Compactness estimates for difference schemes for conservation laws with discontinuous flux Open Access

Abstract

1. Introduction

The main part of this paper investigates a finite difference algorithm as it applies to the Cauchy problem for scalar conservation laws with the form

where |$(x,t)\in{\mathbb{R}}\times{\mathbb{R}}_+$|⁠; |$u(x,t)$| is the scalar unknown function; and |$u_0(x),k(x,t),f(k,u)$| are given functions to be detailed later. Here it suffices to say that for the compactness estimates we need |$k(x,t)\in BV({\mathbb{R}}\times{\mathbb{R}}_+)$|⁠, |$u\mapsto f(k(x,t),u)$| genuinely nonlinear, and |$u_0(x)$| bounded (see Section 2 for the complete list of assumptions).

The special feature of (1.1) is the nonlinear flux function |$f(k(x,t),u)$| that depends explicitly on the spatial and temporal variables through a discontinuous coefficient |$k(x,t)$|⁠. Conservation laws with discontinuous flux functions are encountered in various applications. For example, in oil reservoirs, rock permeability may vary significantly in different locations, resulting in a discontinuous flux function (see, e.g., Gimse & Risebro, 1992). Similarly, in traffic flow, abrupt changes in vehicle density may occur at bottlenecks, which leads to a discontinuous flux (see, e.g., Bürger et al., 2009).

The study of conservation laws with discontinuous flux has been a heavily investigated area for the past three decades. This is partly due to its links to various applications, but also because it possesses several non-trivial mathematical properties. These properties include the existence of several |$L^1$| stable semigroups, which are based on different entropy conditions, as well as the lack of uniform bounds on the total variation (i.e., |$BV$| estimates). As a result, it constitutes a non-trivial generalization of scalar conservation laws (Kružkov, 1970). While a comprehensive review of the extensive literature is beyond the scope of this paper, we provide a few select references for interested readers and direct them to the reference lists in these papers: Klingenberg & Risebro (1995); Karlsen et al. (2003); Karlsen & Towers (2004); Adimurthi & Gowda (2005); Bürger & Karlsen (2008); Andreianov et al. (2011); Bressan et al. (2019); Andreianov & Sylla (2023).

Proving the existence of solutions for conservation laws is associated with establishing strong a priori estimates for approximate solutions. Classically, this involves bounding the total variation of the approximate solutions, independent of the approximation parameter. However, bounding the total variation when the flux is discontinuous is in general impossible. To address this issue, several alternative convergence approaches have been applied over the years, including singular mapping techniques as well as compensated compactness and other advanced weak convergence methods (Klingenberg & Risebro, 1995; Karlsen et al., 2003; Karlsen & Towers, 2004; Holden et al., 2009; Panov, 2010; Erceg et al., 2023) (this is just a few examples). Specifically, the weak convergence methods (see, e.g., Panov, 2010; Erceg et al., 2023) are profound and involve a significant amount of functional analysis, making them technically challenging to comprehend.

The aim of this paper is to derive quantitative |$L^1$| translation estimates that can be utilized for various applications, such as demonstrating the convergence of finite difference approximations. Let us consider a time step |$\varDelta t$| and a grid size parameter |$\varDelta x$|⁠, which collectively form a parameter pair |$\varDelta = (\varDelta x, \varDelta t)$|⁠. We use the notation |$\left \{u^{\varDelta }\right \}_{\varDelta> 0}$| to represent the approximate solutions. The translation estimates, which maintain uniformity across all |$\varDelta $|⁠, are defined as follows:

This estimate holds true for any temporal (⁠|$\tau> 0$|⁠) and spatial (⁠|$h> 0$|⁠) translations. Here, |$\mu _t,\mu _x$| are parameters in the interval |$(0,1)$| that quantify the ‘degree of compactness’, where |$\mu _t=\mu _x=1$| corresponds to uniformly bounded total variation.

In (1.2), |$\chi $| is a weight function that can be employed to control the growth of solutions as they approach infinity in |$x$|⁠. An example of a weight function is |$\chi (x)=(1+\left |x\right |{{}}^2)^{-N}$|⁠, where |$N>1/2$|⁠, see (2.15) and also Karlsen (2023). If |$\textrm{supp} \left (u^{\varDelta }\right )$| is contained within a compact set |$[0,T]\times [-R,R]\subset{\mathbb{R}}^2$|⁠, independent of |$\varDelta $|⁠, we may set |$\chi \equiv 1$|⁠, which is what we do for the rest of the introduction!

