Uniform tail estimates and L^p(ℝ^N)-convergence for finite-difference approximations of nonlinear diffusion equations

1. Introduction

The purpose of this paper is to improve convergence results in $C ([0, T]; L_{l o c}^{p} (R^{N}))$ to $C ([0, T]; L^{p} (R^{N}))$ for numerical schemes of generalized porous medium equations in the context of bounded and integrable solutions. In detail, we study

(GPME)

where $u = u (x, t)$ is the solution, $φ : R \to R$ is nondecreasing and $ϑ$ -Hölder continuous with $ϑ \in (0, 1]$ , $g = g (x, t)$ some right-hand side, $u_{0} \in L^{1} \cap L^{\infty}$ , and $T > 0$ . The general operator $L$ is given as

(1.1)

L = L^{σ, μ} := L^{σ} + L^{μ}

where $L^{σ} : C_{c}^{\infty} (R^{N}) \to L^{1} (R^{N})$ is a possibly degenerate local diffusion operator

(1.2)

L^{σ} [ψ] (x) := t r (σ σ^{T} D^{2} ψ (x))

where $σ = (σ_{1}, \dots, σ_{P}) \in R^{N \times P}$ , $P \in N$ , and $σ_{i} \in R^{N}$ , and the anomalous or nonlocal diffusion operator $L^{μ}$ is defined for any $ψ \in C_{c}^{\infty} (R^{N})$ as

(1.3)

L^{μ} [ψ] (x) = \int_{R^{N} ∖ {0}} (ψ (x + z) - ψ (x) - z \cdot D ψ (x) 1_{| z | \leq 1}) d μ (z),

where $D$ is the gradient, $1_{| z | \leq 1}$

a characteristic function, and

μ

a nonnegative and symmetric Radon measure satisfying at least the usual Lévy measure condition (see Section 3). In this paper, to simplify, we will always choose either

σ \equiv 0

σ σ^{T} = I_{N}

. That is, the local operator is given as

L^{σ} = c Δ := c \sum_{i = 1}^{N} \partial_{x_{i} x_{i}}^{2}

where

c \in {0, 1}

Remark 1.1.

Since $μ$ will be symmetric, ${lim}_{r \to 0^{+}} \int_{r < | z | \leq 1} z d μ (z) = 0$ , and we have an equivalent definition of $L^{μ}$ in (1.3) in terms of a principal value integral:

$L^{μ} [ψ] (x) = lim r \to 0^{+} \int_{| z | > r} (ψ (x + z) - ψ (x)) d μ (z) = P . V . \int_{| z | > 0} (ψ (x + z) - ψ (x)) d μ (z) .$
We will also comment on the case when (GPME) has a convection term in addition, see Section 5.2.

Equations of the form (GPME) (and also variants with convection) appear in numerous applications. We selectively mention reservoir simulation, sedimentation processes, and traffic flow in the local case [25, 8, 39]; cardiac electrophysiology and semiconductor growth in the nonlocal case [7, 40]

; and flows of fluids through porous media, mathematical finance, information theory, and phase transitions in both cases

[38, 12, 11, 36, 6]. Important examples are strongly degenerate Stefan problems (cf. [17, 18]) with

φ (u) = max {0, u - L}

L > 0

, and the full range of porous medium and fast diffusion equations (cf. [12]) with

φ (u) = u | u |^{m - 1}

m > 0

. The class of operators

L^{σ, μ}

coincides with the generators of symmetric Lévy processes [4, 35, 1]. This includes e.g. the Laplacian

Δ

, fractional Laplacians

- (- Δ)^{\frac{α}{2}}

α \in (0, 2)

, relativistic Schrödinger type operators

m^{α} I - (m^{2} I - Δ)^{\frac{α}{2}}

α \in (0, 2)

and

m > 0

, tempered stable processes [11], and even discretizations of all of these. Since

σ

and

μ

may be degenerate or even identically zero, equation (GPME) can be purely nonlocal, purely local, or a combination.

Compactness results depend on the type of equation under study, and the properties available for such an equation. They are essential in the context of existence, continuous dependence, and asymptotic behaviour. For the latter example, this is particularly the case when considering the rescaled solution in the “four-step method” introduced by Kamin and Vázquez in [33]. In all of these cases, an approximate solution $u_{n}$ of the equation under study is considered. Then one shows compactness of the family formed by $u_{n}$ in order to be able to find a limit function. The limit must of course be a solution of the original equation (or variants of it). In this paper, the approximate solution will always come from a finite-difference scheme for (GPME).

