On the generalized Buckley-Leverett equation

In this paper we study the generalized Buckley-Leverett equation with nonlocal regularizing terms. One of these regularizing terms is diffusive, while the other one is conservative. We prove that if the regularizing terms have order higher than one (combined), there exists a global strong solution for arbitrarily large initial data. In the case where the regularizing terms have combined order one, we prove the global existence of solution under some size restriction for the initial data. Moreover, in the case where the conservative regularizing term vanishes, regardless of the order of the diffusion and under certain hypothesis on the initial data, we also prove the global existence of strong solution and we obtain some new entropy balances. Finally, we provide numerics suggesting that, if the order of the diffusion is $0<\alpha<1$, a finite time blow up of the solution is possible.


I. INTRODUCTION
In this paper we study the case of the Buckley-Leverett equation with generalized regularizing terms provided by fractional powers of the laplacian with initial data 0 ≤ u(x, 0) = u 0 (x) ≤ 1, and where M > 0 is a fixed constant.
Here Ω is either Ω = R or Ω = T. Let us immediately emphasize that u 0 (x) ≤ 1 is not a smallness condition, since, in applications, u denotes a certain proportion (compare the following literature outline). Equation (1) is a nonlocal regularization of the classical Buckley-Leverett equation The nonlinearity in Equation (1) is regularized in two different ways: first, due to the diffusive term −νΛ α u, and second due to the conservative term −µΛ β ∂ t u.
Equation (2) was derived by Buckley and Leverett in Ref. 4 and it has been well studied since then (see LeVeque 27 and Mikelić and Paoli 30 ). This equation is used to describe a two-phase flow in a porous medium. For example, oil and water flow in soil or rock. In this situation u represents the saturation of water and M > 0 is the water-over-oil viscosity ratio. Equation (2) is a prototype of a) Electronic mail: jb@impan.pl b) Electronic mail: rgranero@math.ucdavis.edu c) Electronic mail: kluli@math.ucdavis.edu

A. Aim and outline
The purpose of this paper is to study (1). We are mainly interested in the global existence of solutions together with their qualitative behaviour as well as in the finite time singularities.
We provide details of our results in Subsection I B. Subsection I C contains notation, including the definition of a weak solution and certain preliminaries. Section II provides new entropy inequalities for the fractional laplacian that are interesting by themselves, therefore these inequalities are stated for an arbitrary dimension d. Sections II-VIII contain proofs of our results. Finally, in Section IX, we provide some numerical results suggesting the existence of finite time singularities for the cases 0 < α < 1 and µ = 0. These numerics also suggest that in the critical case α = 1 the solution exists globally. This is in agreement with the results for the Burgers equation with fractional dissipation by Kiselev et al., 25 and Dong et al. 14 Let us remark that, when the term µΛ β ∂ t u is added to the equation, even for α = β = 0.25 there is no evidence of blow-up. Consequently, our numerics appear to discard a finite time blow-up scenario when µ > 0.
To the best of our knowledge, all our results are new.

