The insulated and perfect conductivity problems arising from high-contrast composite materials are considered in all dimensions. The solution and its gradient, respectively, represent the electric potential and field. The novelty of this paper lies in finding exact solutions for the insulated and perfect conductivity problems with concentric balls. Our results show that there appears no electric field concentration for the insulated conductivity problem, while the electric field for the perfect conductivity problem exhibits sharp singularity with respect to the small distance between interfacial boundaries of the interior and exterior balls. This discrepancy reveals that concentric balls is the optimal structure of insulated composites, but not for superconducting composites.
Citation: Zhiwen Zhao. Exact solutions for the insulated and perfect conductivity problems with concentric balls[J]. Mathematics in Engineering, 2023, 5(3): 1-11. doi: 10.3934/mine.2023060
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The insulated and perfect conductivity problems arising from high-contrast composite materials are considered in all dimensions. The solution and its gradient, respectively, represent the electric potential and field. The novelty of this paper lies in finding exact solutions for the insulated and perfect conductivity problems with concentric balls. Our results show that there appears no electric field concentration for the insulated conductivity problem, while the electric field for the perfect conductivity problem exhibits sharp singularity with respect to the small distance between interfacial boundaries of the interior and exterior balls. This discrepancy reveals that concentric balls is the optimal structure of insulated composites, but not for superconducting composites.
In this paper, we consider the following two-component Fornberg-Whitham (FW) system for a fluid
{ut+uux=(1−∂2x)−1∂x(ρ−u)ρt+(ρu)x=0(u,ρ)(0,x)=(u0,ρ0)(x) | (1.1) |
where x∈T=R/2πZ, t∈R+. Here, u=u(x,t) is the horizontal velocity of the fluid, and ρ=ρ(x,t) is the height of the fluid surface above a horizontal bottom. This system was first proposed in [5], and local well-posedness and non-uniform dependence on the initial data were established in Sobolev spaces Hs(R)×Hs−1(R) for s>32 in [11,12].
Local well-posedness in Besov spaces Bsp,r(R)×Bs−1p,r(R) of (1.1) was established in [4] for s>max{2+1p,52}. Besov spaces Bsp,r are a class of functions of interest in the study of nonlinear partial differential equations, as they are based on Sobolev spaces and introduce a measure of generalized Hölder regularity through the index r, along with the Sobolev index of differentiability s and the Lebesgue integrability index p. If s and p are fixed, the spaces Bsp,r grow larger with increasing r. In [4], the FW system was shown to be well-posed in the sense of Hadamard by establishing the existence and uniqueness of the solution to the system (1.1) and then proving continuity of the data-to-solution map when the initial data belong to Bsp,r(R)×Bs−1p,r(R) for s>max{2+1p,52}.
In this paper, our objective is to prove non-uniform dependence on periodic initial data for the two-component FW system (1.1) in Bsp,r(T)×Bs−1p,r(T) for s>max{2+1p,52}. We work with periodic initial data, as that simplifies our choice of approximate solutions and the resulting estimates. Setting Λ=1−∂2x, we rewrite (1.1) as
{ut+uux=Λ−1∂x(ρ−u)ρt+uρx+ρux=0(u,ρ)(0,x)=(u0,ρ0)(x) | (1.2) |
where x∈T=R/2πZ and t∈R+.
The paper is organized as follows: In Section 2, we recall the standard definitions and properties of Besov spaces, linear transport equations, the operator Λ, and the two-component FW system. In Section 3, we prove non-uniform dependence on initial data for the FW system (1.2) when the initial data belong to Bsp,r(T)×Bs−1p,r(T) for s>max{2+1p,52}. For this proof, we use a technique previously seen in the study of non-uniform continuity of data-to-solution maps for other nonlinear PDEs, for instance in [6,7,8,10,12]. We construct two sequences of approximate solutions such that the initial data for these sequences converge to each other in Bsp,r(T)×Bs−1p,r(T). Non-uniform dependence is then established by proving that the approximate and hence the exact solutions remain bounded away from each other for any positive time t>0. This idea was first explored by Kato in [9] to show that the data-to-solution map for Burgers' equation is not Hölder continuous in the Hs norm with s>3/2 for any Hölder exponent.
