On the Convergence of the Nonlocal Nonlinear Model to The

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utt = uxx + g(u)xx (1.2) modeling the non-dispersive propagation of longitudinal waves in an elastic bar of infinite length, where a necessary and sufficient condition for it to be hyperbolic is g′(u) > −1. The present work aims to establish rigorously that, in the limit of vanishing nonlocality (that is, as β approaches the Dirac delta function), the solution of (1.1) with a given initial data converges strongly to the corresponding solution of (1.2) in a suitable Sobolev norm. More explicitly, we show that, for long waves of small-but-finite amplitude, the difference between the corresponding solutions of (1.1) and (1.2) with the same initial data remains small over a long time scale. As a concrete example of the convergence result from the class (1.1) to (1.2), we consider the continuous limit of the Fermi–Pasta–Ulam- Tsingou (FPUT) lattice model describing the dynamics of the one-dimensional doubly- infinite chain of identical masses with nearest neighbor interactions. The FPUT lattice dynamics model is also a member of the class (1.1), which corresponds to the triangular kernel to be defined later. We show that, as the lattice spacing approaches zero, the leading approximation of the discrete lattice model approaches the continuous model (1.2). The nonlocal nonlinear wave equation (1.1) was introduced in [4] to model a one- dimensional motion of a nonlocally and nonlinearly elastic medium. The dependent variable u(x, t) is the strain (the spatial derivative of displacement) at the point x at time t, the given function g(u) is the nonlinear part of the (local) stress and (1.1) is the dimensionless equation of motion for the strain (i.e. Newton’s second law of motion for constant mass density and in the absence of body forces). The theory of nonlocal elasticity [12] is an active and interesting subject of research in solid mechanics. In recent years, it has been widely used to predict a more realistic picture of material behavior at the micro/nano length scales. Its main difference with classical elasticity relies on the assumption that the stress at a given point is a function of the strain field at every point in the body, that is, the nonlocal stress is assumed to be the spatial average of the local stress field over the body weighted with a kernel. This fact is reflected in (1.1) through the convolution integral term which introduces a length scale into the model. So this length scale is responsible for dispersion of wave solutions to (1.1). It is worth mentioning that the class (1.1) of nonlocal nonlinear wave equations covers a variety of well-known dispersive wave equations, for instance the improved Boussinesq equation 1 −|x| utt − uxx − uxxtt = g(u)xx corresponding to the exponential kernel β(x) = 2 e is the most widely known member of the class. Depending on the weakness of dispersion, the class (1.1) provides a hierarchy of nonlinear dispersive wave equations which serve as approximations of (1.2). It is natural to ask whether, if the dispersive effect is sufficiently weak, solutions of (1.2) can be approximated using appropriate solutions of (1.1). So, in this paper, we concentrate on proving the convergence of the nonlocal nonlinear model (1.1) to the classical elasticity equation (1.2) on a long time interval in the zero-dispersion limit. A collection of articles [5, 7–11] focuses on various aspects of mathematical analysis of the initial-value problems associated with (1.1). In those studies it is assumed that the Fourier transform β of the kernel function β satisfies the ellipticity and boundedness b 2 condition 2 r 0 ≤ β(ξ) ≤ C(1 + ξ )− 2 for some r ≥ 0. (1.3) Here we remark that the necessaryb condition for wave propagation is given by β(ξ) ≥ 0 and that general dispersive properties of waves are characterized by the kernel function. b The exponent r in (1.3) denotes the rate of decay of β as |ξ| → ∞ and it is determined by the smoothness of β (faster decay reflects a greater smoothness). The limiting case b r = 0 corresponds to the case when β is integrable or more generally a finite measure on R. We note that β is real and hence β should be an even function. The local well-posedness of the Cauchy problem for (1.1) was proved in [4] under b the strong smoothness assumption r ≥ 2. This constraint is equivalent to say that the operator β′′ ∗ (·) to be bounded on Hs(R). As a result, (1.1) becomes a Banach space valued ODE. In [11] the well-posedness result was extended to nonsmooth kernels satisfying 0 ≤ r < 2 for which (1.1) is genuinely hyperbolic; hence requires a totally different approach from that in the r ≥ 2 case. Another improvement in [11] was a long-time existence result in the small amplitude regime. The main tool used there was a Nash-Moser iteration scheme necessitated by the loss of derivative in the energy estimate. In this work we first give an alternative proof for the local well-posedness of the Cauchy problem for (1.1) with r ≥ 0. The improvement here is due to the construction of a quasilinear energy functional allowing a better energy estimate which does not involve a loss of derivative. This allows us to use a Picard iteration scheme to prove the local well-posedness result for (1.1). This result is also stronger than the one in [11] as it does not require extra smoothness of the initial data, which was due to the Nash-Moser scheme there. Furthermore, this improved energy estimate allows us to establish the blow-up criteria for nonsmooth kernels, extending of the blow-up result obtained in [4] smooth kernels (i.e. for the ODE case). We show that the maximal existence time is ∞ controlled by the L norms of u, ux and ut (that may lead to wave breaking) in contrast to the L∞ norm of u in the r ≥ 2 case. In the present work we prove that the limit of (1.1) is (1.2) in the sense that solutions of (1.1) approximate to the solution of (1.2) in the limit of vanishing nonlocality. This is inspired by the comparison result established in [7] where two solutions corresponding to two different kernels were compared in the long-wave limit. Here we concentrate on the particular case where one of the kernels is the Dirac delta function corresponding to the zero-dispersion case. In other words we compare (1.1) for an arbitrary kernel with the classical elasticity equation. The main step is to obtain uniform estimates of solutions with respect to the parameter characterizing dispersion and to estimate the difference between the solutions using the energy estimate. Nevertheless, the fact that our energy estimate does not involve loss of derivative allows us to get a stronger convergence result than that in [7]. In this work we will consider the kernels that satisfy