To prove compactness in $C ([0, T]; L^{p} (R^{N}))$ with $p \in [1, \infty)$ , we employ the well-known Arzelà-Ascoli and Kolmogorov-Riesz compactness theorems (cf. Appendix A). A systematic approach to these theorems are presented in Section 2. Compared to some previous results in this direction (see e.g. Lemma 2.2 in [34] and Theorem A.8 in [31]), we are actually going to deduce equitightness (uniform tail control) for approximate solutions of (GPME). Then we are able to avoid all unnecessary tricks to fulfil the requirements of both Arzelà-Ascoli and Kolmogorov-Riesz, and instead present a minimal and efficient compactness argument. However, we still use the uniform boundedness ( $L^{\infty}$ -stability) of the solutions to make sure that some of the estimates needed hold. A possible improvement of our results is thus to study (GPME) in a pure $L^{1}$ -framework.

As far as we know, equitightness results for (GPME) have not been presented under our general assumptions before (especially in the nonlocal case), and these results are really the core of this paper. Such estimates are deduced by taking, roughly speaking, $x \mapsto 1_{| x | > R}$ as a test function in the very weak formulation of (GPME). This gives uniform tail control of the approximative solution (cf. e.g. [12, Proof of Proposition 10.2]). In examples when (GPME) conserves mass, we can summarize our equitightness results by saying that such estimates always holds in the local case, and also in the nonlocal setting when we assume that the nonlocal operator is comparable to a fractional order operator at infinity.

Finite-difference methods were developed in the full generality of (GPME) in [15, 16]; some early works are can be found in [20, 23]. We also refer to [21] (see also Part II of [22]) in the purely local case. In the case of (GPME) with an additional convection term, we mention e.g. [34, 31, 10].

The numerical schemes which will be presented below include most of the mentioned works on local and nonlocal cases. However, none of the above showed convergence in $C ([0, T]; L^{p} (R^{N}))$ (but some could still deduce that the limit itself belonged to that space, see also Section 5.1). Our equitightness estimates therefore improves convergence results already present in the literature. Conservative and monotone finite-difference schemes for scalar conservation laws are discussed in [31, Theorem 3.8], and they immediately fall into our $C ([0, T]; L^{p} (R^{N}))$ -framework. By adding a possibly nonlinear local diffusion term to such equations, we obtain convection-diffusion equations with local diffusion. Such equations have been studied in the context of finite-difference approximations in [34, Theorem 4.2]. See also Theorem 3.9 and Corollary 3.10 in [27]. Again, we can improve the convergence from the respective $L_{l o c}^{1} (Q_{T})$ and $C ([0, T]; L_{l o c}^{1} (R^{N}))$ to $C ([0, T]; L^{1} (R^{N}))$ since $L^{1}$ -equitightness holds. In the nonlocal diffusion setting, finite-difference schemes has just recently been analyzed in rigorous detail and generality in [15, 16], see also [20, 23]. The former two references obtain compactness/convergence results in $C ([0, T]; L_{l o c}^{1} (R^{N}))$ , and the latter in $L_{l o c}^{1} (Q_{T})$ . Our framework thus improves the compactness/convergence results of all four papers.

Outline

We start by reviewing known compactness theorems in Section 2. Assumptions and extensions are discussed. Main results are provided in Section 3 As the nonlocal operator $L^{μ}$ will be the hardest term to control uniformly at infinity, we discuss its discretization, needed assumptions, and related estimates in that section as well. Section 4 is reserved for proofs, and Section 5 for extensions (including the case of the convection) and comments. Important well-known results are presented in Appendices A and B, and finally, some auxiliary results regarding equitightness can be found in Appendix C.

Notation

Derivatives are denoted by $^{'}$ , $\frac{d}{d t}$ , $\partial_{x_{i}}$ , and $D ψ$ and $D^{2} ψ$ denote the $x$ -gradient and Hessian matrix of $ψ$ . $ρ_{δ}$ with $δ > 0$ will denote a standard mollifier.

We use standard notation for $L^{p}$ , $W^{p, q}$ , and $C_{b}$ . Moreover, $C_{c}^{\infty}$ is the space of smooth functions with compact support, and $C ([0, T]; L_{l o c}^{p} (R^{N}))$ the space of measurable functions $ψ : [0, T] \to L_{l o c}^{p} (R^{N})$ such that $ψ (t) \in L_{l o c}^{p} (R^{N})$ for every $t \in [0, T]$ , ${sup}_{t \in [0, T]} ∥ ψ (t) ∥_{L^{p} (K)} < \infty$ , and $∥ ψ (t) - ψ (s) ∥_{L^{p} (K)} \to 0$ when $t \to s$ for all compact $K \subset R^{N}$ and $t, s \in [0, T]$ . In a similar way we also define $C ([0, T]; L^{p} (R^{N}))$ . Note that the notion of $R^{N} \times (0, T) ∋ (x, t) \mapsto ψ (x, t) \in C ([0, T]; L_{l o c}^{p} (R^{N}))$ is a subtle one. In fact, we mean that $ψ$ has an a.e.-version which is continuous $[0, T] \to L_{l o c}^{p} (R^{N})$ . See e.g. p. 726 in [21] for more details. The space $C^{0, ϑ} (R)$ with $ϑ \in (0, 1]$ is identified as the Banach space with norm $∥ \cdot ∥_{C^{0, ϑ} (R)} := ∥ \cdot ∥_{C_{b} (R)} + | \cdot |_{C^{0, ϑ} (R)}$ where