We refer to translation estimates like (1.2) as quantitative compactness estimates. They can be used to derive convergence results via the well-known Kolmogorov–Riesz–Fréchet characterization of precompact subsets of |$L^1$| in terms of the uniform continuity of translations in |$L^1$|⁠, see, e.g., Brezis (2010, Theorem 4.26). Beyond implying convergence, quantitative compactness estimates can be used to derive continuous dependence and error estimates (Kuznetsov, 1976; Bouchut & Perthame, 1998).

Our approach is technically elementary and relies on discrete interaction estimates and the existence of one uniformly convex entropy. We draw inspiration from previous work of Golse and Perthame (Golse & Perthame, 2013), who developed a quantitative compensated compactness framework for establishing Besov space regularity of solutions to homogenous conservation laws. We adapt their approach and apply it to establish quantitative compactness estimates for sequences of approximate solutions. A related approach has recently been employed in the study of vanishing viscosity approximations of stochastic conservation laws, as described in Karlsen (2023). While the details are specifically outlined for the Lax-Friedrichs scheme, the same framework can be applied to other difference schemes as well. The obtained estimates are new also in the homogenous case |$k\equiv 1$| (scalar conservation law).

To clarify further, let us consider the Lax-Friedrichs scheme for the Cauchy problem (1.1), which can be represented by the following equation (Karlsen & Towers, 2004):

Here, |$U_j^n$| approximates the exact solution |$u$| at the grid point |$(j\varDelta x,n\varDelta t)$|⁠, where |$n$| and |$j$| are integers, |$\lambda = \varDelta t/\varDelta x$| and |$f_j^n=f\big (k_{j}^n,U_{j}^n\big )$|⁠. The temporal and spatial discretization parameters are linked through a CFL condition (2.8), so that |$\varDelta t\sim \varDelta x$|⁠.

Assuming reasonable conditions on |$f$|⁠, |$k\in BV$| and |$u_0\in L^\infty $|⁠, we can establish the following two a priori estimates (see next section):

These estimates are the only bounds that remain robust with respect to the regularity of the coefficient |$k(x,t)$|⁠. If |$k\equiv 1$|⁠, the approximations are of bounded total variation (Holden & Risebro, 2015). The second bound is referred to as a dissipation (or entropy stability) estimate, which is known to hold for many numerical schemes for conservation laws (Eymard et al., 2000; Tadmor, 2003). These estimates are sometimes referred to as weak |$H^1$| estimates since they suggest that |$\iint \left |\partial _x u^{\varDelta }\right |{{}}^2\, \textrm{d}x\, \textrm{d}t\lesssim 1/\left |\varDelta \right |$|⁠. The estimate for the Lax-Friedrichs scheme (1.3) with variable and discontinuous |$k$| was derived in Karlsen & Towers (2004). In this paper, we present a slightly generalized variant of the estimate.

Given the genuinely nonlinear flux |$f(k,u)$|⁠, in the sense of (2.7), consider an entropy/entropy flux pair |$\bigl (S(k,u),Q(k,u)\bigr )$|⁠. Set |$S_j^n = S\big (k_j^n,U_j^n\big )$| and |$Q_j^n = Q\big (k_j^n,U_j^n\big )$|⁠. The Lax-Friedrichs scheme satisfies the following entropy balance (to be derived later):

where, by (1.4) and |$k\in BV$|⁠, we have |$\varDelta x \sum _n\sum _j \big |\varPsi _j^n\big | \lesssim 1$|⁠. Note that the entropy production |$\varPsi _j^n$| is a signed measure (even if |$S=S(u)$| is convex), which is distinct from the homogenous conservation law case where it is negative.

Let |$\nu $| be an arbitrary nonnegative integer and introduce the ‘spatial difference’ quantities

The appearance of |$2\nu $|⁠, as opposed to simply |$\nu $|⁠, in the above equations is due to that we use a staggered mesh, where the various grid functions are only defined at grid points where |$j+n$| is an even integer. Then it is straightforward to deduce from (1.3) and (1.5) that the following |$2\times 2$| system of finite difference equations hold:

where |$C_{A,j}^n\equiv 0$| and |$C_{D,j}^n= \varPsi _{j+2\nu }^n- \varPsi _j^n$|⁠. We are also going to need the following more regular quantities obtained by applying ‘inverse-difference’ operators to |$A_j^n$|⁠, |$D_j^n$|⁠:

Indeed, one can prove that

recalling that |$\nu $| is a nonnegative integer.