This section is a review of relevant definitions and results on Besov spaces, linear transport equations, the operator Λ, and the two-component FW system (1.2). We begin by listing some useful notation to be used throughout Section 3.
For any x,y∈R,
● x≲y denotes x≤αy for some constant α.
● x≈y denotes x=βy for some constant β.
● x≳y denotes x≥γy for some constant γ.
We recall the construction of a dyadic partition of unity from [8]. Consider a smooth bump function χ such that suppχ=[−43,43] and χ=1 on [−34,34]. For ξ>0, set φ−1(ξ)=χ, φ0(ξ)=χ(ξ2)−χ(ξ) and φq(ξ)=φ0(2−qξ). Then, suppφq=[34⋅2q,83⋅2q] and ∑q≥−1φq(ξ)=1. Using this partition, a Littlewood-Paley decomposition of any periodic distribution u is defined in [3] as follows:
Definition 2.1 (Littlewood-Paley decomposition). For any u∈D′(T) with the Fourier series u(x)=∑j∈Zˆujeijx where ˆuj=12π2π∫0e−ijyu(y)dy, its Littlewood-Paley decomposition is given by u=∑q≥−1Δqu, where Δqu are periodic dyadic blocks defined for all q∈Z as
Δqu=∑j∈Zφq(j)ˆujeijx. |
Using this Littlewood-Paley decomposition, Besov spaces on T are defined in [3] as follows:
Definition 2.2 (Besov spaces). Let s∈R and p, r∈[1,∞]. Then the Besov spaces of functions are defined as
Bsp,r≡Bsp,r(T)={u∈D′(T)|‖u‖Bsp,r<∞}, |
where
‖u‖Bsp,r={(∑q≥−1(2sq‖Δqu‖Lp)r)1/rif1≤r<∞supq≥−12sq‖Δqu‖Lpifr=∞. |
Following are some properties proved in [1, Section 2.8] and [3, Section 1.3] that facilitate the study of nonlinear partial differential equations in Besov spaces.
Lemma 2.3. Let s,sj∈R for j=1,2 and 1≤p,r≤∞. Then the following properties hold:
(1) Topological property: Bsp,r is a Banach space continuously embedded in D′(T).
(2) Algebraic property: For all s>0, Bsp,r∩L∞ is a Banach algebra.
(3) Interpolation: If f∈Bs1p,r∩Bs2p,r and θ∈[0,1], then f∈Bθs1+(1−θ)s2p,r and
‖f‖Bθs1+(1−θ)s2p,r≤‖f‖θBs1p,r‖f‖1−θBs2p,r. |
(4) Embedding: Bs1p,r↪Bs2p,r whenever s1≥s2. In particular, Bsp,r↪Bs−1p,r for all s∈R.
Remark on (2) in Lemma 2.3: When s>1p (or s≥1p and r=1), Bsp,r↪L∞. We will use the fact that for 0<s<1p, the result is still true as long as the functions are bounded.
Given a linear transport equation, Proposition A.1 in [2] proves the following estimate for its solution size in Besov spaces:
Proposition 2.4. Consider the linear transport equation
{∂tf+v∂xf=Ff(x,0)=f0(x) | (2.1) |
where f0∈Bsp,r(T), F∈L1((0,T);Bsp,r(T)) and v is such that ∂xv∈L1((0,T);Bs−1p,r(T)). Suppose f∈L∞((0,T);Bsp,r(T))∩C([0,T];D′(T)) is a solution to (2.1). Let 1≤p,r≤∞. If either s≠1+1p, or s=1+1p and r=1, then for a positive constant C that depends on s, p, and r, we have
‖f(t)‖Bsp,r≤eCV(t)(‖f0‖Bsp,r+C∫t0e−CV(τ)‖F(τ)‖Bsp,rdτ) |
where
V(t)=∫t0‖∂xv(τ)‖B1/pp,r∩L∞dτifs<1+1p |
and
V(t)=∫t0‖∂xv(τ)‖Bs−1p,rdτotherwise. |
For r<∞, f∈C([0,T],Bsp,r(T)), and if r=∞, then f∈C([0,T],Bs′p,1(T)) for all s′<s.
