Dynamics of dislocation densities in a bounded channel. Part I: smooth solutions to a singular coupled parabolic system

We study a coupled system of two parabolic equations in one space dimension. This system is singular because of the presence of one term with the inverse of the gradient of the solution. Our system describes an approximate model of the dynamics of dislocation densities in a bounded channel submitted to an exterior applied stress. The system of equations is written on a bounded interval with Dirichlet conditions and requires a special attention to the boundary. The proof of existence and uniqueness is done under the use of two main tools: a certain comparison principle on the gradient of the solution, and a parabolic Kozono-Taniuchi inequality


Setting of the problem
In this paper, we are concerned in the study of the following singular parabolic system: is the open and bounded interval of R.
The goal is to show the long-time existence and uniqueness of a smooth solution of (1.1), (1.2) and (1.3). Our motivation comes from a problem of studying the dynamics of dislocation densities in a constrained channel submitted to an exterior applied stress. In fact, system (1.1) can be seen as an approximate model of the one described in [21], where the model presented in [21] reads: with τ representing the exterior stress field. System (1.4) can be deduced from (1.1), by letting ε = 0; spatially differentiating the resulting system; and by considering ρ ± x = θ ± , ρ = ρ + − ρ − , κ = ρ + + ρ − . (1.5) Here θ + and θ − represent the densities of the positive and negative dislocations respectively (see [33,25] for a physical study of dislocations).
The next challenge (that will be the motivation of another work by the authors) is to show some kind of convergence of the solution (ρ ε , κ ε ) of (1.1) to the solution of (1.4) as ε → 0.

Brief review of the literature
Parabolic problems involving singular terms have been widely studied in various aspects. Degenerate and singular parabolic equations have been extensively studied by DiBenedetto et al. (see for instance [12,13,14,15,10] and the references therein). The authors regard the solutions of singular or degenerate parabolic equations with measurable coefficients whose prototype is: u t − div |∇u| p−2 ∇u = 0, p > 2 or 1 < p < 2.
The study includes local Hölder continuity of bounded weak solutions, local and global boundedness of weak solutions and local intrinsic and global Harnack estimates. Other parabolic equations of the type u t − ∆u m = 0, 0 < m < 1, are examined in [12,16,17]. These equations are singular at points where u = 0. In [16], the authors investigate, for special range of m, the behavior of the solution near the points of singularity. In particular, they show that nonnegative solutions are analytic in the space variables and at least Lipschitz continuous in time. However, in [17], an intrinsic Harnack estimate for nonnegative weak solutions is established for some optimal range of the parameter m. Other class of singular parabolic equations are of the form: b is a certain constant. Such an equation is related to axially symmetric problems and also occurs in probability theory. A wide study of (1.12), including existence, uniqueness and representation theorems for the solution are proved (Dirichlet and Neumann boundary conditions are treated as well). In addition, differentiability and regularity properties are investigated (for the references, see [11,37,2,9]). A more general form of (1.12), including semilinear equations, is treated in [32,7,8,29].
An important type of equations that can be indirectly related to our system are semilinear parabolic equations: u t = ∆u + |u| p−1 u, p > 1. (1.13) Many authors have studied the blow-up phenomena for solutions of the above equation (see for instance [38,31,30,22,35,36]). This study includes uniform estimates at the blow-up time, as well as the investigation of of upper bounds for the initial blow-up rate. Equation (1.13) can be somehow related to the first equation of (1.1), but with a singularity of the form 1/κ. This can be formally seen if we first suppose that u ≥ 0, and then we apply the following change of variables u = 1/v. In this case, equation (1.13) becomes: and hence if p = 3, we obtain: (1.14) Since the solution u of (1.13) may blow-up at a finite time t = T , then v may vanishes at t = T , and therefore equation (1.14) faces similar singularity to that of the first equation of (1.1), but in terms of the solution itself.

Strategy of the proof
The existence and uniqueness is made by using a fixed point argument after a slight artificial modification in the denominator κ x of the first equation of (1.1) in order to avoid dividing by zero. We will first show the short time existence, proving in particular that κ x (x, t) ≥ γ 2 (t) + ρ 2 x (x, t) > 0, (1.15) for initial conditions satisfying: with some suitable γ(t) > 0. The only, but dangerous, inconvenience is that the function γ depends strongly on ρ xxx ∞ , roughly speaking: where ρ xxx ∞ does not have, a priori, a good control independent of γ. Here where a logarithmic estimate interferes (see Section 2, Theorem 2.16) to obtain an upper bound of ρ xxx ∞ of the form where E is an exponential function in time, and m ∈ N. This allows, with (1.16), to have a good lower bound on γ independent of ρ xxx ∞ . After that, due to some a priori estimates, we will move to show the global time existence. One key point here is that ρx κx ≤ 1 which somehow linearize the first equation of (1.1), and then allows the global existence.

Organization of the paper
This paper is organized as follows: In Section 2, we present the tools needed throughout this work; this includes a brief recall on the L p , C α and the BM O theory for parabolic equations. In Section 3, we show a comparison principle associated to (1.1) that will play a crucial rule in the long time existence of the solution as well as the positivity of κ x . In section 4, we present a result of short time existence, uniqueness and regularity of a solution (ρ, κ) of an artificially modified system of (1.1). Section 5 is devoted to give some exponential bounds of the solution given in section 4. In section 6, we show a control of the W 2,1 2 norm of ρ xxx . In a similar way, we show a control of the BM O norm of ρ xxx in section 7. In section 8, we use a Kozono-Taniuchi parabolic type inequality to control the L ∞ norm of ρ xxx . Thanks to this L ∞ control, we will improve the comparison principle of section 3. In Section 9, we prove our main result: Theorem 1.1. Finally, sections 10, 11 are appendices where we present the proofs of some standard results.