Next, we introduce the interaction functional

which is a measure of future potential interaction at time level |$n$| of the finite difference solutions. This is a discrete version of a functional referred to as the Varadhan functional by Tartar (Tartar, 2008, p. 182) and Golse & Perthame (Golse & Perthame, 2013).

In this paper, we establish the following discrete interaction identity for the |$2\times 2$| system (1.6):

To obtain an evolution difference equation satisfied by the interaction potential |$I^n$|⁠, we compute the temporal difference |$\frac{I^{n+1}-I^n}{\varDelta t}$|⁠. This identity is then multiplied by |$\varDelta t$| and summed over |$n$|⁠, resulting in the equation given by (1.7). The derivation of (1.7) employs a discrete chain rule, as well as the difference equations for |$A_j^n$| and |$D_j^n$|⁠. A more complicated form of (1.7) holds when there is a weight function |$\chi $| present. The interaction identity (1.7) can be interpreted as a discrete form of the interaction identity (7) introduced in Golse & Perthame (2013), as well as the stochastic interaction identity (3.12) in Karlsen (2023) (with |$\sigma =0$|⁠). It should be noted that the term |${\mathcal{E}}$| appearing on the right-hand side of (4.12) is solely a consequence of the discretization process and does not have a corresponding counterpart in the interaction identities of Golse & Perthame (2013); Karlsen (2023). For the detailed form of |${\mathcal{E}}$|⁠, see Lemma 4.1.

The terms on the right-hand side of (1.7) can all be bounded by a |$\varDelta $|-independent constant |$C$| times |$2\nu \varDelta x$|⁠. The next step is to convert (1.7) into a quantitative compactness estimate, which requires exploiting the genuine nonlinearity of |$f(k,u)$| in |$u$|⁠. The precise assumption can be found in (2.7), and a corresponding assumption for homogeneous conservation laws can be found in Golse & Perthame (2013, Section 5). A relevant special case occurs if |$u\mapsto f(k,u)$| is uniformly convex and |$S=f$|⁠. Then

and therefore

A similar estimate can be deduced for the temporal differences. Along with the compact support assumption (or if we use a weight function) and Hölder’s inequality, this translates into the |$L^1$| translation estimate given by (1.2) with |$h=\tau =2\nu \varDelta x$| and |$\mu _t=\mu _x=1/4$|⁠, which is the main result of this paper.

In summary, the above outline provides an overview of the quantitative compactness approach. To delve deeper, we will present detailed proofs incorporating a weight function in the following sections. We demonstrate the effectiveness of the quantitative compactness approach on the Lax-Friedrichs scheme. However, with slight modifications to the discrete interaction identity given by (1.7), the same approach can be applied to other classical schemes, provided they are uniformly bounded in |$L^\infty $| and satisfy a weak |$H^1$| (dissipation) estimate, that is to say, they satisfy modified forms of the a priori estimates described in (1.4). This allows for elementary convergence proofs, especially in cases where obtaining total variation estimates is challenging.

Our results provide an existence theorem for the Cauchy problem (1.1). The uniqueness question for conservation laws with discontinuous flux is another problem entirely, even in the most basic setting where the flux has a single spatial discontinuity and no temporal discontinuity (the so-called two-flux problem). For example, it turns out that the two-flux problem generally has infinitely many definitions of entropy solution, each one of which generates its own distinct |$L^1$| contraction semigroup (Adimurthi & Gowda, 2005). In Karlsen & Towers (2004) we used results from Karlsen et al. (2003) to prove a uniqueness result applicable to the Lax-Friedrichs scheme of this paper in the special case where |$k$| is piecewise Lipschitz continuous, meaning that all of the discontinuities of |$k$| occur along Lipschitz continuous curves in the |$(x,t)$|-plane. This uniqueness result was proven under the additional assumption that all of the flux discontinuities satisfy a certain ‘crossing condition’. At least for the simple two-flux version of the problem (1.1), the solution generated by the Lax-Friedrichs scheme of this paper corresponds to the so-called vanishing viscosity solution (Andreianov et al., 2011). In the absence of the crossing condition it is not known whether the (subsequential) limit of the Lax-Friedrichs scheme is the vanishing viscosity solution. Finally, given (1.8), it is worth noting that the solution |$u$| derived as the limit of the Lax-Friedrichs scheme, exhibits Besov space regularity as given by

This regularity aligns with the known regularization effect established in Golse & Perthame (2013, Theorem 5.1) for homogeneous equations with a single convex entropy.