| ψ |_{C^{0, ϑ} (R)} := sup x, y \in R {\frac{| ψ (x) - ψ (y) |}{| x - y |^{ϑ}}} .

When $ϑ = 1$ , we simply get $C^{0, ϑ} = W^{1, \infty}$ .

Remark 1.2.

From now on, we will study convergence in $C ([0, T]; L^{r} (R^{N}))$ (abbrev. $C_{t} (L_{x}^{r})$ ) with $r \in [1, \infty)$ . We use $r$ because we want it to be a unique identifier.

2. On compactness and convergence in $C ([0, T]; L^{r} (R^{N}))$

Let us give an overview of the properties needed to deduce compactness and convergence in $C_{t} (L_{x}^{r})$ with $r \in [1, \infty)$ .

2.1. Necessary and sufficient conditions for compactness

Consider a sequence ${u_{n}}_{n \in N} \subset C_{t} (L_{x}^{r})$ , and assume that ${u_{n}}_{n \in N}$ satisfies:

(Equitight in space pointwise in time) For all $t \in [0, T]$ ,

$lim R \to \infty sup n \int_{| x | > R} | u_{n} (x, t) |^{r} d x = 0.$
(Equicontinuous in space pointwise in time) For all $t \in [0, T]$ , there exists a modulus of continuity $λ$ such that

$sup n ∥ u_{n} (\cdot + ξ, t) - u_{n} (\cdot, t) ∥_{L^{r} (R^{N})} \leq λ (| ξ |) .$
(Equicontinuous in time) For all $t, s \in [0, T]$ , there exists a modulus of continuity $λ$ such that

$sup n ∥ u_{n} (\cdot, t) - u_{n} (\cdot, s) ∥_{L^{r} (R^{N})} \leq λ (| t - s |) .$

Properties (I)–(III) are exactly the necessary and sufficient conditions needed to obtain compactness in $C_{t} (L_{x}^{r})$ (cf. the Arzelà-Ascoli and Kolmogorov-Riesz compactness theorems A.1 and A.3).

Theorem 2.1 (Compactness).

Assume $r \in [1, \infty)$ , and let

{u_{n}}_{n \in N} \subset C ([0, T]; L^{r} (R^{N})) .

The following are equivalent:

The sequence ${u_{n}}_{n \in N}$ satisfies properties (I)–(III).
There exists a subsequence ${u_{n_{k}}}_{k \in N}$ and a $u \in C ([0, T]; L^{r} (R^{N}))$ such that

$u_{n_{k}} \to u in C ([0, T]; L^{r} (R^{N})) as k \to \infty .$

Remark 2.2.

The classical Kolmogorov-Riesz compactness theorem requires the family of functions to be equibounded, that is, for all $t \in [0, T]$ ,

sup n ∥ u_{n} (\cdot, t) ∥_{L^{r} (R^{N})} < \infty .

However, in [30], it has been pointed out that such a property follows from properties (I)–(II).

Proof.

The fact that the sequence $x \mapsto {u_{n} (x, t)}_{n \in N}$ satisfies (I)–(II) is, by the Kolmogorov-Riesz compactness theorem (cf. Theorem A.3), equivalent with ${x \mapsto {u_{n} (x, t)}_{n \in N}}$ being relatively compact in $L^{r} (R^{N})$ for all $t \in [0, T]$ . Finally, since the sequence $t \mapsto {u_{n} (x, t)}_{n \in N}$ satisfies (III), the proof is completed by an application of the Arzelà-Ascoli compactness theorem (cf. Theorem A.1). ∎

2.2. Consequences

Recall that, in our context, ${u_{n}}_{n \in N}$ will be a sequence of, e.g., numerical approximations of some function $u$ which could, e.g., be a distributional/very weak, entropy, energy, strong, mild, or classical solution of (GPME). The next properties therefore relate the above compactness with results for the underlying equation.

(Consistent approximation) Assume ${u_{n}}_{n \in N}$ is a consistent approximation of some problem (P), i.e.,

$u_{n} \to u in C ([0, T]; L^{r} (R^{N})) as n \to \infty ⟹ u solves (P) in some sense.$

Corollary 2.3 (Existence by consistent approximation).