Let Λ=1−∂2x; then, for any test function g, the Fourier transform of Λ−1g is given by F(Λ−1g)=11+ξ2ˆg(ξ). Moreover, for any s∈R, Λ−1∂x is continuous from Bs−1p,r to Bsp,r; that is, for all h∈Bs−1p,r, there exists a constant κ>0 depending on s, p, and r such that
‖Λ−1∂xh‖Bsp,r≤κ‖h‖Bs−1p,r. | (2.2) |
The well-posedness of the two-component FW system (1.2) in Besov spaces was established on the real line in [4] with the following result:
Theorem 2.5. Let s>max{2+1p,52}, p∈[1,∞], r∈[1,∞] and (u0,ρ0)∈Bsp,r(R)×Bs−1p,r(R). Then the system (1.2) has a unique solution (u,ρ)∈C([0,T];Bsp,r(R)×Bs−1p,r(R)), where the doubling time T is given by
T=C(‖u0‖Bsp,r+‖ρ0‖Bs−1p,r)2, |
with C being a constant that depends on s, p, and r, and the solution size is estimated as
‖(u,ρ)‖Bsp,r×Bs−1p,r≤2(‖u0‖Bsp,r+‖ρ0‖Bs−1p,r). |
Moreover, the data-to-solution map is continuous.
Since we work with Bsp,r(T)×Bs−1p,r(T) in this paper, we state the following:
Corollary 2.6. Theorem 2.5 holds when R is replaced by T.
Proof. The existence of a solution to (1.2) is proved by altering the mollifier used to prove Theorem 2.5. This adaptation of the mollifier was done for the single Fornberg-Whitham equation in [7, Section 3.1]. Uniqueness and continuous dependence on periodic initial data for the system (1.2) are established by approximation arguments similar to those in [4, Sections 3.2 and 3.3].
In this section, we establish non-uniform dependence on initial data in the periodic case for the two-component FW system (1.2) in Besov spaces.
Theorem 3.1. Let s>max{2+1p,52} and r∈[1,∞]. The data-to-solution map (u0,ρ0)↦(u(t),ρ(t)) of the Cauchy problem (1.2) is not uniformly continuous from any bounded subset of Bsp,r(T)×Bs−1p,r(T) into C([0,T];Bsp,r(T))×C([0,T];Bs−1p,r(T)) where T is given by Theorem 2.5. In particular, there exist two sequences of solutions {(uω,n,ρω,n)} with ω=±1 such that the following hold:
(i) limn→∞(‖u1,n(0)−u−1,n(0)‖Bsp,r+‖ρ1,n(0)−ρ−1,n(0)‖Bs−1p,r)=0.
(ii) lim infn→∞(‖u1,n−u−1,n‖Bsp,r+‖ρ1,n−ρ−1,n‖Bs−1p,r)≳|sint|.
Proof. For n∈N, we consider two sequences of functions {(uω,n,ρω,n)} with ω=±1, defined by
{uω,n=−ωn+1nssin(nx+ωt)ρω,n=1n+1nssin(nx+ωt). |
We take initial data
{u0ω,n=uω,n(0)=−ωn+1nssinnxρ0ω,n=ρω,n(0)=1n+1nssinnx. |
Let the solutions to the FW system (1.2) with these initial data be denoted by (uω,n,ρω,n). At t=0, we have
limn→∞(‖u01,n−u0−1,n‖Bsp,r+‖ρ01,n−ρ0−1,n‖Bs−1p,r)=limn→∞2‖n−1‖Bsp,r=0, |
which proves part (i) of Theorem 3.1.
