The purpose of this paper is to improve convergence results in to for numerical schemes of generalized porous medium equations in the context of bounded and integrable solutions. In detail, we study
where is the solution, is nondecreasing and -Hölder continuous with , some right-hand side, , and . The general operator is given as
where is a possibly degenerate local diffusion operator
where , , and , and the anomalous or nonlocal diffusion operator is defined for any as
where is the gradient,
a characteristic function, anda 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 or . That is, the local operator is given as where .
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 , , and the full range of porous medium and fast diffusion equations (cf. ) with , . The class of operators coincides with the generators of symmetric Lévy processes [4, 35, 1]. This includes e.g. the Laplacian , fractional Laplacians , , relativistic Schrödinger type operators , and , tempered stable processes , and even discretizations of all of these. Since and 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 . In all of these cases, an approximate solution of the equation under study is considered. Then one shows compactness of the family formed by 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 with , 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  and Theorem A.8 in ), 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 (-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 -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, 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  (see also Part II of ) 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 (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 -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 . Again, we can improve the convergence from the respective and to since -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 , and the latter in . Our framework thus improves the compactness/convergence results of all four papers.
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 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.
Derivatives are denoted by , , , and and denote the -gradient and Hessian matrix of . with will denote a standard mollifier.
We use standard notation for , , and . Moreover, is the space of smooth functions with compact support, and the space of measurable functions such that for every , , and when for all compact and . In a similar way we also define . Note that the notion of is a subtle one. In fact, we mean that has an a.e.-version which is continuous . See e.g. p. 726 in  for more details. The space with is identified as the Banach space with norm where
When , we simply get .
From now on, we will study convergence in (abbrev. ) with . We use because we want it to be a unique identifier.
2. On compactness and convergence in
Let us give an overview of the properties needed to deduce compactness and convergence in with .
2.1. Necessary and sufficient conditions for compactness
Consider a sequence , and assume that satisfies:
(Equitight in space pointwise in time) For all ,
(Equicontinuous in space pointwise in time) For all , there exists a modulus of continuity such that
(Equicontinuous in time) For all , there exists a modulus of continuity such that
Theorem 2.1 (Compactness).
The fact that the sequence satisfies (I)–(II) is, by the Kolmogorov-Riesz compactness theorem (cf. Theorem A.3), equivalent with being relatively compact in for all . Finally, since the sequence satisfies (III), the proof is completed by an application of the Arzelà-Ascoli compactness theorem (cf. Theorem A.1). ∎
Recall that, in our context, will be a sequence of, e.g., numerical approximations of some function 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 is a consistent approximation of some problem (P), i.e.,
Corollary 2.3 (Existence by consistent approximation).
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 of (P).
Proposition 2.4 (Convergence by uniqueness).
Assume by contradiction that does not converge to in . Then there is a subsequence and an such that for every . By compactness (Theorem 2.1) there is a further subsequence converging in to a function which is a solution of (P) (Corollary 2.3). However, uniqueness tells us that , and we have a contradiction. The whole sequence thus converges to in . ∎
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.
(Equitight in space uniformly in time) For all ,
(Equicontinuous in time) For all and all compact , there exists a modulus of continuity such that
Variants of Arzelà-Ascoli and Kolmogorov-Riesz.
Generalizations of the Arzelà-Ascoli compactness theorem to -spaces can be found in Simon’s well-cited paper . There he discusses compactness of functions which are in time with values in some Banach space
. 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.
Regarding Kolmogorov-Riesz on compact sets, the Helly compactness theorem can be used as a particular version in the -setting, see [29, Section 6].
3. Main results
3.1. Assumptions and concept of solution
Consider the following typical assumptions on (GPME):
In this paper, we restrict ourselves to , where
and is a general symmetric Lévy operator under the usual assumption:
We will work with very weak solutions of (GPME):
Definition 3.1 (Very weak solutions).
Let and . Then is a very weak solution of (GPME) if, for all , and
3.2. Approximation through numerical method
To define our numerical scheme, we need to introduce a discrete grid in . Consider a sequence of numbers defining a nonuniform gird in time such that and let . The time steps are then denoted by
Moreover, let the space step , and consider the discrete subset of given by
We are now ready to define our numerical scheme. Since and do not necessarily have pointwise values, we set, for ,
Then we seek a function which solves
Note that the above scheme is implicit in the diffusion term. This is done to ensure a simple theoretical analysis for merely Hölder continuous . If we choose an explicit scheme, we need to rely on a CFL-type stability condition which involves , and this condition will blow up if 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 and will now be discussed. For the partial derivative in time, we use the simple backward difference:
The finite-difference discretization of is well-known and given by
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 given by
where and .
We can also write
where is the dirac delta measure centered at (cf. ).
We then need the following discrete versions of (3.1):
Equitightness and convergence in with
of numerical schemes will be presented for a suitable interpolant which extends the discrete solutions to the whole space. For that reason, let us define the piecewise constant space interpolant of as
and the piecewise linear in time and piecewise constant in space space-time interpolant of as
Our results make use of a family of functions of the form
where is some fixed function such that
Observe that for all and . Moreover:
We note that gives the least restrictive convergence of in as .
Theorem 3.7 (Equitightness estimate).
We interpret when .
The proof, see Section 4, basically consists of choosing as a test function in Definition 3.1. Since we need to rely on in as . 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:
|In addition to (3.2), also assume that, for ,|
Corollary 3.9 (Equitightness).
More generally, we also have equitightness when has weights satisfying (3.2) and
The latter is satisfied for e.g. such that (3.1) holds and for and some . 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 .
Proof of Corollary 3.9.
Let us summarize the direct computations done in Appendix C:
in as if when , and when (see Corollary C.5).
Now, we can deduce a condition on in order to have convergence as , and then conclude by Theorem 3.7. Choose (hence, ), and require that:
Hence, we get a restriction on the lower bound of :