B. Results
First, let us provide a result concerning the global existence of weak solutions for (1), corresponding to rough initial data, i.e., merely 0 ≤ u 0 ≤ 1 a.e., as well as concerning new entropy balances (that are needed in the existence part of the result, but are interesting by themselves). Proposition 1. Let 0 ≤ u 0 ≤ 1, u 0 ∈ L 1 (Ω) ∩ L ∞ (Ω) be the initial data for (1) with ν > 0, 0 < α < 2, µ = 0 and M > 0. Then there exists a global weak solution such that Furthermore, if u is an L 2 (0,T; H 1 (Ω)) solution to (1), then the following entropy inequalities hold  and Let us remark that the terms  t 0  Ω Λ α u(s) log(u(s))dxds will provide a L 2 t bound on a fractional derivative of the solutions. The proof of Proposition 1 will be established in Section III.
Our next results concern the qualitative behaviour of smooth solutions. In case Ω = T, we denote the average of u 0 by Proposition 2. Let u be the classical solution to (1) with initial data 0 ≤ u 0 ≤ 1, where ν > 0, 0 < α ≤ 2 and M > 0. Then, Our main results address the problem of global existence of smooth solutions. More precisely, we have results for three cases, depending on the values of the parameters α and β.
1. the subcritical case: the higher space derivative is in the dissipative term, i.e., 1 < max{α, β} ≤ 2. Here we show global existence of smooth solutions with no restrictions on the initial data. Compare Theorem 1. 2. the critical case: the transport term exactly balances the regularizing terms, i.e., 1 = max{α, β}.
Here, for µ > 0 and β = 1, we prove global existence of smooth solutions without any size restriction on the initial data. In the other cases we need certain smallness conditions. Namely, for µ = 0, ν > 0 and α = 1, we obtain global existence of smooth solutions for initial data satisfying a smallness restriction on the lower order norm L ∞ ; this smallness restriction is explicit in terms of ν and M. Finally, in the case µ > 0, α = 1 and 0 < β < 1, we obtain the global existence for initial data satisfying a smallness condition in H 1+β 2 . The smallness restriction is here slightly less explicit, but easily computable. See Theorem 2. 3. the supercritical case: the higher space derivative is in the transport term, i.e., 0 ≤ α < 1 and µ = 0. Even here, for Ω = T, we are able to prove global existence of smooth solutions for smooth, periodic initial data satisfying an explicit smallness restriction on the Lipschitz norm The remaining open problems are in the critical and supercritical regime. In particular, our results do not apply to the case where max{α, β} < 1, µ > 0, and there is no large data, global results for the critical case with µ = 0, ν > 0, α = 1. In the context of the latter, let us observe that on one hand, there are certain new methods available for nonlinear problems with nonlocal critical dissipation, like the method of moduli of continuity by Kiselev et al., 25 the fine-tuned DeGiorgi method by Caffarelli and Vasseur 6 or the method of the nonlinear maximum principles by Constantin and Vicol 10 (see also Constantin et al. 9 ). But on the other hand, our nonlinearity is more complex than the typical ones. Now, let us state the main theorems. First, we study the subcritical case max{α, β} > 1.
Moreover, for t ≤ T, the solution satisfies
For the critical case, let us define the following constants. Definition 1. Let γ * be a constant such that and let γ be any fixed number such that 0 < γ < γ * .
Next, let C S be the Sobolev's constant corresponding to the embedding H 1+β 2 ↩→ L ∞ .
We have Theorem 2. Let 0 ≤ u 0 ≤ 1, u 0 ∈ H s (Ω), s ≥ 1 be the initial data for (1) with M > 0. Then (1) has a global solution u(t) ∈ C([0,T], H s (Ω)) ∩ L 2 (0,T; H s+0.5 (Ω)) ∀ T < ∞ that satisfies the energy balance Under the following conditions: (i) Either ν ≥ 0, 0 ≤ α ≤ 2 and µ > 0, β = 1 (conservative regularization with no smallness conditions on the data). (ii) Or ν > 0, α = 1, µ = 0 and the initial data is such that In this case the solution satisfies the maximum principle (iii) Or ν > 0, α = 1, µ > 0, 0 < β < 1 and the initial data is such that Then, the solution satisfies the maximum principle . Remark 1. The fact that independently of the value of u, allows for a global result relying on a condition related to M, ν, and µ. However, we are interested in results that deal with every possible value of the physical parameters present in the problem.
In our opinion, there are two reasons, at least in the case µ = 0, why the smallness condition (6) may be seen as a rather mild restriction. The first one is that the size restriction affects a lower norm, merely L ∞ , keeping the higher seminorms as large as desired. The second one is that, given M and ν, the constant γ * can be easily computed. For instance, if we further assume γ * ≤ 1, the expression for γ * is explicit: The last case, namely where 0 < α < 1, is harder because the leading term in the equation is the transport term. However, under certain conditions, we can prove the global existence of solutions. Before we can state the relevant result, we need some notation.
Definition 2. Let γ and M be given, positive constants. Define Σ(γ) as follows: Next, let γ * be a small enough constant such that Then we have the following.

and the energy balance
In the above theorem, we impose domain restrictions and stronger smallness assumptions. These domain restrictions are due to the better behavior of the fractional laplacian in a bounded domain. The size restrictions on data are again on a lower order norm (Lipschitz) and with a rather explicit constant.
Finally we obtain the standard finite time blow up for certain initial data in some Hölder seminorm.
Proposition 3. Fix a constant M > 0 and consider µ = 0, min{ν, α} = 0. Then, there exist 0 ≤ u 0 ≤ 1 ∈ H 2 (Ω) and T * < ∞ such that the corresponding solution, u(t), of Equation (2) has a finite time singularity in C δ for 0 < δ ≪ 1, i.e., lim sup The proof of this result is obtained by a virial-type argument. However, we remark that it can also be obtained by means of pointwise arguments (see Castro and Córdoba 7 for an application of these pointwise arguments to prove blow up). These virial-type arguments have been used for several transport equations even in the case of nonlocal velocities (see Córdoba et al., 12 Dong et al., 14 Li and Rodrigo, 28 and Li et al. 29 ). In this case, the transport term is highly nonlinear and this method fails in the case of viscosity ν > 0, 0 < α ≪ 1.