Assume $r \in [1, \infty)$ . Let ${u_{n}}_{n \in N} \subset C ([0, T]; L^{r} (R^{N}))$ be a sequence satisfying properties (I)–(IV). Then there exist a solution $u \in C ([0, T]; L^{r} (R^{N}))$ of (P).

Proof.

By (I)–(III), any subsequence ${u_{n_{k}}}_{k \in N}$ defined in Theorem 2.1 has a further subsequence converging to some function $u \in C ([0, T]; L^{r} (R^{N}))$ . The consistency given by (IV) tells us that the limit $u$ is a solution of (P). ∎

We end this discussion, by noting that full convergence of the sequence relies on uniqueness of the problem (P):

(Uniqueness) There is at most one solution $u \in C ([0, T]; L^{r} (R^{N}))$ of (P).

Proposition 2.4 (Convergence by uniqueness).

Assume $r \in [1, \infty)$ . Let ${u_{n}}_{n \in N} \subset C ([0, T]; L^{r} (R^{N}))$ be a sequence satisfying properties (I)–(V). Then

un→uin \quadC([0,T];Lr(RN))\quad as \quadn→∞,

where $u$ is the unique solution of (P).

Proof.

Assume by contradiction that ${u_{n}}_{n \in N}$ does not converge to $u$ in $C_{t} (L_{x}^{r})$ . Then there is a subsequence and an $ε > 0$ such that ${sup}_{t \in [0, T]} ∥ u - u_{n_{k}} ∥_{L^{r} (R^{N})} > ε$ for every $k \in N$ . By compactness (Theorem 2.1) there is a further subsequence ${u_{n_{k_{j}}}}_{j \in N}$ converging in $C_{t} (L_{x}^{r})$ to a function $~ u$ which is a solution of (P) (Corollary 2.3). However, uniqueness tells us that $u = ~ u$ , and we have a contradiction. The whole sequence ${u_{n}}_{n \in N}$ thus converges to $u$ in $C_{t} (L_{x}^{r})$ . ∎

2.3. Some variants

We will now discuss some variants of the above conditions which we will use in the paper, and also a comparison with other compactness results.

Equitight in space uniformly in time.

Now, we replace (I) and (III) by:

(Equitight in space uniformly in time) For all $t \in [0, T]$ ,
(Equicontinuous in time) For all $t, s \in [0, T]$ and all compact $K \subset R^{N}$ , there exists a modulus of continuity $λ_{K}$ such that

$sup n \int_{K} | u_{n} (\cdot, t) - u_{n} (\cdot, s) | d x \leq λ_{K} (| t - s |) .$

We immediately see that (i) implies (I). Moreover:

Lemma 2.5.

Assume $r \in [1, \infty)$ . If the sequence ${u_{n}}_{n \in N} \subset C ([0, T]; L^{r} (R^{N}))$ satisfies (i) and (iii), then for all $ε > 0$ there exists $η > 0$ such that for all $n$ and all $0 \leq t \leq s \leq T$

(2.1)

| t - s | < η ⟹ \int_{R^{N}} | u_{n} (x, t) - u_{n} (x, s) |^{r} d x < ε .

Remark 2.6.

The condition (2.1) is a weaker version of the original (III). However, the second item of the Arzelà-Ascoli compactness theorem A.1 is still fulfilled, and hence, Theorem 2.1 still holds under the assumptions (i), (II), and (iii).

Proof.

Fix $ε > 0$ and the compact set $K$ such that $B_{R} (0) \subset K \subset R^{N}$ . Then

\begin{matrix} sup n \int_{R^{N}} | u_{n} (x, t) - u_{n} (x, s) |^{r} d x \leq sup n \int_{K} | u_{n} (x, t) - u_{n} (x, s) |^{r} d x + 2 sup n sup t \in [0, T] \int_{| x | > R} | u_{n} (x, t) |^{r} d x \end{matrix}

By (i), we choose $R > 0$ such that

2 sup n sup t \in [0, T] \int_{| x | > R} | u_{n} (x, t) |^{r} d x < \frac{ε}{2},

and then by (iii), we take $η > 0$ such that

sup n \int_{K} | u_{n} (x, t) - u_{n} (x, s) |^{r} d x < \frac{ε}{2} .

The proof is complete. ∎

Variants of Arzelà-Ascoli and Kolmogorov-Riesz.

Generalizations of the Arzelà-Ascoli compactness theorem to $L^{p}$ -spaces can be found in Simon’s well-cited paper [37]. There he discusses compactness of functions which are $L^{p}$ in time with values in some Banach space $B$

. Sections 8 and 9 of that paper also contain what is commonly known as the Aubin-Lions lemma. Such an approach is different than what we do here, and is probably more suited in an energy-like setting. A nonlocal version is given by Theorem 3.1 in

[32].