The structure of this paper is as follows: Section 2 presents the assumptions related to the data of the problem and precisely defines the Lax-Friedrich scheme. This section also introduces our main result. In Section 3, we establish preliminary |$L^\infty $| and weak |$H^1$| estimates. Section 4 proves the spatial translation estimate, while Section 5 details the temporal estimate. By bringing together the spatial and temporal estimates, we provide the proof of our main result.

2. Lax-Friedrichs scheme and main result

We begin by listing some assumptions on |$u_0,k,f$| which will be needed.

Regarding the initial function we assume

For the discontinuous coefficient |$k:{\mathbb{R}}\times{\mathbb{R}}_+\to{\mathbb{R}}$| we assume that

Regarding the flux function |$f:[\alpha ,\beta ]\times [a,b]\to{\mathbb{R}}$| we assume that

We need also an assumption on |$f$| that guarantees that the Lax-Friedrichs approximations stay uniformly bounded. For example, we can require

which in fact implies that the interval |$[a,b]$| becomes an invariant region.

We use the notation |$\partial _k G$| and |$\partial _u G$| to denote the first order partial derivatives of |$G(\cdot ,\cdot )$| with respect to the first and second variables. Letting |${\mathcal{U}}:= [\alpha ,\beta ] \times [a,b]$|⁠, we will use the following abbreviations:

We are given functions |$S$| and |$Q$| (referred to earlier) that are assumed to form an entropy/entropy flux pair |$(S(k,u),Q(k,u))$|⁠, meaning that

We assume that

We make the following genuine nonlinearity assumptions about |$f$| and |$S$|⁠. For |$k \in [\alpha ,\beta ]$|⁠,

for some constants |$C_f, C_S>0$|⁠.

Next we describe the Lax-Friedrichs scheme. Let |$\varDelta x>0$| and |$\varDelta t>0$| denote the spatial and temporal discretization parameters, which are chosen so that they always obey the CFL condition

Here |$\kappa $| is a positive parameter, which we can choose to be very small so that the allowable time step is reduced only negligibly. We will work under the standing assumption that the space step |$\varDelta x$| and the time step |$\varDelta t$| are comparable, i.e., there are constants |$c_1,c_2>0$| such that |$c_1\le \frac{\varDelta t}{\varDelta x}\le c_2$|⁠. Therefore, when we declare that a constant |$C$| is independent of |$\varDelta x$| (or |$\varDelta t$|⁠), it implies that |$C$| is also independent of |$\varDelta = (\varDelta x, \varDelta t)$|⁠.

The time domain |$[0,\infty )$| is discretized via |$t^n = n\varDelta t$| for |$n\in{\mathbb{Z}}_+^0:=\{0,1,\ldots \}$| (⁠|${\mathbb{Z}}_+:=\{1,2,\ldots \}$|⁠), resulting in time strips |$[t^n,t^{n+1})$|⁠. The spatial domain |${\mathbb{R}}$| is divided into cells |$[x_{j-1},x_{j+1})$| with centers at the points |$x_j =j \varDelta x$| for |$j \in{\mathbb{Z}}$|⁠. Let |$\rho _j(x)$| be the characteristic function for the interval |$[x_{j-1},x_{j+1})$| and |$\rho _j^n$| the characteristic function for the rectangle |$[x_{j-1},x_{j+1}) \times [t^n,t^{n+1})$|⁠.

The finite difference scheme then generates, for each mesh size |$\varDelta = (\varDelta x,\varDelta t)$|⁠, with |$\varDelta x$| and |$\varDelta t$| taking values in sequences tending to zero, a piecewise constant approximation

where the values |$\big \{U_j^n:(j,n)\in{\mathbb{Z}}\times{\mathbb{Z}}_+^0, j+n=\textrm{even}\big \}$| remain to be defined.