To prove part (ii), first we estimate ‖(u0ω,n,ρ0ω,n)‖Bγp,r×Bγ−1p,r and ‖(uω,n,ρω,n)‖Bγp,r×Bγ−1p,r for any γ>0 and r<∞. Using the triangle inequality, we have
‖(u0ω,n,ρ0ω,n)‖Bγp,r×Bγ−1p,r≤2‖n−1‖Bγp,r+n−s‖sinnx‖Bγp,r+n1−s‖sinnx‖Bγ−1p,r. | (3.1) |
By Definition 2.2,
‖sinnx‖Bγp,r=(∑q≥−12γqr‖Δqsinnx‖rLp)1r. | (3.2) |
From Definition 2.1, as shown in the Appendix, we have ‖Δqsin(nx)‖Lp=φq(n), where 0<φq(n)≤1 for all q such that 1ln(2)ln(38n)≤q≤1ln(2)ln(43n) and φq(n)=0 otherwise. Hence, (3.2) implies that for any γ>0,
‖sin(nx)‖Bγp,r≤(1ln(2)ln(43n)∑q=1ln(2)ln(38n)(2q)γr)1r. |
As 2q≤43n for every term in the summation, from the above, we obtain
‖sin(nx)‖Bγp,r≤(1ln(2)ln(43n)∑q=1ln(2)ln(38n)(43n)γr)1r=(1ln(2)[ln(43n)−ln(38n)])1r(43n)γ=(1ln(2)ln(329))1r(43)γnγ=Cγnγ. | (3.3) |
Here and in what follows, Cγ is a generic constant that depends only on γ for fixed p and r. Similarly, it follows that for any γ>0,
‖cos(nx)‖Bγp,r≤Cγnγ. | (3.4) |
By Definition 2.1,
Δqn−1=φq(0)n−1={n−1ifq=−10otherwise. |
So, ‖n−1‖Bγp,r=(∑q≥−12γqr‖Δqn−1‖rLp)1r=2−γn−1. Using this and (3.3), it follows from (3.1) that
‖(u0ω,n,ρ0ω,n)‖Bγp,r×Bγ−1p,r≤21−γn−1+Cγnγn−s+Cγnγ−1n1−s≤Cγmax{n−1,nγ−s}. | (3.5) |
Since (uω,n,ρω,n) is a phase shift of (u0ω,n,ρ0ω,n), we have
‖(uω,n,ρω,n)‖Bγp,r×Bγ−1p,r≤Cγmax{n−1,nγ−s}. | (3.6) |
If r=∞, (3.5) and (3.6) follow immediately from Definition 2.2.
We complete the proof of Theorem 3.1 by establishing (ii) for {(uω,n,ρω,n)}, taking advantage of the following lemma, whose proof follows the proof of Theorem 3.1. Lemma 3.2 establishes that for each n and ω, (uω,n,ρω,n) approximates (uω,n,ρω,n) in Bsp,r(T)×Bs−1p,r(T) uniformly on [0,T] for some T>0.
Lemma 3.2. Let Eω,n=(Eω,n1,Eω,n2) where Eω,n1=uω,n−uω,n and Eω,n2=ρω,n−ρω,n, with ω=±1. Then for all t∈(0,T), where T is given by Theorem 2.5, ‖Eω,n(t)‖Bsp,r×Bs−1p,r=‖Eω,n1(t)‖Bsp,r+‖Eω,n2(t)‖Bs−1p,r→0 as n→∞.