Regarding Kolmogorov-Riesz on compact sets, the Helly compactness theorem can be used as a particular version in the $B V$ -setting, see [29, Section 6].

3. Main results

3.1. Assumptions and concept of solution

Consider the following typical assumptions on (GPME):

( $A_{φ}$ )		$φ \in C_{l o c}^{0, ϑ} (R, R), ϑ \in (0, 1], is nondecreasing .$
( $A_{g}$ )		$g is measurable and \int_{0}^{T} (∥ g (\cdot, t) ∥_{L^{1} (R^{N})} + ∥ g (\cdot, t) ∥_{L^{\infty} (R^{N})}) d t < \infty .$
( $A_{u_{0}}$ )		$u_{0} \in L^{1} (R^{N}) \cap L^{\infty} (R^{N}) .$

In this paper, we restrict ourselves to $L^{σ, μ} = L^{σ} + L^{μ}$ , where

(

A_{σ}

)

L^{σ} = c Δ := c N \sum i = 1 \partial_{x_{i} x_{i}}^{2} where c \in {0, 1},

and $L^{μ}$ is a general symmetric Lévy operator under the usual assumption:

		$μ is a nonnegative symmetric Radon measure on R^{N} ∖ {0} satisfying$

Remark 3.1.

Assumption ( $A_{g}$ ) is the same as saying that

$g \in L^{1} (0, T; L^{1} (R^{N}) \cap L^{\infty} (R^{N}))$

where the space is understood as iterated $L^{P}$ -spaces in the sense of [2]. Of course, we have that $L^{1} (0, T; L^{1} (R^{N})) = L^{1} (Q_{T})$ .
Without loss of generality, we can assume $φ (0) = 0$ (by adding constants to $φ$ in (GPME)).
As solutions of (GPME) will be bounded, we can always assume that $φ \in C^{0, ϑ}$ in ( $A_{φ}$ ).

We will work with very weak solutions of (GPME):

Definition 3.1 (Very weak solutions).

Let $u_{0} \in L_{l o c}^{1} (R^{N})$ and $g \in L_{l o c}^{1} (R^{N})$ . Then $u \in L_{l o c}^{1} (Q_{T})$ is a very weak solution of (GPME) if, for all $ψ \in C_{c}^{\infty} (R^{N} \times [0, T))$ , $φ (u) L^{σ, μ} [ψ] \in L^{1} (Q_{T})$ and

\iint_{Q_{T}} (u \partial_{t} ψ + φ (u) L^{σ, μ} [ψ] + g ψ) d x d t + \int_{R^{N}} u_{0} (x) ψ (x, 0) d x = 0.

Note that under the assumptions ( $A_{φ}$ )–(3.1), very weak solutions of (GPME) are unique in $L^{1} \cap L^{\infty}$ by [13, Theorem 3.1]. By the same reference,

∥ u (\cdot, t) ∥_{L^{\infty} (R^{N})} \leq ∥ u_{0} ∥_{L^{\infty} (R^{N})} and u \in C ([0, T]; L_{l o c}^{1} (R^{N})),

and hence, they are moreover in $C ([0, T]; L_{l o c}^{r} (R^{N}))$ for all $r \in [1, \infty)$ .

3.2. Approximation through numerical method

To define our numerical scheme, we need to introduce a discrete grid in $R^{N} \times (0, T)$ . Consider a sequence of numbers defining a nonuniform gird in time $T_{Δ t}^{T} := {t_{j}}_{j = 0}^{J}$ such that $0 = t_{0} < t_{1} < \dots < t_{J} = T$ and let $J := {1, \dots, J}$ . The time steps are then denoted by

Δ t_{j} = t_{j} - t_{j - 1} > 0 f o r e v e r y $ j \in J $ a n d Δ t = max j \in J {Δ t_{j}} .

Moreover, let the space step $h > 0$ , and consider the discrete subset of $R^{N}$ given by

G_{h} := h Z^{N} = {x_{β} := h β : β \in Z^{N}} .

We are now ready to define our numerical scheme. Since $u_{0}$ and $g$ do not necessarily have pointwise values, we set, for $R_{h} := h (- \frac{1}{2}, \frac{1}{2}]$ ,

(3.1)

U_{β}^{0} := \frac{1}{h^{N}} \int_{x_{β} + R_{h}} u_{0} (x) d x and G_{β}^{j} := \frac{1}{h^{N} Δ t_{j}} \int_{t_{j - 1}}^{t_{j}} \int_{x_{β} + R_{h}} g (x, τ) d x d τ .

