta (x)\) is the Dirac delta function and \(I_2\) is the \(2\times 2\) identity matrix. In addition, define \(G^*=G^*(x,t)\in {\mathbb {R}}^{2\times 2}\) by
The next theorem is of fundamental importance in our analysis.
Theorem 3.2
([10, Theorem 5.8] and [11, Theorem 1.3]) For any \(k\ge 0\), we have
where \(\delta ^{(k)}(x)\) is the k-th derivative of the Dirac delta function and \(Q_j=Q_j(t)\) is a \(2\times 2\) polynomial matrix. In particular,
Moreover, with \(\gamma _i=(-1)^i \nu /(4c)\), we have
Here O(1) is a bounded scalar function.
For the analysis of \(u_i\), we need pointwise estimates for
We note that
where \(\delta _{ij}\) is the Kronecker delta. Then Theorem 3.2 implies
where
Moreover, we have
We next write down an integral equation for \(v_i\) defined by (26). Let
where \(\Xi _i\) and N are defined by (24) and (30), respectively. Then by Duhamel’s principle, we obtain
Lemma 3.4
The function \(v_i\) defined by (26) satisfies the integral equation
Here \(\gamma _i=(-1)^i \nu /(4c)\) and \(i'=3-i\).
We set
and
Lemma 3.4 may then be written as
In the next two sections, we prove pointwise estimates for \({\mathcal {I}}_i(x,t)\) and \({\mathcal {N}}_i(x,t)\).
3.2.2 Contribution from the Initial Data
Our goal in this section is to prove the following pointwise estimates for \({\mathcal {I}}_i(x,t)\) defined by (34):
Lemma 3.5
For any \(n\ge 1\), there exist positive constants \(\delta _n\) and \(C_n\) such that if (12) holds, then we have
for all \(x\in {\mathbb {R}}\) and \(t\ge 0\).
Proof
We assume \(t\ge 1\) below (the case when \(t<1\) is easier to handle). Let
and
Then of course \({\mathcal {I}}_i(x,t)={\mathcal {I}}_{i,1}(x,t)+{\mathcal {I}}_{i,2}(x,t)\).
We first show
For this purpose, set
and \(\eta =(\eta _1 \, \eta _2)^{T}\). We then have
By the definition of \(M_i\), see (6), we have
This and (12) imply
We first consider Case (i) \(|x-\lambda _i t|\le (t+1)^{1/2}\). In this case, integration by parts and (31) yield
We next consider Case (ii) \((t+1)^{1/2}<|x-\lambda _i t|<t+1\) with \(x-\lambda _i t>0\) (the case when \(x-\lambda _i t<0\) is similar). Again, by integration by parts,
We finally consider Case (iii) \(|x-\lambda _i t|\ge t+1\). For brevity, we assume \(x-\lambda _i t>0\). In this case, by (12), we have
We next show
Writing
and applying (33), we see that it suffices to show that
are all bounded by \(C\delta \Psi _{i;n}(x,t)\). First, this is trivial for D(x, t). Next, since \(A(x,t)=\gamma _{i'}\partial _x {\mathcal {I}}_{i',1}(x,t)\), modifying the calculations above for \({\mathcal {I}}_{i,1}(x,t)\) yield the bound for A(x, t). The bound for B(x, t) is also obtained in a way similar to that for \({\mathcal {I}}_{i,1}(x,t)\) (except that we don’t need \(\eta _i\) in the analysis).
Let us finally consider C(x, t). First, Case (i) \(|x-\lambda _{i'}t|\le (t+1)^{1/2}\) is easy:
Case (ii) \(|x-\lambda _{i'}t|>(t+1)^{1/2}\) with \(x-\lambda _{i'}t>0\) is as follows:
The case when \(x-\lambda _{i'}t<0\) is similar. This ends the proof of the lemma. \(\quad \square \)
3.2.3 Contribution from the Nonlinear Terms
Our goal in this section is to prove the following pointwise estimates for \({\mathcal {N}}_i(x,t)\) defined by (35):
Lemma 3.6
For any \(n\ge 1\), there exist positive constants \(\delta _n\) and \(C_n\) such that if (12) holds, then we have
for all \(x\in {\mathbb {R}}\) and \(t\ge 0\).
To prove this lemma, we first prove some preparatory lemmas. To state these, we introduce the notation
for a function \(f=f(x,t)\).