We show that (u−1,n,ρ−1,n) and (u1,n,ρ1,n) stay bounded away from each other for any t>0. Since
‖u1,n−u−1,n‖Bsp,r≥‖u1,n−u−1,n‖Bsp,r−‖u1,n−u1,n‖Bsp,r−‖u−1,n−u−1,n‖Bsp,r | (3.7) |
and
‖ρ1,n−ρ−1,n‖Bs−1p,r≥‖ρ1,n−ρ−1,n‖Bs−1p,r−‖ρ1,n−ρ1,n‖Bs−1p,r−‖ρ−1,n−ρ−1,n‖Bs−1p,r, | (3.8) |
adding (3.7) and (3.8) we obtain
‖u1,n−u−1,n‖Bsp,r+‖ρ1,n−ρ−1,n‖Bs−1p,r≥‖u1,n−u−1,n‖Bsp,r+‖ρ1,n−ρ−1,n‖Bs−1p,r−‖E1,n(t)‖Bsp,r×Bs−1p,r−‖E−1,n(t)‖Bsp,r×Bs−1p,r≥n−s(‖sin(nx+t)−sin(nx−t)‖Bsp,r+‖sin(nx+t)−sin(nx−t)‖Bs−1p,r)−2‖n−1‖Bsp,r−‖E1,n(t)‖Bsp,r×Bs−1p,r−‖E−1,n(t)‖Bsp,r×Bs−1p,r=2n−s(‖cos(nx)‖Bsp,r|sin(t)|+‖cos(nx)‖Bs−1p,r|sin(t)|)−21−γn−1−‖E1,n(t)‖Bsp,r×Bs−1p,r−‖E−1,n(t)‖Bsp,r×Bs−1p,r. | (3.9) |
By Definition 2.2, if r=∞, we immediately have
‖cos(nx)‖Bsp,r≥Csns, | (3.10) |
where Cs is a constant that depends only on s for a given p. For 1≤r<∞, there is a similar estimate, whose proof is given in the Appendix. Also, by Lemma 3.2, we have ‖Eω,n(t)‖Bsp,r×Bs−1p,r→0 for ω=±1, as n→∞. Using this and (3.10), it follows from (3.9) that
lim infn→∞(‖u1,n−u−1,n‖Bsp,r+‖ρ1,n−ρ−1,n‖Bs−1p,r)≥2Cs(lim infn→∞|sin(t)|+lim infn→∞n−1|sin(t)|)≈|sin(t)|>0. |
This proves part (ii) of Theorem 3.1 and completes the proof of non-uniform dependence on initial data for the two-component FW system (1.2) in Bsp,r(T)×Bs−1p,r(T) for s>max{2+1p,52}.
Now we prove Lemma 3.2.
Proof. (Lemma 3.2) We show that ‖Eω,n(t)‖Bγp,r×Bγ−1p,r→0 as n→∞ for any γ with max{s−32,1+1p}<γ<s−1, and then interpolate between such a γ and a value δ>s. Recall that Eω,n1=uω,n−uω,n and Eω,n2=ρω,n−ρω,n. It can be seen that Eω,n1 and Eω,n2 vanish at t=0 and that they satisfy the equations
{∂tEω,n1+uω,n∂xEω,n1=−Eω,n1∂xuω,n+Λ−1∂x(Eω,n2−Eω,n1)−R1∂tEω,n2+uω,n∂xEω,n2=−Eω,n2∂xuω,n−ρω,n∂xEω,n1−Eω,n1∂xρω,n−R2. | (3.11) |
Here, R1 and R2 are the approximate solutions for the FW system, that is,
{R1=∂tuω,n+uω,n∂xuω,n−Λ−1∂x(ρω,n−uω,n)R2=∂tρω,n+∂x(ρω,nuω,n). |
● Estimate for ‖R1‖Bγp,r: Using the definitions of uω,n and ρω,n, we have
R1=∂tuω,n+uω,n∂xuω,n−Λ−1∂x(ρω,n−uω,n)=12n2s−1sin(2(nx+ωt)). |
Then by (3.3),
‖R1‖Bγp,r≤Cγnγ−2s+1. |
● Estimate for ‖R2‖Bγ−1p,r: Using the definitions of uω,n and ρω,n,