0 ≤ β(ξ) ≤ C (1.4) b with C = kβkL∞ . We note that any kernel satisfying (1.3) with some r ≥ 0 also satisfies the weaker estimate (1.4). In other words, we consider all integrable kernels as well as all b finite measures on R. This is the largest class of kernels that we will consider throughout this work. Clearly, one can obtain sharper results assuming extra smoothness on β (i.e. r> 0). We will indicate those cases in our local existence theorems. 3 The plan of this paper is as follows. In Section 2 we write (1.1) as a first-order nonlinear system and prove an energy estimate for strong solutions of the corresponding linearized system. In Sections 3 and 4 we prove the well-posedness of the Cauchy problem for the linearized system and the nonlinear system, respectively. Section 5 is concerned with the dependence of the existence time of the solutions on the initial data and the size of the nonlinear term, which makes possible to extend the local well-posedness result to long time intervals. In the same section, as a by-product, we prove the result stating ∞ that the finite-time blow-up of solutions can occur only in the L norms of u, ux and ut. In Section 6, we are able to prove the main result of the paper stating that, in limit of vanishing nonlocality (i.e. in the zero-dispersion limit), small-but-finite amplitude solutions of the Cauchy problem for (1.1) converge to the corresponding solution of (1.2). In Section 7 we provide a rigorous justification of the convergence of the FPUT discrete lattice dynamic equation corresponding to the triangular kernel to the classical elasticity equation as the lattice spacing approaches zero. Throughout the paper, we use the standard notation for function spaces. The Fourier −iξx transform u of u is defined by u(ξ) = R u(x)e dx. The norm of u in the Lebesgue p space L (R) (1 ≤ p ≤ ∞) is representedR by k·kLp . To denote the inner product of u and v in Lb2 space, the symbol bhu, vi is used. The notation Hs = Hs(R) (for s ∈ R) is 2 used to denote the L -based Sobolev space of order s on R, with the norm kukHs = 2 s 2 1/2 R(1 + ξ ) |u(ξ)| dξ . C is a generic positive constant. Partial differentiations are denoted R by Dx etc. Further, to simplify the presentation in the rest of the paper, we introduce theb notations Xs and Y s1,s2 to refer the spaces defined by

Xs = C [0,T ],Hs , Y s1,s2 = C [0,T ],Hs1 ∩ C1 [0,T ],Hs2 , (1.5) for ﬁxed T > 0. Here the spaces Xs and Y s1,s2 are endowed with the usual norms

kukXs = sup ku(t)kHs , kukY s1,s2 = sup ku(t)kHs1 + sup kut(t)kHs2 , (1.6) 0≤t≤T 0≤t≤T 0≤t≤T respectively.

2. An energy estimate

The proof of the local well-posedness of (1.1) in Sobolev spaces Hs is based on the energy estimates and Gronwall’s inequality. In this section, we construct an energy estimate for the ﬁrst-order inhomogeneous linearized system related to (1.1). Throughout this section we assume that β satisﬁes (1.4). s 2 s/2 s In the rest of the work, we will adopt the notations Λ = (1 − Dx) and [Λ ,f]g = Λs(fg) − fΛsg. Furthermore, to prove our estimates below and in the next sections, we will need the following commutator estimates [14]:

s ∞ s−1 ∞ Lemma 2.1. Let s> 0. Then for all f,g satisfying f ∈ H , Dxf ∈ L , g ∈ H ∩L ,

s ∞ s−1 s ∞ [Λ ,f]g L2 ≤ C kDxfkL kgkH + kfkH kgkL . In particular, when s> 3/2 , due to the Sobolev embeddings Hs−1 ⊂ L∞, for all f,g ∈ Hs

s s s [Λ ,f]Dxg L2 ≤ CkfkH kgkH . 4

We first define the pseudo-differential operator K by

1/2 Ku = F −1 k(ξ)u(ξ) with k(ξ)= β(ξ) (2.1) b s b 1/2 for any β satisfying (1.4). Clearly, K is bounded on H with operator norm C = kβkL∞ . Then the Cauchy problem b

ut = Kvx, u(x, 0) = u0(x), (2.2) ′ vt = K (1 + g (u))ux , v(x, 0) = v0(x) (2.3) is equivalent to the Cauchy problem deﬁned by (1.1) and the initial data u(x, 0) = u0(x), ut(x, 0) = K(v0(x))x. We now study the corresponding linearized problem

ut = Kvx, u(x, 0) = u0(x), (2.4)

vt = K (1 + w)ux , v(x, 0) = v0(x). (2.5) Here w is a given ﬁxed function and the hyperbolicity of the linearized system (2.4)-(2.5) is guaranteed by the condition w(x, t) > −1. We deﬁne the Hs-energy functional of (2.4)-(2.5) by

2 1 s 2 s 2 Es (t)= (1 + w(x, t))(Λ u(x, t)) + (Λ v(x, t)) dx, (2.6) 2 ZR s 2 s/2 where Λ = (1 − Dx) . Note that when w is assumed to satisfy