Lemma 3.7
Let \(n\ge 1\) and \(\varepsilon =\max (M_1,M_2)\). If \(\varepsilon \) is sufficiently small, we have
Proof
We only prove the lemma for \(i=1\) (the other case is similar). By Lemma 3.3, we have
Since
Lemma A.7 implies
Next, note that (3) and (25) imply
where \(L_2=\partial _t-c\partial _x-(\nu /2)\partial _{x}^{2}\). Using these, similar to the bound for \(\zeta _{1;n}\) in the proof of Lemma 3.1, we can show that
Combining these, we obtain the lemma. \(\quad \square \)
We next show the following:
Lemma 3.8
Let \(n\ge 1\) and \(\varepsilon =\max (M_1,M_2)\). If \(\varepsilon \) is sufficiently small, we have
Proof
We only prove the lemma for \(i=1\) (the other case is similar). Applying Lemma A.1 yields
where
and
By Lemmas A.2 and A.3, we obtain
For \(I_{22}(x,t)\), we apply Lemma A.1 (without the integral on \([0,t^{1/2}]\)), which yields
where
By some tedious calculations, we obtain
and
Since \(|L_2 \theta _{2}^{2}(x,t)|\le C\varepsilon ^2 \Theta _4(x,t;-c,\nu ^*)\), Lemma A.3 yields
And since \(|L_{2}^{2}\theta _{2}^{2}(x,t)|\le C\varepsilon ^2 \Theta _6(x,t;-c,\nu ^*)\), Lemma A.4 implies
This proves the lemma. \(\quad \square \)
Similarly, we can now show the following:
Lemma 3.9
Let \(n\ge 1\) and \(\varepsilon =\max (M_1,M_2)\). If \(\varepsilon \) is sufficiently small, we have
We move on to prove the following:
Lemma 3.10
Let \(n\ge 1\). If \(\delta \) defined by (12) is sufficiently small, we have
Here P(t) is defined by (27).
Proof
We only prove the lemma for \(i=1\) (the other case is similar). Set \(f=\theta _2 v_2\). Then Lemma A.1 implies
where
and
By Lemmas A.2 and A.3, we obtain
To bound \(I_{22}(x,t)\), we first note that
Then set
and
We note that
By [10, Theorem 2.6 and Remark 2.8], we have
Footnote 2 In addition, applying Taylor’s theorem, we see that
These imply
and
Using these and integration by parts, we get
Applying Lemmas A.7 and A.9, we obtain
This proves the lemma. \(\quad \square \)
The lemma below can be shown in a similar manner.
Lemma 3.11
Let \(n\ge 1\). If \(\delta \) defined by (12) is sufficiently small, we have
Set
Of course \(n=n_a+n_n+n_c\). Correspondingly, set
and
Then \({\mathcal {N}}_i(x,t)={\mathcal {N}}_{i,a}(x,t)+{\mathcal {N}}_{i,b}(x,t)+{\mathcal {N}}_{i,c}(x,t)\); see (35).
We next prove the following:
Lemma 3.12
Let \(n\ge 1\). If \(\delta \) defined by (12) is sufficiently small, we have
Proof
Let \(i=1\) (the case of \(i=2\) is similar). By integration by parts, we have
By some tedious calculations, we can show that
Then Lemmas 3.7–3.11, A.6, and A.7 yield
It remains to show that
where
We define the decomposition \(I(x,t)=I_1(x,t)+I_2(x,t)\) by
and
We first consider \(I_1(x,t)\). By (32), to show that \(|I_1(x,t)|\) is bounded by \(C(\delta +P(t))^2 \Psi _{1;n}(x,t)\), it suffices to prove the same bound for
and
The term corresponding to \(\delta ^{(1)}(x)\) is not needed since \(q_{10}=(1/2 \, -1/2)^{T}\). Noting that
Lemmas A.6, A.7, and A.10 imply the desired bounds for the two integrals above.