Then we seek a function $U : G_{h} \times T_{Δ t}^{T} \to R$ which solves

(3.2)

\begin{matrix} U_{β}^{j} = U_{β}^{j - 1} + Δ t_{j} L^{h} [φ (U_{\cdot}^{j})]_{β} + Δ t_{j} G_{β}^{j} \forall (β, j) \in (Z^{N}, J) . \end{matrix}

Remark 3.2.

Note that the above scheme is implicit in the diffusion term. This is done to ensure a simple theoretical analysis for merely Hölder continuous $φ$ . If we choose an explicit scheme, we need to rely on a CFL-type stability condition which involves $| φ |_{C^{0, 1}}^{- 1}$ , and this condition will blow up if $φ$ is not Lipschitz continuous. In the pure Hölder case, we would then need a further approximation of the nonlinearity. We refer the reader to [15, 16] for details.

The discrete variants of $\partial_{t}$ and $L^{σ, μ}$ will now be discussed. For the partial derivative in time, we use the simple backward difference:

\frac{ψ (t_{j}) - ψ (t_{j - 1})}{Δ t_{j}} for all ψ : T_{Δ t}^{T} \to R .

The finite-difference discretization of $Δ$ is well-known and given by

Δ_{h} ψ (x) = N \sum i = 1 \frac{ψ (x + h e_{i}) + ψ (x - h e_{i}) - 2 ψ (x)}{h^{2}} .

Recently, such approximation was shown to be in the class of Lévy operators for a certain finite measure [14, 13, 15, 16]. We thus present a unified approach to numerical discretizations for local and nonlocal operators. In fact, by choosing the correct weights, we recover either local, nonlocal, or combinations of both. Hence, consider the family of bounded, symmetric, and monotone operators $L^{h} : ℓ^{1} (G_{h}) \to R$ given by

L^{h} [ψ] (x_{β}) = L^{h} [ψ]_{β} = \sum γ \neq 0 (ψ (x_{β} + z_{γ}) - ψ (x_{β})) ω_{γ, h} % f o r a l l $ ψ : G_{h} \to R $,

where $z_{- γ} = - z_{γ} \in G_{h}$ and $\sum_{γ \neq 0} ω_{γ, h} < \infty$ .

Remark 3.3.

We can also write

L^{h} [ψ]_{β} = \int_{R^{N}} (ψ (x_{β} + z) - ψ (x_{β})) d ν_{h} (z) w i t h ν_{h} (z) = \sum γ \neq 0 δ_{z_{γ}} (z) w_{γ, h},

where $δ_{z_{γ}}$ is the dirac delta measure centered at $z_{γ}$ (cf. [16]).

We then need the following discrete versions of (3.1):

		$For all h > 0 and γ \neq 0, ω_{γ, h} = ω_{- γ, h} \geq 0 and$
		$sup 0 < h < 1 {\sum h \leq \| z_{γ} \| \leq 1 \| z_{γ} \|^{2} ω_{γ, h} + \sum \| z_{γ} \| > 1 ω_{γ, h}} < \infty .$

Remark 3.4.

The parallel of the above assumptions with (3.1) might be easier to see in terms of $ν_{h}$ :

sup 0 < h < 1 {\int_{0 < | z | \leq 1} | z |^{2} d ν_{h} (z) + \int_{| z | > 1} d ν_{h} (z)} < \infty

In Appendix C.3, we also provide conditions for checking this assumption through operator consistency (see (3.6) below) and assumptions on the limit operator $L^{σ, μ}$ .

Equitightness and convergence in $C ([0, T]; L^{r} (R^{N}))$ with $r \in [1, \infty)$

of numerical schemes will be presented for a suitable interpolant which extends the discrete solutions to the whole space

R^{N} \times (0, T)

. For that reason, let us define the piecewise constant space interpolant

¯ ¯¯ ¯ U : R^{N} \to R

U : G_{h} \to R

¯ ¯¯ ¯ U (x) = \sum x_{β} \in G_{h} 1_{x_{β} + R_{h}} (x) U_{β},

and the piecewise linear in time and piecewise constant in space space-time interpolant $˜ ¯ ¯¯ ¯ U : R^{N} \times [0, T] \to R$ of $U : G_{h} \times T_{Δ t}^{T} \to R$ as

(3.3)

\begin{matrix} ˜ ¯ ¯¯ ¯ U (x, t) & =^{0} (x) 1_{{t_{0}}} (t) + \sum t_{j} \in T_{Δ t}^{T} ∖ t_{0} 1_{(t_{j - 1}, t_{j}]} (t) (^{j - 1} (x) + \frac{t - t_{j - 1}}{t_{j} - t_{j - 1}} (^{j} (x) -^{j - 1} (x))) . \end{matrix}

3.3. Equitightness

Our results make use of a family of functions $X$ of the form

(3.4)

X_{R} (x) := X (\frac{x}{R}) with R > 0,

where $X$ is some fixed function such that

(3.5)

Observe that $D^{k} X \in C_{c} (R^{N})$ for all $k \in N$ and $1 - X \in C_{c} (R^{N})$ . Moreover:

Lemma 3.5.