2 2 2 where c1 and c2 are constants, Es (t) is equivalent to the norm ku(t)kHs + kv(t)kHs . Lemma 2.2. Let s> 3/2, Xs = C [0,T ],Hs , Y s,s−1 = C [0,T ],Hs ∩C1 [0,T ],Hs−1 . s s,s−1 Let u0, v0 ∈ H and w ∈ Y with 0 < c1 ≤ 1+ w(x, t) for all (x, t)∈ R × [0,T ]. Suppose u, v ∈ Xs satisfy the inhomogeneous system

ut = Kvx + F1, u(x, 0) = u0(x), (2.8)

vt = K (1 + w)ux + F2 + G, v(x, 0) = v0(x) (2.9)

s on R × [0,T ], where G satisﬁes kG(t)kHs ≤ Cku(t)kHs and F1, F2 ∈ X . Then we have the estimate

Ct Ct ku(t)kHs + kv(t)kHs ≤ ku0kHs + kv0kHs e + C(kF1kXs + kF2kXs )(e − 1) (2.10) for 0 ≤ t ≤ T , where C is a constant dependent on kwkY s,s−1 . Proof. Throughout the proof we will use the abbreviation w = 1+ w to simplify the presentation. Diﬀerentiating (2.6) and using (2.8)-(2.9) we get e d 2 1 s 2 s s s s Es (t) = wt(Λ u) + w(Λ ut)(Λ u)+(Λ vt)(Λ v) dx dt ZR 2 1 s 2 e s s s s s s = wt(Λ u) + w(Λ F1)(Λ u)+(Λ F2)(Λ v)+(Λ G)(Λ v) ZR 2 s es s s + w(Λ Kvx)(Λ u)+(Λ K(wux))(Λ v) dx. (2.11) 5 e e We handle the last two terms on the right-hand-side of (2.11) separately:

s s s s I(t) = w(Λ Kvx), Λ u + Λ K(wux), Λ v = D (wΛsKv) − w (Λ sKv), Λsu + Λ sK(wu ), Λsv e x x e x = − w(ΛsKv), Λsu − w (ΛsKv ), Λ su + Λs(wu ) , ΛsKv e ex x e x = − w (ΛsKv), Λsu + ΛsKv, −w(Λsu )+Λ s(wu ) ex e x e x = − w (ΛsKv), Λsu + ΛsKv, [Λs, w]u , (2.12) x e x e s s s s with [Λ , w]ux = [Λ , w]ux = Λ (wux) − wΛ ux, where we have used the facts that the s 2 operators Λ , K and Dx all commute and K is self-adjoint on L . Thus, (2.11) takes the form e

d 2 1 s 2 s s s s Es (t) = wt(Λ u) − wx(Λ Kv)(Λ u)+(Λ Kv)[Λ , w]ux dt ZR 2 s se s s s s + w(Λ F1)(Λ u)+(Λ F2)(Λ v)+(Λ G)(Λ v) dx.

Since kKuk s ≤ Ckuk s− r ≤eCkuk s , we deduce from the above equation that H H 2 H

d 2 2 s E ≤ C kw k ∞ kuk s + kw k ∞ kvk s kuk s + kvk s [Λ , w]u dt s t L H x L H H H x L2

+ kwkL∞ kF1kHs kukHs + kF2kHs kvkHs + kGkHs kvkHs . (2.13) Using the relation kG(te)kHs ≤ Cku(t)kHs and the commutator estimate of Lemma 2.1 s s s [Λ , w]ux L2 ≤ CkwkH kukH , we get

d 2 2 E ≤ C kw k ∞ kuk s + kw k ∞ kvk s kuk s + 1+ kwk s kvk s kuk s dt s t L H x L H H H H H + kwkL∞ kF1kHs kukHs + kF2kHs kvkHs . (2.14) For s> 3/2, we have e

sup kwt(t)kL∞ + kwx(t)kL∞ + kw(t)kHs +1 ≤ C1, 0≤t≤T and sup kw(t)kL∞ ≤ C2 0≤t≤T where C1 and C2 are constants dependinge on kwkY s,s−1 . So, using the fact that Es(t) is equivalent to the norm ku(t)kHs + kv(t)kHs , (2.14) becomes

d 2 2 E (t) ≤ C E (t)+ C kF k s + kF k s E (t). dt s 1 s 2 1 H 2 H s By Gronwall’s inequality we get,

C1t C1t Es(t) ≤Es(0)e + C2 kF1kXs + kF2kXs e − 1 . This completes the proof of (2.10). 6 3. Well-posedness for the linear system

In this section we establish the well-posedness of the Cauchy problem (2.4)-(2.5) deﬁned for the linearized system.

r Lemma 3.1. Suppose β satisﬁes (1.3) for some r ≥ 0. Let s> 5/2, σ = min{s,s+ 2 −1}, s s s,σ s 1 σ s X = C [0,T ],H and Y = C [0,T ],H ∩ C [0,T ],H . Let u0, v0 ∈ H and s,σ w ∈ Y with 0 < c1 ≤ 1+ w(x, t) for all (x, t) ∈ R × [0,T ]. Then there exist unique u, v ∈ Y s,σ satisfying (2.4)-(2.5) on R × [0,T ]. Proof. For the existence proof of the linearized system, we follow the standard hyperbolic approach [15] in which a solution of (2.4)-(2.5) is obtained as a limit of solutions to