We define |$\big \{U_j^0: j=\textrm{even}\big \}$| by

Given |$\big \{U_j^n:j+n=\textrm{even}\big \}$|⁠, we define next |$\big \{U_j^{n+1}:j+n=\textrm{odd}\big \}$|⁠. Let |$(K,U)=(K,U)(x,t)$| denote a weak solution of the |$2\times 2$| system

with Riemann initial data

where the coefficient |$k(x,t)$| has been discretized via the piecewise constant approximation

Recall that |$k(x,t)$| is a |$BV$| function. For every |$t$|⁠, the function |$k(t,\cdot )$| can be considered as a precise representative, being defined everywhere and normalized through right-continuity. Consequently, we are justified in setting |$k_j^n = k(x_j,\hat{t}_n)$| in (2.12), where |$\hat{t}_n$| is an arbitrary point lying in the interval |$[t^n,t^{n+1})$| (for example, |$\hat{t}_n=t^n$|⁠). We then define

Integrating the weak formulation of (2.11) over the control volume |$[x_{j-1},x_{j+1})\times (0,\varDelta t)$| gives

After a direct evaluation of the integrals for |$\varDelta t$| small, we obtain the staggered Lax-Friedrichs scheme

Notice that in this paper we restrict our attention to the sublattice

which means that |$\big \{U_j^0: j=\textrm{even}\big \}$|⁠, |$\big \{U_j^1: j=\textrm{odd}\big \}$|⁠, |$\big \{U_j^2: j=\textrm{even}\big \}$|⁠, etc. are calculated. The following abbreviations can be used to shorten some of the expressions that will arise. For fixed |$n \in \{0, 1, \ldots , N\}$|⁠,

Note that

while

We work with the so-called ‘even’ sublattice described above in the interest of conceptual simplicity. Note that when one applies the Lax-Friedrichs scheme on the standard lattice, what is generated is two uncoupled numerical solutions, one solution on the even sublattice, and one solution on the odd sublattice. Our analysis for the even sublattice would then apply to each of those solutions separately.

The following theorem is our main result:

The validation of (2.14) is directly derived from the results in Section 4 (Proposition 4.5) and Section 5 (Proposition 5.7).

We will conclude this section by outlining the key concepts underpinning the proof of the spatial part of Theorem 2.1. Our discussion will be focused on a homogenous equation, augmented with artificial viscosity, to provide a clear outline of the underlying ideas.

For any fixed |$\varepsilon>0$|⁠, let |$u_{\varepsilon }\in C^2$| satisfy the equations

where |$f,\eta \in C^2$| are (for example) uniformly convex and

assuming that |$\left \|u_{\varepsilon }\right \|_{L^\infty ({\mathbb{R}}_+\times{\mathbb{R}})}, \left \|u_{\varepsilon }\right \|_{L^\infty ({\mathbb{R}}_+;L^p({\mathbb{R}}))}\lesssim 1$|⁠, |$p=1,2$|⁠. To prevent ambiguity and ensure clarity, we denote the entropy/entropy flux pair as |$(\eta , q)$|⁠, distinguishing them from the functions |$(S,Q)$| used for (1.1). We denote by |$\varDelta _h W(x,t):=W(t,x+h)-W(x,t)$| the spatial difference operator with step size |$h$|⁠. Set

and

Then the system (2.17) takes the form

where, applying Lemma 3.4 and Remark 3.5 with |$S=\eta $| and |$Q=q$|⁠,

We need the (spatial) anti-derivatives of |$a$| and |$d$|⁠:

which crucially are uniformly (in |$\varepsilon $|⁠) Lipschitz continuous.

Denote by |$I(t)$| the (spatial) interaction functional:

A straightforward calculation will confirm that the following (spatial) interaction identify holds:

Since we assumed |$a,d\in L^\infty _tL^1_x$|⁠,

Hence, by integrating the above identify in |$t\in [0,T]$|⁠, |$T>0$|⁠, we arrive at

where

Utilizing the assumed bounds on |$u_{\varepsilon }$|⁠, we can estimate these three terms following the approach outlined in Karlsen (2023). The final result is that |$\left |J_i\right | \lesssim h$| for |$i=1,2,3$|⁠, which implies that

A similar temporal translation estimate can be derived for |$u_{\varepsilon }$|⁠. This brings us to the end of the outline detailing the proof for the vanishing viscosity approximation |$u_{\varepsilon }$|⁠.

3. Preliminary results

The following result is taken from Karlsen & Towers (2004, Lemma 4.1).