Assume $k \in N \cup {0}$ . The function defined by (3.4)–(3.5) satisfies:

$X_{R} \to 0$ pointwise as $R \to \infty$ .
If $\frac{N}{k} < p \leq \infty$ , then $∥ D^{k} X_{R} ∥_{L^{p} (R^{N})} = R^{\frac{N}{p} - k} ∥ D^{k} X ∥_{L^{p} (R^{N})} \to 0$ as $R \to \infty$ .

Remark 3.6.

We note that $p = \infty$ gives the least restrictive convergence of $D^{k} X_{R} \to 0$ in $L^{p}$ as $R \to \infty$ .

Theorem 3.7 (Equitightness estimate).

Assume ( $A_{φ}$ ), ( $A_{g}$ ), ( $A_{u_{0}}$ ), (3.2), and $r \in [1, \infty)$ . Let $p \in (1, 1 / (1 - ϑ)]$ and $q$ be such that $p^{- 1} + q^{- 1} = 1$ (hence, $q \in [1 / ϑ, \infty)$ ), and let $U_{β}^{j}$ be an a.e.-solution of (3.2) corresponding to $U_{β}^{0}$ and $G_{β}^{j}$ , and ${˜ ¯ ¯¯ ¯ U}_{h}$ the corresponding interpolant. Then

\begin{matrix} sup t \in [0, T] \int_{| x | > R} | {˜ ¯ ¯¯ ¯ U}_{h} (x, t) |^{r} d x \leq M_{u_{0}, g}^{r - 1} (∥ u_{0} X_{R} ∥_{L^{1} (R^{N})} + ∥ g X_{R} ∥_{L^{1} (0, T; L^{1} (R^{N}))} + C_{u_{0}, g, φ} T ∥ L^{h} [X_{R}] ∥_{L^{p} (R^{N})}) \end{matrix}

where

M_{u_{0}, g} := ∥ u_{0} ∥_{L^{\infty} (R^{N})} + ∥ g ∥_{L^{1} (0, T; L^{\infty} (R^{N}))} .

and

C_{φ, u_{0}, g} := | φ |_{C^{0, ϑ}} (M_{u_{0}, g})^{ϑ - \frac{}{1} q} (∥ u_{0} ∥_{L^{1} (R^{N})} + ∥ g ∥_{L^{1} (Q_{T})})^{\frac{1}{q}} .

Remark 3.8.

We interpret $1 / (ϑ - 1) = \infty$ when $ϑ = 1$ .

The proof, see Section 4, basically consists of choosing $X_{R}$ as a test function in Definition 3.1. Since $φ (u) \in L^{q}$ we need to rely on $L^{h} [X_{R}] \to 0$ in $L^{p}$ as $R \to \infty$ . This puts a restriction on the class of operators, as can be seen in the next result. To write down the result properly, we will consider weights satisfying one of the following:

		$For all h > 0 and γ \neq 0, ω_{γ, h} = ω_{- γ, h} \geq 0 and$

		In addition to (3.2), also assume that, for $α \in (0, 2)$ ,
		$sup R > 1 sup 0 < h < 1 {R^{α - 2} \sum 1 < \| z_{γ} \| \leq R \| z_{γ} \|^{2} ω_{γ, h} + R^{α} \sum \| z_{γ} \| > R ω_{γ, h}} \leq C .$

Corollary 3.9 (Equitightness).

Assume ( $A_{φ}$ ), ( $A_{g}$ ), ( $A_{u_{0}}$ ), and $r \in [1, \infty)$ . Let $U_{β}^{j}$ be an a.e.-solution of (3.2) corresponding to $U_{β}^{0}$ and $G_{β}^{j}$ , and ${˜ ¯ ¯¯ ¯ U}_{h}$ the corresponding interpolant. Then

\begin{matrix} lim R \to \infty sup 0 < h < 1 sup t \in [0, T] \int_{| x | > R} | {˜ ¯ ¯¯ ¯ U}_{h} (x, t) |^{r} d x = 0. \end{matrix}

if any of the following holds:

When $L^{h} = Δ_{h}$ (with weights satisfying (3.2)) and $\frac{(N - 2)^{+}}{N} < ϑ \leq 1$ .
When $L^{h}$ has weights satisfying (3.3) and $\frac{N - 1}{N} < ϑ \leq 1$ .
When $L^{h}$ has weights satisfying (3.3) and $\frac{(N - α)^{+}}{N} < ϑ \leq 1$ .