h h h h ut = J Kvx , u (x, 0) = u0(x), (3.1) h h h h vt = J K (1 + w)ux , v (x, 0) = v0(x). (3.2) Here J h is a Friederichs mollifier given by 1 x − y J hϕ(x)= η( )ϕ(y)dy h ZR h ∞ R with some nonnegative η ∈ C0 ( ) and R η(x)dx = 1. Due to the regularizing effect of J h, (3.1)-(3.2) is an Xs-valued ODER system and hence has solution uh, vh ∈ Xs. Moreover, since J hK satisfies the same bounds as K, the energy estimate of Lemma 2.2 will hold uniformly in h. It follows that (uh, vh) is uniformly bounded in Xs. We will hn hn show that for any sequence hn → 0, the solutions (u , v ) form Cauchy sequences in a lower norm. The key ingredient in the proof is the mollifier estimate [6]

hn hm kJ ϕ − J ϕkHs−1 ≤ |hn − hm|kϕkHs . (3.3)

hn hm On the other hand we have the direct estimate kJ ϕ − J ϕkHs ≤ CkϕkHs . Using the Gagliardo–Nirenberg type inequality [2] for any 0 <α< 1 we have

hn hm α kJ ϕ − J ϕkHs−α ≤ C|hn − hm| kϕkHs . (3.4)

Given s > 5/2 we ﬁrst choose 0 <α< 1 so that s − α > 5/2. We start by writing hn hm hn hm (3.1)-(3.2) for hn and hm and setting p = u − u , q = v − v . Then, (3.1)-(3.2) imply that the diﬀerences p, q satisfy the initial value problem

hn hn hm hm pt = J Kqx + (J − J )Kvx , p(x, 0)=0, hn hn hm hm qt = J K (1 + w)px + (J − J )K (1 + w)ux , q(x, 0)=0. The above problem is in the form of (2.8)-(2.9) if we take

hn hm hm hn hm hm F1 = (J − J )Kvx , F2 = (J − J )K (1 + w)ux , G =0. By (3.4) we have

h h s−1−α h h Hence limh→0(u , v ) = (u, v) exists in X . On the other hand, (u , v ) being bounded in Xs will have a weak limit in Xs. Uniqueness of the limit shows regularity, namely that u, v ∈ Xs indeed satisﬁes (2.4)-(2.5). The second part of the assertion, σ r ut, vt ∈ C [0,T ],H (with σ = s + 2 − 1 when r < 2), follows directly from the smoothing eﬀect of K in (3.1)-(3.2). Finally, uniqueness for the linear system follows directly from the energy estimate.

4. Local well-posedness for the nonlinear system

Once having proved the well-posedness of the linearized system, the next stage is to prove the local well-posedness of the nonlinear system (2.2)-(2.3). In this section we use Picard’s iterations to prove the local well-posedness of (2.2)-(2.3). In the process, we will make use of the nonlinear estimates (see [1, 3]): Lemma 4.1. Let h ∈ C∞(R) with h(0) = 0. Then, for any s ≥ 0 and u, v ∈ L∞ ∩ Hs,

s 1. h(u) ∈ H with kh(u)kHs ≤ C1kukHs where C1 depends on h and kukL∞ .

2. kh(u) − h(v)kHs ≤ C2ku − vkHs where C2 depends on h and kukL∞ , kvkL∞ , kukHs , and kvkHs .

To employ the energy estimate of Lemma 2.2, we want the initial value u(x, 0) = u0(x) to satisfy ′ 0

n+1 n+1 n+1 ut = Kvx , u (x, 0) = u0(x), n+1 ′ n n+1 n+1 vt = K (1 + g (u ))ux , v (x, 0) = v0(x)

0 0 with (u , v ) = (u0, v0). By the energy estimate (2.10) of Lemma 2.2 for F1 = F2 = G =0 log 2 and t ≤ T0 = C we have

1 1 Ct γ Ct ku (t)k s + kv (t)k s ≤ ku k s + kv k s e ≤ e ≤ γ H H 0 H 0 H 2 ′ n n with C = C kg (u0)kHs ≤ C(γ). A similar estimate will also work for all (u , v ), ′ n log 2 hence we have 0

n+1 n+1 n+1 pt = Kqx , p (x, 0)=0, n+1 ′ n n+1 ′ n ′ n−1 n n+1 qt = K (1 + g (u ))px + K (g (u ) − g (u ))ux , q (x, 0)=0. 8 ′ n Applying Lemma 2.2 for s − 1 rather than s, with F1 = G = 0 and F2 = K g (u ) − ′ n−1 n g (u ) ux, we get n+1 n+1 ′ n ′ n−1 n Ct s−1 s−1 kp (t)kH + kq (t)kH ≤ C K g (u (t)) − g (u (t)) ux(t) Hs−1 (e − 1) ′ n ′ n−1 n Ct ≤ C g (u (t)) − g (u (t)) ux(t) H s−1 (e − 1) n n−1 Ct ≤ C u (t) − u (t) Hs−1 (e − 1) ,

CT1 1 where we have used Lemma 4.1. Choosing T1 < T0 now so that e − 1 ≤ 2C we see that for t ≤ T1,