R2=∂tρω,n+∂x(ρω,nuω,n)=1nscos(nx+ωt)+1n2s−1sin(2(nx+ωt)). |
So from (3.3) and (3.4), it follows that
‖R2‖Bγ−1p,r≤Cγ(n−snγ−1+n1−2snγ−1)≤Cγnγ−s−1. |
Therefore,
‖R1‖Bγp,r+‖R2‖Bγ−1p,r≲nγ−s−1. | (3.12) |
Since Eω,n1(t) and Eω,n2(t) satisfy the linear transport equations (3.11), to estimate the error ‖Eω,n(t)‖Bγp,r×Bγ−1p,r, we apply Proposition 2.4 to obtain
‖Eω,n1(t)‖Bγp,r≤K1eK1V1(t)∫t0e−K1V1(τ)‖F1(τ)‖Bγp,rdτ | (3.13) |
and
‖Eω,n2(t)‖Bγ−1p,r≤K2eK2V2(t)∫t0e−K2V2(τ)‖F2(τ)‖Bγ−1p,rdτ | (3.14) |
where K1, K2 are positive constants depending on γ and
F1(t)=−Eω,n1∂xuω,n+Λ−1∂x(Eω,n2−Eω,n1)−R1, | (3.15) |
F2(t)=−Eω,n2∂xuω,n−ρω,n∂xEω,n1−Eω,n1∂xρω,n−R2. | (3.16) |
V1(t)=∫t0‖∂xuω,n(τ)‖Bγ−1p,rdτ, |
V2(t)={∫t0‖∂xuω,n(τ)‖B1/pp,r∩L∞dτifγ<2+1p∫t0‖∂xuω,n(τ)‖Bγ−2p,rdτotherwise. |
Since max{s−32,1+1p}<γ<s−1, we have
V1(t)≲nγ−st≤n−1t andV2(t)≤C∫t0‖uω,n(τ)‖Bγp,rdτ | (3.17) |
for some constant C that depends on γ, p, and r. By Theorem 2.5 and Eq (3.5), it follows that
V2(t)≤2C∫t0‖(u0ω,n,ρ0ω,n)‖Bγp,r×Bγ−1p,rdτ≲n−1t. | (3.18) |
Let K=max{K1,K2}. Using (3.17) and (3.18), we combine (3.13) and (3.14) to obtain
‖Eω,n1(t)‖Bγp,r+‖Eω,n2(t)‖Bγ−1p,r≲∫t0eK(t−τ)/n(‖F1(τ)‖Bγp,r+‖F2(τ)‖Bγ−1p,r)dτ. | (3.19) |
● Estimate for ‖F1(τ)‖Bγp,r: From (3.15), as Bγp,r is a Banach algebra, we have
‖F1‖Bγp,r≤‖Eω,n1‖Bγp,r‖∂xuω,n‖Bγp,r+‖Λ−1∂x(Eω,n2−Eω,n1)‖Bγp,r+‖R1‖Bγp,r≤‖Eω,n1‖Bγp,r‖uω,n‖Bγ+1p,r+‖Λ−1∂x(Eω,n2−Eω,n1)‖Bγp,r+‖R1‖Bγp,r. | (3.20) |
From (2.2),
‖Λ−1∂x(Eω,n2−Eω,n1)‖Bγp,r≤κ‖Eω,n2−Eω,n1‖Bγ−1p,r≤M(‖Eω,n1‖Bγp,r+‖Eω,n2‖Bγ−1p,r) | (3.21) |
where M is a constant depending on γ,p, and r. By Theorem 2.5, we have
‖uω,n‖Bγ+1p,r≤2‖(u0ω,n,ρ0ω,n)‖Bγ+1p,r×Bγp,r, |
so by (3.5), ‖uω,n‖Bγ+1p,r≤2Cγmax{n−1,nγ+1−s}. As γ>max{s−32,1+1p},
‖uω,n‖Bγ+1p,r≲nγ+1−s. | (3.22) |
Using (3.21) and (3.22), from (3.20), we obtain
‖F1(τ)‖Bγp,r≲(M+nγ+1−s)‖Eω,n1(τ)‖Bγp,r+M‖Eω,n2(τ)‖Bγ−1p,r+‖R1(τ)‖Bγp,r. | (3.23) |
● Estimate for ‖F2(τ)‖Bγ−1p,r: We may use the algebra property, item (2) of Lemma 2.3, for Bγ−1p,r since γ−1>max{s−52,1p}>0 and the functions we are dealing with are bounded. Then, from (3.16),