Remark 3.10.

More generally, we also have equitightness when $L^{h}$ has weights satisfying (3.2) and

$sup 0 < h < 1 {R^{\frac{N}{p} - 2} \sum 1 < | z_{γ} | \leq R | z_{γ} |^{2} ω_{γ, h} + R^{\frac{N}{p}} \sum | z_{γ} | > R ω_{γ, h}} = o_{R} (1) .$
Let us also provide the counterpart of (3.3) and (3.3) for $μ$ :

$μ is a nonnegative symmetric Radon measure on R^{N} ∖ {0} satisfying$

$\int_{0 < | z | \leq 1} | z |^{2} d μ (z) + \int_{| z | > 1} | z | d μ (z) < \infty,$

and

$μ satisfies (???) and, for α \in (0, 2),$

The latter is satisfied for e.g. $μ$ such that (3.1) holds and $d μ (z) \leq | z |^{- (N + α)} d z$ for $| z | > 1$ and some $α \in (0, 2)$ . We again refer to Appendix C.3 for practical ways of checking (3.3) and (3.3) through operator consistency (see (3.6) below) and assumptions on the limit operator $L^{σ, μ}$ .
Assumptions (b) and (b) are moment like conditions. They imply that $μ 1_{| z | > 1}$ has $s$ moments with $s = 1$ or $s \in (0, α)$ , respectively. This is equivalent with the underlying Lévy process having moments of order $s$ , see e.g. Section 2.3 in [24].

Proof of Corollary 3.9.

Let us summarize the direct computations done in Appendix C:

$Δ_{h} X_{R} \to 0$ in $L^{p}$ as $R \to \infty$ if $p > \frac{N}{2}$ when $N \geq 2$ , and $p > 1$ when $N = 1$ (see Corollary C.5).
Under (3.3), $L^{h} [X_{R}] \to 0$ in $L^{p}$ as $R \to \infty$ if $p > N$ (see Corollary C.6).
Under (3.3), $L^{h} [X_{R}] \to 0$ in $L^{p}$ as $R \to \infty$ if $p > N / α$ when $N \geq 2$ , $p > 1 / α$ when $α \in (0, 1)$ and $N = 1$ , and $p > 1$ when $α \in [1, 2)$ and $N = 1$ (see Corollary C.6 and Example C.1).

Now, we can deduce a condition on $ϑ$ in order to have convergence as $R \to \infty$ , and then conclude by Theorem 3.7. Choose $p = 1 / (1 - ϑ)$ (hence, $q = 1 / ϑ$ ), and require that:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 1 < \frac{1}{1 - ϑ} & in Case (i) for N = 1, \frac{N}{2} < \frac{1}{1 - ϑ} & in Case (i) for N \geq 2, N < \frac{1}{1 - ϑ} & in Case (ii) for N \geq 1, \frac{N}{α} < \frac{1}{1 - ϑ} & in Case (iii) for N \geq 2 % , \frac{1}{α} < \frac{1}{1 - ϑ} & in Case (iii) for N = 1 and% α \in (0, 1), 1 < \frac{1}{1 - ϑ} & in Case (iii) for N = 1 and α \in [1, 2) . \end{matrix}

Hence, we get a restriction on the lower bound of $ϑ$ :

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \end{matrix}

		$μ is a nonnegative symmetric Radon measure on R^{N} ∖ {0} satisfying$
		$\int_{0 < \| z \| \leq 1} \| z \|^{2} d μ (z) + \int_{\| z \| > 1} \| z \| d μ (z) < \infty,$

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1. Introduction

Remark 1.1.

Outline

Notation

Remark 1.2.

2. On compactness and convergence in C([0,T];Lr(RN))

2.1. Necessary and sufficient conditions for compactness

Theorem 2.1 (Compactness).

Remark 2.2.

Proof.

2.2. Consequences

Corollary 2.3 (Existence by consistent approximation).

Proof.

Proposition 2.4 (Convergence by uniqueness).

Proof.

2.3. Some variants

Equitight in space uniformly in time.

Lemma 2.5.

Remark 2.6.

Proof.

Variants of Arzelà-Ascoli and Kolmogorov-Riesz.

3. Main results

3.1. Assumptions and concept of solution

Remark 3.1.

Definition 3.1 (Very weak solutions).

3.2. Approximation through numerical method

Remark 3.2.

Remark 3.3.

Remark 3.4.

3.3. Equitightness

Lemma 3.5.

Remark 3.6.

Theorem 3.7 (Equitightness estimate).

Remark 3.8.

Corollary 3.9 (Equitightness).

Remark 3.10.

Proof of Corollary 3.9.

2. On compactness and convergence in $C ([0, T]; L^{r} (R^{N}))$