n+1 n+1 1 n n C kp (t)k s−1 + kq (t)k s−1 ≤ kp (t)k s−1 + kq (t)k s−1 ≤···≤ . H H 2 H H 2n This shows that (un, vn) forms a Cauchy sequence in Hs−1 and the limit (u, v) will be a solution in Hs−1. Finally, as was done in the proof of Lemma 3.1, we obtain regularity, s namely that u, v ∈ C [0,T1],H . Noting that the operator K has a smoothing effect s+ r −1 of order r/2, we get ut, vt ∈ C [0 ,T1],H 2 when r < 2. This proves the theorem below: Theorem 4.2. Suppose β satisfies (1.3) for some r ≥ 0. Let s > 5/2, σ = min{s,s + r s s s 2 − 1}. Suppose u0, v0 ∈ H with sufficiently small ku0kH + kv0kH . Then there exists s,σ T1 > 0 so that the Cauchy problem (2.2)-(2.3) has a unique solution u, v ∈ Y = s 1 σ C [0,T1],H ∩ C [0,T1],H . We conclude by stating the local well-posedness result for our original equation (1.1). Noting the relationship between the initial values for (1.1) and (2.2)-(2.3), we have: Corollary 4.3. Suppose β satisfies (1.3) for some r ≥ 0. Let s > 5/2, σ = min{s,s + r s s s 2 − 1}. Suppose u0, v0 ∈ H with sufficiently small ku0kH + kv0kH . Then there exists T1 > 0 so that the Cauchy problem for (1.1), with initial data u(x, 0) = u0(x), s,σ s 1 σ ut(x, 0) = K(v0(x))x has a unique solution u ∈ Y = C [0,T1],H ∩ C [0,T1],H . Remark 4.4. We finally note that when K is invertible, namely β(ξ) ≥ c(1+ξ2)−r/2, the condition u (x, 0) = K(v (x)) amounts to u (x, 0)=(w (x)) with arbitrary w ∈ Hσ. t 0 x t 1 x b 1

5. Blow up and existence time of solutions

In this section we will first determine conditions for finite-time blow-up of solutions to (2.2)-(2.3)and then we will highlight the dependence of the life span of solutions upon the initial data. All the results in this section correspond to similar results for our original problem (1.1). When r ≥ 2, it was shown in [4] that blow-up can only happen in the L∞ norm of u; namely, as long as ku(t)kL∞ stays finite, there is global existence for any s> 1/2. This is due to the fact that when r ≥ 2, (2.2)-(2.3) is an Hs-valued ODE system. In the general case r ≥ 0, the condition that ku(t)kL∞ stays finite will not suffice as there may be also wave breaking. The phenomenon where ku(t)kL∞ remains bounded but kux(t)kL∞ and/or kut(t)kL∞ become unbounded will be called as wave breaking. 9 s Consider the solution (u, v) of (2.2)-(2.3) with the initial data u0, v0 in H , s> 5/2. ′ Then, obviously, the pair (u, v) satisfies (2.8)-(2.9) with w = g (u) and F1 = F2 = G = 0. From the estimate (2.13) in the proof of Lemma 2.2 we have

d 2 2 s E ≤ C kw k ∞ kuk s + kw k ∞ kvk s kuk s + kvk s [Λ , w]u . dt s t L H x L H H H x L2

When w = g′(u), with the use of the stronger commutator estimate

s ∞ s−1 s ∞ [Λ , w]ux L2 ≤ C kwxkL kuxkH + kwkH kuxkL in Lemma 2.1, this gives d 2 ′ ∞ 2 ′ ∞ Es ≤ C k(g (u))tkL kukHs + k(g (u))xkL kukHs kvkHs dt ′ ′ + k(g (u))xkL∞ kukHs + kg (u)kHs kuxkL∞ kvkHs . (5.1) Now suppose that for some T > 0, ku(t)kL∞ ≤ M and kut(t)kL∞ + kux(t)kL∞ ≤ M1 for all t ∈ [0,T ]. But with C(M) = sup |g′(p)| + |g′′(p)| |p|≤M ′ ′ ′ kg (u)kL∞ ≤ C(M) and k(g (u))tkL∞ + k(g (u))xkL∞ ≤ C(M) kutkL∞ + kuxkL∞ ≤ ′ C(M)M1, and kg (u)kHs ≤ C(M)kukHs for t ∈ [0,T ], where we have used Lemma 4.1. So we get d E2(t) ≤ C(M)M E2(t), dt s 1 s and Gronwall’s inequality implies

2 2 2 C(M)M1t ku(t)kHs + kv(t)kHs ≈Es (t) ≤ Ce .

∞ ∞ ∞ 2 This shows that whenever ku(t)kL , kut(t)kL and kux(t)kL stay finite, ku(t)kHs + 2 kv(t)kHs also stays finite; so to investigate blow-up versus global existence, one needs to ∞ control the L norms of u, ux and ut. In other words; Lemma 5.1. Let s> 5/2. Then the solution of (2.2)-(2.3), with initial data in Hs, will blow up in finite time if and only if

ku(t)kL∞ + kut(t)kL∞ + kux(t)kL∞ blows up in ﬁnite time. Next we will consider the existence time of solutions to (2.2)-(2.3) in the case of a power-type nonlinearity, g(u) = αun+1 (n ≥ 1); namely we show how existence time depends on the initial data. Then, with w = g′(u) = (n + 1)αun we have from (5.1):

d 2 n 2 n n E ≤ C k u k ∞ kuk s + k u k ∞ kuk s kvk s + kuk s ku k ∞ kvk s . (5.2) dt s t L H x L H H H x L H The ﬁrst term in (5.2) is estimated as follows

n n−1 n−1 ∞ ∞ ∞ s−1 k u tkL ≤ CkukL kutkL ≤ CkukHs−1 kutkH . 10 But kutkHs−1 = kKvxkHs−1 ≤ CkvxkHs−1 ≤kvkHs , so

n n−1 n ∞ s k u tkL ≤ CkukHs kvkH ≤ CEs . Similarly, n n n n ∞ s−1 k u xkL ≤ Ck u xkH ≤ CkukHs ≤ CEs . Adding up, d E2(t) ≤ CEn+2(t). dt s s This in turn implies that