‖F2‖Bγ−1p,r≤‖Eω,n2‖Bγ−1p,r‖∂xuω,n‖Bγ−1p,r+‖ρω,n‖Bγ−1p,r‖∂xEω,n1‖Bγ−1p,r+‖∂xρω,n‖Bγ−1p,r‖Eω,n1‖Bγ−1p,r+‖R2‖Bγ−1p,r≲n−1‖Eω,n1‖Bγp,r+‖Eω,n2‖Bγ−1p,r‖uω,n‖Bγp,r+‖R2‖Bγ−1p,r. | (3.24) |
By Corollary 2.6, ‖uω,n‖Bγp,r≤2‖(u0ω,n,ρ0ω,n)‖Bγp,r×Bγ−1p,r, which implies
‖uω,n‖Bγp,r≤2Cγmax{n−1,nγ−s} |
by (3.5). As γ<s−1, ‖uω,n‖Bγp,r≲n−1. Using this in (3.24) yields
‖F2(τ)‖Bγ−1p,r≲n−1‖Eω,n1(τ)‖Bγp,r+n−1‖Eω,n2(τ)‖Bγ−1p,r+‖R2(τ)‖Bγ−1p,r. | (3.25) |
Adding (3.23) and (3.25) gives
‖F1(τ)‖Bγp,r+‖F2(τ)‖Bγ−1p,r≲(M+nγ+1−s)(‖Eω,n1(τ)‖Bγp,r+‖Eω,n2(τ)‖Bγ−1p,r)+‖R1(τ)‖Bγp,r+‖R2(τ)‖Bγ−1p,r. | (3.26) |
Substituting (3.26) into (3.19), we obtain
‖Eω,n(t)‖Bγp,r×Bγ−1p,r≲f(t)+∫t0g(τ)‖Eω,n(τ)‖Bγp,r×Bγ−1p,rdτ | (3.27) |
where
f(t)≈∫t0eK(t−τ)/n(‖R1(τ)‖Bγp,r+‖R2(τ)‖Bγ−1p,r)dτ | (3.28) |
and
g(τ)≈(M+nγ+1−s)eK(t−τ)/n≤(M+1)eK(t−τ)/n. | (3.29) |
Using Grönwall's inequality, from (3.27) we obtain
‖Eω,n(t)‖Bγp,r×Bγ−1p,r≲f(t)+∫t0g(τ)f(τ)e∫tτg(z)dzdτ. | (3.30) |
Using (3.12) along with (3.28) and (3.29), from (3.30), we obtain
‖Eω,n(t)‖Bγp,r×Bγ−1p,r≲nγ−s−1, | (3.31) |
which means that ‖Eω,n(t)‖Bγp,r×Bγ−1p,r→0 as n→∞ for any max{s−32,1+1p}<γ<s−1.
On the other hand, if δ∈(s,s+1), then noting that the solution with the given data is in Bδp,r×Bδ−1p,r for any δ we have, for 0<t<T (from Theorem 2.5)
‖Eω,n(t)‖Bδp,r×Bδ−1p,r≤‖(uω,n,ρω,n)‖Bδp,r×Bδ−1p,r+‖(uω,n,ρω,n)‖Bδp,r×Bδ−1p,r≤2‖(u0ω,n,ρ0ω,n)‖Bδp,r×Bδ−1p,r+‖(uω,n,ρω,n)‖Bδp,r×Bδ−1p,r, | (3.32) |
where we have used the solution size estimate in Theorem 2.5. Now, for δ<s+1, Eqs (3.5) and (3.6) imply that ‖(u0ω,n,ρ0ω,n)‖Bδp,r×Bδ−1p,r≤Cδnδ−s and ‖(uω,n,ρω,n)‖Bδp,r×Bδ−1p,r≤Cδnδ−s, where Cδ denotes a constant that depends only on δ, for a given p and r. So (3.32) yields
‖Eω,n(t)‖Bδp,r×Bδ−1p,r≲nδ−s. | (3.33) |
We use the interpolation property, item (3) from Lemma 2.3, with θ=δ−sδ−γ, to obtain
‖Eω,n(t)‖Bsp,r×Bs−1p,r≤‖Eω,n(t)‖θBγp,r×Bγ−1p,r‖Eω,n(t)‖1−θBδp,r×Bδ−1p,r. | (3.34) |
From (3.34), using (3.31) and (3.33), we obtain
‖Eω,n(t)‖Bsp,r×Bs−1p,r≲(nγ−s−1)δ−sδ−γ(nδ−s)s−γδ−γ=n−θ. | (3.35) |
As θ∈(0,1), (3.35) implies that ‖Eω,n(t)‖Bsp,r×Bs−1p,r→0 as n→∞ for any s>max{2+1p,52}. This completes the proof of Lemma 3.2.