In Karlsen & Towers (2004) we employed the single entropy |$S(u) = u^2/2$|⁠. We wish to allow for an entropy of the form |$S(k,u)$|⁠. The first part (inequality (3.1)) of the lemma that follows is a generalization of Karlsen & Towers (2004, Lemma 4.3), which we prove by modifying the proof in Karlsen & Towers (2004). The second part of the lemma (inequality (3.2)) is new, and is required for the analysis that follows.

In what follows, we consider an arbitrary weight function |$\chi \in C^1({\mathbb{R}}) \cap L^1({\mathbb{R}})$| satisfying the properties in (2.15). We will use the following abbreviations:

Also, the following facts will be helpful. For a fixed integer |$i$|⁠, if |$\{Z_j^n \}$| is bounded, then

and

The following result is the version of Karlsen & Towers (2004, Lemma 4.3) that is appropriate for the assumptions of this paper.  

The following lemma is a mild adaptation of Golse & Perthame (2013, Lemma 5.2). It will be used in the upcoming two sections.

4. Estimate of spatial translates

This section focuses on stating and proving Proposition 4.5, crucial for deriving Theorem 2.1 as mentioned earlier. To facilitate this, we will first lay the groundwork with a series of technical lemmas. A condensed version of the proof, particularly addressing the viscosity approximation, is presented at the end of Section 2 to guide the upcoming calculations.

For now we assume that |$h = 2 \nu $|⁠, where |$\nu $| is a nonnegative integer. In what follows |$\varDelta _h$| and |$\varDelta _{-h}$| denote the spatial finite difference operators,

We will also use the difference operators |$\varDelta ^u_h$| and |$\varDelta ^h_h$| defined by

with similar definitions for |$\varDelta ^u_{-h} f_j^n$| and |$\varDelta ^k_{-h} f_j^n$|⁠. Note the identity

We will employ the inequalities

The immediate goal is to prove Lemma 4.4 below. We will use the following key fact, which is the content of Lemma 3.4:

where

We write the Lax-Friedrichs scheme and entropy inequality in the form

where, by (3.2) of Lemma 3.2,

With the assumption that |$k \in BV({\mathbb{R}}\times{\mathbb{R}}_+)$|⁠, along with (3.5) of Lemma 3.3,

where |$C_3(T)$| is a constant that is independent of |$\varDelta x$|⁠.

We multiply both equations of (4.5) by |$\chi _j \varDelta _h$|⁠, and after some algebra, arrive at the following |$2\times 2$| discrete system:

where

Define

where

The following identities hold:

where we are using the notational convention

with a similar definition for sums where |$\varOmega _n$| is replaced by |$\varOmega _n^{\prime}$|⁠.

The interaction identity (4.12) below can be viewed as a discrete, Lax-Friedrichs version of the interaction identity (7) in Golse & Perthame (2013), and also the stochastic interaction identity (3.12) of Karlsen (2023) (with |$\sigma =0$|⁠). The term |${\mathcal{E}}$| appearing on the right side of (4.12) is purely an artifact of discretization, and does not have a counterpart in the interaction identities of Golse & Perthame (2013) and Karlsen (2023).

We are now in a position to prove the main result of this section, specifically, the spatial part of Theorem 2.1.

5. Estimate of temporal translates

In this section we prove estimates that are the temporal analog of the spatial estimates obtained in Section 4. The section concludes with Proposition 5.7, which effectively provides the temporal part of the translation estimate of Theorem 2.1.

We prove a sequence of lemmas leading up to the proof of Proposition 5.7. We now assume that |$\tau = \theta \varDelta t $|⁠, where |$\theta $| is an even nonegative integer. Define the temporal finite difference operator

Another version of the system (4.7) results by multiplying (4.5) by |$\chi _j \varDelta ^{\tau }$|⁠:

with (4.8) replaced by

i.e., all instances of |$\varDelta _{h}$| in (4.8) have been replaced by |$\varDelta ^{\tau }$|⁠.

Define

The following lemma states another interaction identity, analogous to (4.12). After the operator |$\varDelta ^\tau $| is substituted for |$\varDelta _h$|⁠, the proof is identical to the proof of Lemma 4.1.

Prior to utilizing this interaction identity, it is necessary for us to obtain some estimates, which serve as consequences of (3.5).

We can now prove the main result of this section.

References