Es(0) s s ku(t)kH + kv(t)kH ≈Es(t) ≤ 1 . n n 1 − CtEs (0) In other words, we have the following lemma establishing the relationship between the existence time and the initial data: Lemma 5.2. The solution of (2.2)-(2.3) with g(u) = αun+1 (n ≥ 1) and initial data s u0, v0 in H exists at least for times 1 t = O n . ku0kHs + kv0kHs We next investigate the dependence of the existence time on the size of the nonlinear term. For this aim we consider the problem

ǫ ǫ ǫ ut = Kvx , u (x, 0) = u0(x), (5.3) ǫ ǫ n ǫ n+1 ǫ vt = K u + ǫ (u ) x, v (x, 0) = v0(x), (5.4) parametrized by a small positive parameter ǫ reﬂecting the strength of the nonlinearity. Using again the estimate (2.13) in Lemma 2.2 with F1 = F2 = G = 0 and keeping track of the extra ǫn in front of (uǫ)n+1, we get

ǫ ǫ Es(0) ku (t)kHs + kv (t)kHs ≈Es(t) ≤ 1 . n n n 1 − Ctǫ Es (0) That is, the existence time of the time-dependent solution (u, v) to (5.3)-(5.4) is at least of order O(1/ǫn) and to get the long-time existence of solutions, it suﬃces to take ǫ small enough. So we have the following lemma establishing the long-time existence result: Lemma 5.3. The solution of (5.3)-(5.4) with g(u) = αun+1 (n ≥ 1) and initial data s u0, v0 in H , exists at least for times 1 t = O . n n ǫ ku0kHs + kv0kHs

11 6. Convergence to the equation of classical elasticity

The object of this section is to study the relationship between the family of nonlocal wave equations and the classical (local) elasticity model. We prove that as the kernel β approaches the Dirac delta measure, not only (1.1) approaches formally the classical elasticity equation, (1.2), but also the corresponding solutions converge strongly to the solution of (1.2). The notion of this comparison result is in the sense of that in [7]. We will consider the largest class of kernels, namely those satisfying (1.4). Note that as β is real, β and β are even functions. We will further assume that b b βdx = 1 and x2|β|dx < ∞. (6.1) ZR ZR In the rest of the paper we focus on the power-type nonlinearity g(u) = un+1 (n ≥ 1). Using the transformation (x,t,u) → (x/δ, t/δ, ǫuδ) with small parameters δ > 0 and 1 x ǫ > 0 in (1.1), we now deﬁne the family of kernels as βδ(x) = δ β( δ ) and the family of nonlocal nonlinear dispersive wave equations

δ δ n δ n+1 utt = βδ ∗ u + ǫ (u ) xx. (6.2) Here, the parameter δ acts as a length scale in the problem and serves to measure the intensity of the dispersive eﬀect. We note that, as δ → 0, the kernels βδ will converge to the Dirac measure in the distribution sense. Hence, as the dispersion parameter δ tends to zero, (6.2) will formally approach the classical elasticity equation

n n+1 utt = u + ǫ u xx (6.3) corresponding to the nondispersive case. Understanding this passage from (6.2) to (6.3) is our main motivation here. We shall use uδ and u to denote the solutions to the initial value-problems defined for the nonlocal model (6.2) and the local model (6.3), respectively. We will prove that the difference between the corresponding solutions of (6.2) and (6.3) remains small over a long time interval in a suitable Sobolev norm. That is, we will show that, in the zero-dispersion limit, the solutions of the classical elasticity equation are well approximated by the corresponding solutions of the nonlocal wave equations. Obviously, the solutions we consider are small-but-finite amplitude long wave solutions as it is clear from the form of (6.2). First we note that the relationship between the Fourier transforms of βδ and β is βδ(ξ) = β(δξ). So kβδkL∞ = kβkL∞ and thus the energy estimates of the previous section will be uniform in δ. Hence Lemma 5.3 implies that the solution uδ of the c b c b T Cauchy problem for (6.2) will all exist in a certain time interval [0, ǫn ]. Similarly, when β is the Dirac delta measure corresponding to (6.3), we have existence of solution 1 δ u for times O( ǫn ) hence we can assume that both u and u exist in the same time interval. Moreover, as σ ≥ s − 1 in Corollary 4.3, it is clear that both u and uδ belong T s 1 T s−1 to C [0, ǫn ],H ∩ C [0, ǫn ],H . Our aim is to show that, for suitable initial data, δ T s 1 T s−1 u converges tou in C [0, ǫn ],H ∩ C [0, ǫn ],H . We convert (6.2) to the system

δ δ ut = Kδvx, (6.4) δ δ n δ n+1 vt = Kδ u + ǫ (u ) x, (6.5) 12 −1 where Kδv = F k(δξ)v(ξ) while (1.2) corresponds to b ut = vx, (6.6) n n vt = 1 + (n + 1)ǫ u ux. (6.7) We will use the estimates of Lemma 2.2 to compare the solutions of these two systems. We ﬁrst derive an estimate on the operator Kδ − I. Proposition 6.1. For v ∈ Hs+θ and 0 <θ ≤ 2, we have

θ kKδv − vkHs ≤ Cδ kvkHs+θ . (6.8)

Proof. The assumption ( 6.1) implies that β′′ exists and is bounded. Also, since β is real, β and hence β will be even. Clearly k = (β)1/2 will have the same properties. As b 1 b k(0) = 1 and k′(0) = 0, we have the Taylor estimate |k(ξ) − 1| ≤ kk′′k ∞ ξ2. Joining b b 2 L this estimate with |k(ξ) − 1|≤kkkL∞ + 1, we have for any 0 <θ ≤ 2