When p=r=2, Bs2,2 and Hs are equivalent by [2, Proposition 1.2], and so we obtain the following corollary:
Corollary 3.3. The data-to-solution map for the two-component FW system (1.2) is not uniformly continuous from any bounded subset of Hs(T)×Hs−1(T) into C([0,T];Hs(T))×C([0,T];Hs−1(T)) for s>52.
In this paper, we considered the two-component Fornberg-Whitham (FW) system (1.2) and used a sequential approach to prove that its data-to-solution map is not uniformly continuous for periodic initial data belonging to Besov spaces Bsp,r(T)×Bs−1p,r(T) where s>max{2+1p,52}. As a corollary, this establishes non-uniform dependence on periodic initial data for the FW system (1.2) in Sobolev spaces Hs(T)×Hs−1(T) for s>52.
In this appendix, we provide a lower bound on ‖cos(nx)‖Bsp,r for any s>0 and 1≤r<∞. By Definition 2.2,
‖cos(nx)‖Bsp,r=(∑q≥−12sqr‖Δqcosnx‖rLp)1r. | (4.1) |
By Definition 2.1, Δqcos(nx)=φq(n)einx. Therefore, ‖Δqcos(nx)‖Lp=φq(n), where 0<φq(n)≤1 for all q such that 1ln(2)ln(38n)≤q≤1ln(2)ln(43n) and φq(n)=0 otherwise, (4.1) implies that
‖cos(nx)‖Bsp,r=(1ln(2)ln(43n)∑q=1ln(2)ln(38n)(2q)srφrq(n))1r. |
Since 2q≥38n for all terms in the summation, from the above we have
‖cos(nx)‖Bsp,r≥(38)sns(1ln(2)ln(43n)∑q=1ln(2)ln(38n)φrq(n))1r. | (4.2) |
Recall that φ0(ξ)=χ(ξ2)−χ(ξ) and φq(ξ)=φ0(2−qξ) for any q>−1, where suppχ=[−43,43] and χ=1 on [−34,34]. This means that suppφq=[34⋅2q,83⋅2q] for any q≥1 and furthermore, φq=1 on the interval [43⋅2q,32⋅2q]. In other words, φq(n)=1 for 1ln(2)ln(23n)≤q≤1ln(2)ln(34n). Therefore, from (4.2) we have
‖cos(nx)‖Bsp,r≥(38)sns(1ln(2)ln(34n)∑q=1ln(2)ln(23n)1)1r=(38)sns(1ln(2)[ln(34n)−ln(23n)])1r=(1ln(2)ln(98))1r(38)sns=Csns, |
where Cs is a constant that depends only on s, for a given p and r. The same estimate holds for ‖sin(nx)‖Bsp,r as well.
All authors contributed equally towards conceptualization, formal analysis, investigation and methodology in this project; Writing of the original draft was done by Prerona Dutta; thereafter all authors together completed the review and editing process. All authors have read and approved the final version of the manuscript for publication.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
We thank John Holmes, at The Ohio State University, for his valuable suggestions on this project. We would also like to thank the anonymous referees for their comments which greatly helped in improving the paper overall.
All authors declare no conflicts of interest in this paper.
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