Singular integral operators
We denote the usual Fourier transform of u byû. Given a function u : Ω → R, we write Λ α u = (−∆) α/2 u for the fractional laplacian, i.e., This operator admits the kernel representation if the function is periodic and if the function is flat at infinity. Notice that we have where Γ(·) denotes the classical Γ function.

Functional spaces
We write H s (Ω d ) for the usual L 2 -based Sobolev spaces with norm |∂ ⌊s⌋

Entropy functionals
For a given function u ≥ 0, we define the following entropy functionals These two entropies have an associated Fisher information, The third entropy that we are using reads with its Fisher information

Notation
Recall that we denote the mean of a function by Let us introduce f and a as follows: and Finally, let us introduce the notation for the mollifiers. For ϵ > 0, we write J ϵ for the heat kernel at time t = ϵ and define

Weak solutions to (1) and their local existence
We start this section with Definition 3. Let µ, ν ≥ 0 and 0 < T < ∞ be a fixed positive parameter. The function with initial data u ϵ,δ (0) = u 0 . Standard energy estimates give us uniform bounds. Then we can pass to the limits ϵ, δ → 0. The proof of the continuation criteria can be obtained by energy methods.

II. THE ENTROPY INEQUALITIES
In this section we provide the proof of three entropy inequalities that, in our opinion, may be of independent interest. Proposition 4. Let u be a given function and 0 < α < 2, 0 < ϵ < α/2 be two fixed constants. Then provided that the right hand sides are meaningful.
Proof. Let us fix i = 1. First, we symmetrize Furthermore, since (a − b) log a b ≥ 0, every term in the series is positive, i.e., for every γ ∈ Z d , we have  ) dy dx ≥ 0.
In particular Let us consider first the case 0 ≤ u ∈ L 1 . We have This latter integral is similar to the Riesz potential. Due to the positivity of u, we have We have Consequently, we get The case 0 ≤ u ∈ L ∞ was first proved by Bae and Granero-Belinchón. 2 For the sake of completeness, we include here a sketch of the proof. Using (24), we have The proof for the case i = 2 is similar.

III. PROOF OF PROPOSITION 1: WEAK SOLUTIONS
We prove the result for Ω = T, but the same proof can be adapted to deal with Ω = R. We consider the regularized problems with the regularized initial data u ϵ (x, 0) = J ϵ * u 0 (x) + ϵ, and ϵ ≤ 1/2. These approximate problems have global classical solution due to the Theorem 3.1 in Ref. 24. Consequently, we focus on obtaining the appropriate ϵ-uniform bounds. By assumption 0 ≤ u(x, 0) ≤ 1. We apply the same technique as Córdoba and Córdoba 11 (see also Refs. 1, 2, 5, 7, 8, 13, and 16-19 for more details and application to other partial differential equations), i.e., we track Due to smoothness of u ϵ , we have that M ϵ (t), M ϵ (t) are Lipschitz, and consequently almost everywhere differentiable. Hence, using together with the kernel expression for Λ α , we have the ϵ-uniform bounds 0 ≤ ϵ ≤ u ϵ (x,t) ≤ 1.5.
By space integration of (25), we get We compute We have and as a consequence  Thus, we conclude As we get a ϵ-uniform estimate. Now we apply (23) from Proposition 4 and u ϵ (x,t) ≤ 1.5 to get Consequently, we have ϵ-uniform bounds and the first entropy inequality (4). For the second entropy inequality (5), we compute We are going to handle both integrals separately. We have where in the second term we have used 2u∂ .
The second integral reads Collecting all these computations we get

IV. PROOF OF PROPOSITION 2: DECAY ESTIMATES
Let us prove first the periodic case. The L 1 norm is preserved. Consequently, the mean propagates. We again apply the technique of tracking M(t) and M(t). Recall The smoothness needed to proceed with M(t) ′ is, for this proposition, an assumption.
Since x, y ∈ T, |x − y | 1+α ≤ (2π) 1+α . Hence, using (11) and (13), we get Consequently, Integrating this ODI, we have Let us turn our attention to the flat at infinity case. Again the L 1 norm propagates. We take a positive number r > 0 (that will be specified below) and define
We choose now thus, recalling (13), we have that for both u( x t ) > 0 and u( x t ) = 0. With the same argument as in the periodic case, we get thus, using explicit value of C α,1 we arrive at

V. PROOF OF THEOREM 1: GLOBAL SOLUTIONS FOR max{α, β} > 1
Equipped with the Lemma 1 and its proof, we can focus on the appropriate energy estimates that ensures global existence (rigorously, we should do this on the level of the regularized problem) Notice that we also have a global bound We split the proof in three parts: the first one is devoted to the proof of the purely parabolic case µ = 0. Then, in step 2 and 3 we consider the cases µ > 0, β < 1 < α and µ > 0, α < 1 < β, respectively.