We next consider \(I_2(x,t)\). We have \(I_2(x,t)=I_{21}(x,t)+I_{22}(x,t)\) with
and
Taking into account (32) and (33), Lemmas A.6, A.7, and A.10 yield \(|I_{21}(x,t)|\le C(\delta +P(t))^2 \Psi _{1;n}(x,t)\) (we divide the domain of temporal integration into [0, t/2] and [t/2, t] then use integration by parts before applying the lemmas). For \(I_{22}(x,t)\), we proceed as follows: using the technique in the proof of Lemma A.1, we obtain
where \(L_i=\partial _t+\lambda _i \partial _x-(\nu /2)\partial _{x}^{2}\). Here we used \(\lim _{t\rightarrow 0}(g_{1j}-g_{1j}^{*})(x,t)=0\). The sum of the first two terms on the right-hand side can be bounded using Lemmas A.6, A.8, and A.10; the sum of the third and the fourth term can be bounded using Lemmas A.6, A.7, A.10, and the relation
To bound the sum of the fifth and the sixth term, noting that
it suffices to show that the following integrals are bounded by \(C(\delta +P(t))^2 \Psi _{1;n}(x,t)\):
and
We can bound A(x, t) using Lemma A.4, B(x, t) and D(x, t) by Lemma A.3, and C(x, t) by Lemma A.6. Finally, we consider E(x, t). Taking into account \(L_2 g_{22}^{*}=0\) and \(\lim _{t\rightarrow 0}\partial _x g_{22}^{*}(x-y,t)=\delta ^{(1)}(x-y)\), integration by parts applied to the operator \(L_2\) yields
Hence \(|E(x,t)|\le C(\delta +P(t))^2 \Psi _{1;n}(x,t)\). This ends the proof of the lemma. \(\square \)
The lemma below can be proved similarly. Note that \({\mathcal {N}}_{i,b}(x,t)\) is related to the nonlinear term \((v-1)u_x\) as opposed to \((v-1)^2\) for \({\mathcal {N}}_{i,a}(x,t)\); see (36). As in the bound of \({\mathcal {N}}_{i,a}(x,t)\), the term \((v-1)\) is dealt with the inequality \(|v_i(x,s)|\le P(t)\Psi _{i;n}(x,s)\) (\(0\le s\le t\)); on the other hand, the first derivative \(u_x\) is handled using [10, Theorem 2.6 and Remark 2.8] as in the proof of Lemma 3.10. The term \({\mathcal {N}}_{i,c}(x,t)\) can be handled in a similar manner.
Lemma 3.13
Let \(n\ge 1\). If \(\delta \) defined by (12) is sufficiently small, we have
Combining Lemmas 3.12 and 3.13, the proof of Lemma 3.6 is complete.
3.2.4 Final Step of the Proof
The remaining step of the proof is standard. By Lemma 3.5 and 3.6, we obtain
for some \(C_1,C_2>0\). Here P(t) is defined by (27). When \(\delta \) is sufficiently small, the line \(y=p\) and the parabola \(y=C_1 \delta +C_2 p^2\) intersect at \(p=p_1\) and \(p_2\), where
Note that \(C_1 \delta \le p_1<p_2\). By (37), we either have \(P(t)\le p_1\) or \(P(t)\ge p_2\). Since P(t) is continuous in t, if \(P(0)\le p_1\), then \(P(t)\le p_1\) for all \(t\ge 0\). By (12), taking \(C_1\) sufficiently large, we indeed have \(P(0)\le C_1 \delta \le p_1\). Therefore, we conclude that
This ends the proof of Theorem 2.1.
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Notes
The decay estimate for \(\partial _{x}^{2}u\) is not explicitly stated in the theorem but is shown in its proof (see [10, p. 107]). Note that this is where the \(H^6\)-regularity is used.
The case of \(\beta =2\) is excluded just for simplicity.
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Acknowledgements
This work was supported by Grant-in-Aid for JSPS Research Fellow (Grant Number 20J00882) and JSPS Grant-in-Aid for Early-Career Scientists (Grant Number 22K13938).
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Appendix A. Lemmas on Convolutions Involving a Heat Kernel
Appendix A. Lemmas on Convolutions Involving a Heat Kernel
Lemma A.1
Suppose that \(f=f(x,t)\) is a \(C^2\)-smooth function on \({\mathbb {R}}\times (0,\infty )\). Let \(\lambda \ne \lambda '\) and \(\nu >0\), and set \(L_{\lambda '}=\partial _t+\lambda ' \partial _x-(\nu /2)\partial _{x}^{2}\). Then for \(t\ge 1\), the function I(x, t) defined by
can be written as
where
and
Proof
Let
Dividing the domain of temporal integration, we get
The first term on the right-hand side is \(I_1(x,t)\). For the second term, integration by parts yields
where \(L_{\lambda }=\partial _s+\lambda \partial _y-(\nu /2)\partial _{y}^{2}\). Here we used \(\lim _{s\rightarrow t}g_{\lambda }(x-y,t-s)=\sqrt{2\pi \nu }\delta (x-y)\). The lemma follows from the equality above by noting that \(L_{\lambda }g_{\lambda }=0\). \(\quad \square \)
In the lemmas below, C and \(\nu ^*\) denote generic large constants. We remind the reader that \(\Theta _{\alpha }(x,t;\lambda ,\mu )\) is defined by (13).