1 θ θ |k(ξ) − 1| = |k(ξ) − 1| − 2 |k(ξ) − 1| 2 ≤ C|ξ|θ, (6.9)

′′ where C is a constant dependent on kkkL∞ and kk kL∞ . Then

2 2 2 s 2 kKδv − vkHs = (k(δξ) − 1) (1 + ξ ) |v(ξ)| dξ ZR

2 2θ 2θ 2 s b 2 2 2θ 2 ≤ C δ |ξ| (1 + ξ ) |v(ξ)| dξ ≤ C δ kvkHs+θ . ZR b

s+θ Theorem 6.2. Suppose β satisﬁes (1.4) and (6.1). Let u0, v0 ∈ H , s> 5/2, 0 <θ ≤ − δ δ T s+θ 1 T s+θ 1 2. Let u , v ,u,v ∈ C [0, ǫn ],H ∩ C [0, ǫn ],H satisfy (6.4)-(6.5) and (6.6)- δ δ (6.7) with the initial data u (x, 0) = u(x, 0) = u0(x) and v (x, 0) = v(x, 0) = v0(x). T Then we have, for all t ∈ [0, ǫn ],

δ δ θ Cǫnt ku (t) − u(t)kHs−1 + kv (t) − v(t)kHs−1 ≤ Cδ te , (6.10) where C is a constant independent of δ and ǫ.

δ δ Proof. By Lemma 2.2, we have uniform bounds on ku kXs+θ , kv kXs+θ , kukXs+θ and δ δ kvkXs+θ . Let p = u − u, q = v − v. Then the diferences p and q solve the system

δ pt = Kδvx − vx δ n δ n δ n n qt = Kδux + (n + 1)ǫ Kδ (u ) ux − ux − (n + 1)ǫ u ux, or equivalently the system

pt = Kδqx + Kδvx − vx n δ n δ n qt = Kδpx + Kδux − ux + (n + 1)ǫ Kδ (u ) px + Kδ (u ) ux δ n δ n n −(u ) ux + (u ) ux − u ux . 13 We have the initial-value problem

pt = Kδqx + F1, p(x, 0)=0, (6.11) n δ n qt = Kδ 1 + (n + 1)ǫ (u ) px + F2 + G, q(x, 0)=0, (6.12) where the residual terms F1, F2 and G are given by

F1 = Kδvx − vx n δ n δ n F2 = Kδux − ux + (n + 1)ǫ Kδ (u ) ux − (u ) ux n δ n n G = (n + 1)ǫ (u ) − u ux. To complete the proof we need to estimate the residual terms. By Proposition 6.1, we have the following bounds on F1, F2 and G:

s−1 kF1(t)kH = Kδvx(t) − vx(t) Hs−1 θ θ θ ≤ Cδ kvx(t)kHs+θ−1 ≤ Cδ kv(t)kHs+θ ≤ Cδ n δ n δ n s−1 kF2(t)kH ≤ Kδux(t) − ux(t) Hs−1 + (n + 1)ǫ Kδ((u ) ux)(t) − ((u ) ux)(t) Hs−1 θ n θ δ n ≤ Cδ + Cǫ δ k (u ) ux (t)kHs+θ−1 θ n θ θ ≤ Cδ + Cǫ δ ku(t)kHs+θ ≤ Cδ n δ n n s−1 kG(t)kH = (n + 1)ǫ (u (t)) − (u(t)) ux(t) Hs−1 n δ n n s−1 ≤ Cǫ (u ( t)) − (u(t)) Hs−1 kux(t)k H n δ n n s ≤ Cǫ (u (t)) − (u(t)) Hs−1 ku(t)kH n δ n s−1 ≤ Cǫ u (t) − u(t) Hs−1 = Cǫ kp(t)kH for some constant C independent of δ. Combining these estimates with the one in (2.13) we get d E2 (t) ≤ CǫnE2 (t)+ CδθE (t). (6.13) dt s−1 s−1 s−1 Applying Gronwall’s inequality yields (6.10). This completes the proof. The above theorem is the main result of our work. It says that the error in the approximation of solutions to (6.3) by the corresponding solutions of (6.2) will be small as δ becomes small enough with some convergence rate θ ∈ (0, 2] and it is controlled on the time scale O(1/ǫn). Also, the convergence becomes slower for larger times.

7. Convergence from the discrete lattice model to the continuous model

As a continuation of the considerations in the previous section, the problem of search- ing the continuous limit of the discrete lattice dynamic model is interesting in its own right. So, our objective in this section is to provide a rigorous justiﬁcation of the convergence of the FPUT equation (which is a member of the class (1.1) for the triangular kernel to be described below) to the classical elasticity equation in the limit of vanishing nonlocality. In other words, we now show that, as the lattice spacing approaches zero, the 14 FPUT discrete lattice model and the classical elasticity model behave qualitatively similarly, provided the initial data is properly imposed. As it is expected, in the long-wave limit the discreteness does not prevail and the predictions of the discrete and continuum models agree to within an order of neglected terms. To this end we consider the triangular kernel deﬁned by