Now we test
Integrating, we obtain Testing against −∂ 2 x u we can perform energy estimates as in Step 1. We obtain 1 2 We use the interpolation inequality

Now we use the interpolation
For α > 1 we have that . Now, we test against −∂ 2 x u. We can conclude as in Step 1. We obtain VI. PROOF OF THEOREM 2: GLOBAL SOLUTION FOR max{α, β} = 1 Step 1: Case ν > 0, α = 1, µ = 0. We do the case s = 1, the other cases being analogous. Testing (1) against Λu and using the self-adjointness, we have Equations (28)- (31). Notice that under the hypothesis Using that we conclude for a small enough 0 < δ. Notice that this δ only depends on M,u 0 , and ν. Next, testing (1) against −∂ 2 x u and integrating by parts, we get (33)-(35). If we integrate by parts in (34), we get The first inequality above uses also (36) and the second the interpolation and, due to Gronwall's inequality together wit (37), we obtain This ends the proof of case (ii) of our thesis.
Step 2: Case ν > 0, α = 1, µ > 0, β < 1. In this case we cannot use the pointwise methods, so we cannot get immediately ∥u(t)∥ L ∞ ≤ ∥u 0 ∥ L ∞. Estimate (27) implies a global bound in H β/2 , but this bound is too weak to give us a pointwise estimate for u. However, as testing (1) against Λu and using the definition of γ and γ * , we have, as in step 1, As a consequence, we obtain the global bound Now we test against −∂ 2 x u and we conclude as in Step 1. Case (iii) is proved.
Step 3: Case ν ≥ 0, µ > 0, β = 1. In this case, (27) implies a global bound in H 0.5 . Then, testing (1) against Λu and using (8), we have As a consequence, we can apply Gronwall's inequality to get a global bound Now we test against −∂ 2 x u and we conclude as in Step 1. Case (i) is proved.

VII. PROOF OF THEOREM 3: GLOBAL SOLUTION IF 0 < α < 1 AND µ = 0
We consider the case s = 2, the other cases being similar. Let us writex t for the point where ∂ x u reaches its maximum, i.e., With a similar argument as in the proof of Proposition 2 (see also Ref. 11), we have Due to the kernel expressions (11) and (13), we have Due to the smallness choice (10) and ∥u(t)∥ L ∞ ≤ ∥u 0 ∥ L ∞, we have Let us writex t for the point where ∂ x u reaches its minimum, i.e., As before, due to the kernel expression (11) and (13), we have Consequently, with the same argument, we have (for negative ∂ x u(x t ,t)) We test Equation (1) against ∂ 4 x u and integrate by parts. We have 1 2 with x udx (41) ds ≤ ∥u 0 ∥ 2 H 2 e c(u 0 , M )t .

VIII. PROOF OF PROPOSITION 3: FINITE TIME SINGULARITIES
First, we study the case ν = 0. Let us take u 0 such that u 0 ≥ 0, u 0 (0) = 0, We argue by contradiction: assume that we have u(t) a global C 2 solution corresponding to u 0 . Recalling the expression a(x) given in (21) we define the characteristic curve y(t), solution to y ′ (t) = a(u( y(t),t)), y(0) = 0 and v(x,t) = u(x + y(t),t).
We obtain the ODI d dt J(t) ≥ δ 1 + M J(t) 2 , and the blow up of J(t) in finite time T * = T * (δ,u 0 , M).
We have proved the case ν = 0, but the proof of the case 0 < ν and α = 0 is analogous and can be easily adapted from here.

IX. NUMERICAL SIMULATIONS
In this section we present our numerical simulations suggesting a finite time blow up in the case ν > 0, 0 < α < 1. To approximate the solution, we discretize using the fast fourier transform with N = 2 14 spatial nodes. The main advantage of this numerical scheme is that the differential operators are multipliers on the Fourier side. Once the spatial part has been discretized, we use a Runge-Kutta scheme to advance in the time variable.
In our simulations, we consider the initial data and values M = ν = 0.5 and µ = 0. Then, we approximate the solution for (1) for different values of the parameter 0 < α ≤ 1. In particular, we study four cases, Interestingly, we observe (see Figure 4) that even for small values of α and β, in the case with µ > 0, there is not evidence of finite time singularities.