Lemma A.2
Let \(\lambda \ne \lambda '\), \(\mu >0\), and \(0<\alpha \le 3\). Then we have
Proof
See the analysis of \(I_1(x,t)\) in the proof of [10, Lemma 3.4]. \(\quad \square \)
Lemma A.3
Let \(\lambda ,\lambda ' \in {\mathbb {R}}\), \(\mu >0\), and \(\alpha >0\) (not necesarily \(\lambda \ne \lambda '\)). Then for \(t\ge 1\), we have
Proof
See the analysis of \(I_{21}(x,t)\) in the proof of [10, Lemma 3.4]. \(\quad \square \)
Lemma A.4
Let \(\lambda \ne \lambda '\), \(\mu >0\), and \(\alpha >1\). Then we have
Proof
See the analysis of \(I_{22}^{(1)}(x,t)\) in the proof of [10, Lemma 3.4]. \(\quad \square \)
Lemma A.5
([10, Lemma 3.2]) Let \(\lambda \in {\mathbb {R}}\), \(\mu >0\), \(\alpha \ge 0\), and \(\beta >0\). Then we have
where \(\gamma =\alpha +\min (\beta ,3)-1\). We also have
where \(\gamma =\min (\alpha ,1)+\beta -1\).
We remind the reader that \(\psi _n(x,t;\lambda )\) is defined in (8).
Lemma A.6
Let \(\lambda \in {\mathbb {R}}\), \(\mu >0\), and \(\alpha ,\beta \ge 0\). Then we have
where \(\gamma _1=\alpha +\min (\beta ,3-\alpha _n)-1\). We also have
where \(\gamma _2=\min (\alpha ,1)+\beta -1\).
Proof
A straightforward (but lengthy) adaptation of the proof of [7, Lemma A.7] proves the lemma. \(\quad \square \)
For \(\lambda ,\lambda ' \in {\mathbb {R}}\) and \(K>0\), let
where \(\textrm{char}\{ S \}\) is the indicator function of a set S.
Lemma A.7
Let \(\lambda \ne \lambda '\), \(\mu >0\), \(\alpha \ge 0\), and \(0\le \beta \le 2\alpha _n\) (\(\beta \ne 2\)).Footnote 3 Then for \(K>0\) large enough, we have
where \(\gamma _1=\alpha +\min (\beta ,3-\alpha _n)-1\) and \(\gamma _{1}'=\alpha +\min (\beta ,2)-1\). We also have
where \(\gamma _2=\min (\alpha ,1)+\beta -1\).
Proof
This is also proved by an adaptation of the proof of [7, Lemma A.8]. \(\quad \square \)
Lemma A.8
Let \(\lambda \ne \lambda '\), \(\mu >0\), \(\alpha \ge 0\), and \(0\le \beta <3-\alpha _n\). Then we have
Proof
This is a simple generalization of [7, Lemma A.2]. \(\quad \square \)
Lemma A.9
Let \(\lambda \ne \lambda '\), \(\mu >0\), and \(\alpha \ge 0\). Then for \(t\ge 4\), we have
Proof
A simple adaptation of the proof of [7, Lemma A.6] proves the lemma. \(\square \)
Lemma A.10
Let \(\lambda \in {\mathbb {R}}\), \(\mu >0\), and \(\alpha \ge 0\). Then
Proof
The lemma can be proved by slightly modifying the proof of [10, Lemma 3.9].
\(\square \)
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Koike, K. Time-Asymptotic Expansion with Pointwise Remainder Estimates for 1D Viscous Compressible Flow. Arch Rational Mech Anal 247, 81 (2023). https://doi.org/10.1007/s00205-023-01914-4
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DOI: https://doi.org/10.1007/s00205-023-01914-4