1 − |x| for |x|≤ 1, βT (x)= 0 for |x| > 1,

T 4 2 ξ T T 1/2 for which β (ξ) = ξ2 sin ( 2 ). Recall that k (ξ) = β (ξ) . Then (6.2) becomes the differential-difference equation c c δ d δ n δ n+1 utt = ∆δ u + ǫ (u ) (7.1) if we use the difference operator 1 ∆dg(u)= g u(x + δ, t) − 2g u(x, t) + g u(x − δ, t) . (7.2) δ δ2 Under the transformation uδ(x, t)= wδ(x + δ, t) − wδ(x, t) /δ, the differential-difference equation (7.1) becomes the famous FPUT equation

n δ d δ ǫ δ δ n+1 δ δ n+1 wtt(x, t) = ∆δ w (x, t)+ n+2 w (x + δ, t) − w (x, t) − w (x, t) − w (x − δ, t) δ (7.3) The discrete lattice model (7.1), or equivalently (7.3), describes longitudinal vibrations of a one-dimensional doubly-inﬁnite chain of identical particles with unit mass. The lattice consists of particles equally spaced at distance δ and it takes into account only nearest neighbor interactions (this is the celebrated FPUT lattice [13] with the potential function 1 2 2 ǫn+2un+2 δ V (u)= 2 ǫ u + n+2 ). In the above equations, w (x, t) stands for the displacement of the particle at x from its equilibrium position at time t while uδ(x, t) corresponds to the strain (the relative displacement between locations of two adjacent particles at x + δ and x). We are interested in investigating asymptotic behavior of (7.1) as δ → 0, which allows us to pass from the discrete description to the continuous description. If we δ T take the initial velocity as ut (x, 0) = Kδ (v0(x))x for some suitable v0, the long-time existence result of Section 5 (Lemma 5.3) will hold. Through the Fourier transform one T gets K u = χ 1 1 ∗ u where χ 1 1 denotes the characteristic function of the interval (− 2 , 2 ) (− 2 , 2 ) 1 1 T 1 T (− , ); K is the convolution with χ δ δ . Then K Dx is the diﬀerence operator. 2 2 δ δ (− 2 , 2 ) δ δ So, for the initial values u (x, 0) = u0(x) and 1 δ δ uδ(x, 0) = KT v (x) = v (x + ) − v (x − ) , (7.4) t δ 0 x δ 0 2 0 2 T (7.1) has a solution over long times. As would be expected, Kδ (v0(x))x is just the T ′ discretized derivative of v0 and limδ→0 Kδ (v0(x))x = v0(x). In fact from the Taylor

15 theorem δ 2 T ′ 1 δ 2 ′′′ Kδ (v0(x))x − v0(x) = ( − |y|) v0 (x + y)dy, 2δ δ 2 Z− 2 δ 2 2 T ′ 2 1 δ 2 ′′′ Kδ (v0(x))x − v0(x) ≤ 2 ( − |y|) v0 (x + y)dy . 4δ Z δ 2 − 2

Then δ 2 2 T ′ 2 1 ′′′ 2 δ 2 4 2 Kδ (v0)x − v0 L2 ≤ 2 kv0 kL2 ( − |y|) dy ≤ Cδ kv0kH3 . 4δ Z δ 2 − 2 T s 2 s Since Kδ commutes with the operator Λ = (1 − Dx) , we have for any s ≥ 1, T ′ T s−1 s−1 ′ 2 s−1 2 3 s+2 Kδ (v0)x − v0 Hs−1 = Kδ Λ (v0)x − Λ v0 L2 ≤ Cδ kΛ v0kH ≤ Cδ kv0kH .

We now compare the solution uδ of (7.1) with initial data 1 δ δ uδ(x, 0) = u (x), uδ(x, 0) = v (x + ) − v (x − ) , (7.5) 0 t δ 0 2 0 2 to the solution u of the classical (local) elasticity equation, (6.3), with the initial data

u(x, 0) = u0(x), ut(x, 0)=(v0(x))x, (7.6) s+2 δ δ where u0, v0 ∈ H . We first define the differences p = u − u and q = ut − ut. Then, as in the previous section, p, q satisfy the system (6.11)-(6.12) with the initial data 1 δ δ p(x, 0)=0, q(x, 0) = v (x + ) − v (x − ) − (v (x)) . δ 0 2 0 2 0 x However, Theorem 6.2 will not directly apply due to nonzero initial data related to q. So, 2 using Gronwalls’ inequality and the fact that Es−1(0) ≈kq(0)kHs−1 ≤ Cδ , we conclude from (6.13) that

δ δ 2 T u (t) − u(t) − + u (t) − u (t) − ≤ Cδ (1 + t) for all t ≤ . Hs 1 t t Hs 1 ǫn

With the above preparatory work, we can now state the following comparison result: s+2 δ δ T s+2 Theorem 7.1. Suppose u0, v0 ∈ H , s > 5/2. Let u , v ,u,v ∈ C [0, ǫn ],H ∩ 1 T s+1 C [0, ǫn ],H satisfy (7.1) and (6.3) with the initial data given by (7.5) and (7.6). T Then we have, for all t ∈ [0, ǫn ] δ δ 2 Cǫnt u (t) − u(t) Hs−1 + ut (t) − ut(t) Hs−1 ≤ Cδ (1 + t)e , where C is a constant independent of δ and ǫ. Remark 7.2. It is worth mentioning that the proper imposition of the initial conditions is crucial in establishing the above comparison result. The comparison result in Theorem 7.1 is based on the initial conditions (7.5) and (7.6) given for uδ and u, respectively. We note from (7.5) and (7.6) that while uδ and u are initially equal to the same function, δ the initial values of their derivatives ut and ut are diﬀerent. This is because the initial δ condition stated in Corollary 4.3 for ut involves the pseudo-diﬀerential operator K. So δ ut (x, 0) must be written properly in order to apply the existence result given in Corollary 4.3 for the second-order equation (1.1). 16 References

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