Tools: theory of parabolic equations
We start with some basic notations and terminology.
• I T is the cylinder I × (0, T );Ī is the closure of I; I T is the closure of I T ; ∂I is the boundary of I.
• S T is the lateral boundary of I T , or more precisely, S T = ∂I × (0, T ).
• ∂ p I T is the parabolic boundary of I T , i.e. ∂ p I T = S T ∪ (I × {t = 0}).
• D s y u = ∂ s u ∂y s , u is a function depending on the parameter y, s ∈ N.
• [l] is the floor part of l ∈ R.
• Q r = Q r (x 0 , t 0 ) is the lower parabolic cylinder given by: • |Ω| is the n-dimentional Lebesgue measure of the open set Ω ⊂ R n .

L p and C α theory of parabolic equations
A major part of this work deals with the following typical problem in parabolic theory: where T > 0 and ε > 0. A wide literature on the existence and uniqueness of solutions of (2.1) in different function spaces could be found for instance in [27], [20] and [28]. We will deal mainly with two types of spaces: 1. The Sobolev space W 2,1 p (I T ), 1 < p < ∞ which is the Banach space consisting of the elements in L p (I T ) having generalized derivatives of the form D r t D s x u, with r and s two non-negative integers satisfying the inequality 2r + s ≤ 2, also in L p (I T ). The norm in this space is defined by the equality 2. The Hölder spaces C ℓ (Ī) and C ℓ,ℓ/2 (I T ), ℓ > 0 a nonintegral positive number.
The Hölder space C ℓ (Ī) is the Banach space of all functions v(x) that are continuous inĪ , together with all derivatives up to order [ℓ], and have a finite norm |v| (ℓ) where v (0) The Hölder space C ℓ,ℓ/2 (I T ) is the Banach space of functions v(x, t) that are continuous in I T , together with all derivatives of the form D r t D s x v for 2r + s < ℓ, and have a finite norm |v| (ℓ) where v (0) x,I T , The above definitions could be found in [27,Section 1]. Now, we write down the compatibilty conditions of order 0 and 1. These compatibility conditions concern the given data φ, Φ and f of problem (2.1).
Compatibility condition of order 0. Let φ ∈ C(Ī) and Φ ∈ C(S T ). We say that the compatibility condition of order 0 is satisfied if (2.9) Compatibility condition of order 1. Let φ ∈ C 2 (Ī), Φ ∈ C 1 (S T ) and f ∈ C(I T ). We say that the compatibility condition of order 1 is satisfied if (2.9) is satisfied and in addition we have: (2.10) We state two results of existence and uniqueness adapted to our special problem. We begin by presenting the solvability of parabolic equations in Hölder spaces. Suppose 0 < α < 2, a non-integral number. Then for any f ∈ C α,α/2 (I T ), satisfying the compatibility condition of order 1 (see (2.9) and (2.10)), problem (2.1) has a unique solution u ∈ C 2+α,1+α/2 (I T ) satisfying the following inequality: The constant appearing in the above Hölder estimate (2.11) can be estimated as follows: where c = c(ε, α) > 0 is a positive constant. In order to obtain (2.12), we consider three cases for the time T .
Case 2, T < 1. We linearly extend the function Φ from [0, T ] to [0, 1], and we extend the function f from I T to In this case, We have the same result of Case T = 1.
Case 3, T > 1. Take n ∈ N such that n ≤ T ≤ n + 1. We obtain the estimate (2.12) on c H by iteration. Let F = |f | (α) . We know that: We use the fact that |u(., j)| I×(j−1,j) , j ∈ N, and 1 ≤ j ≤ n, we first compute for c = c(ε, α) given in Case 1: where for the last line, we have used the fact that c > 1. The other terms of (2.13) can be estimated in a similar way. Since n + 1 ≤ T + 1, the estimate (2.12) directly follows.
We now present the solvability in Sobolev spaces. Recall the norm of fractional Sobolev spaces. If f ∈ W s p (a, b), s > 0 and 1 < p < ∞, then with p = 3/2 (p = 3/2 is called the singular index) satisfying in the case p > 3/2 the compatibility condition of order zero (see (2.9)), there exists a unique solution u ∈ W 2,1 p (I T ) of (2.1) satisfying the following estimate:

Remark 2.4 (Neumann conditions)
An analogous theorem of Theorem 2.3 is valid for problem (2.1), but with Neumann boundary conditions u x = 0 on S T .
The singular index in this case will be p = 3, see [27, Chapter 4, Section 10].
Remark 2.6 (The sense of the compatibility condition stated in Theorem 2.3) Remark that in the case p > 3/2, the two functions φ and Φ presented in (2.15) are continuous up to the boundary, i.e. φ ∈ C(Ī) and Φ ∈ C(S T ). This is due to the fact that we have where n = 1 is the space dimension. In this case the fractional Sobolev embedding [1] gives the result, and a sense of the compatibility condition stated in Theorem 2.3 is then given.
For a better understanding of the spaces stated in the above two theorems, especially fractional Sobolev spaces, we send the reader to [1] or [27]. The dependence of the constant c of Theorem 2.3 on the variable T will be of notable importance and this what is emphasized by the next lemma. the estimate (2.16) can be written as: where c = c(ε, p) > 0 is a positive constant depending only on p and ε.
The proof of this lemma will be done in Appendix A. Moreover, We will frequently make use of the following two lemmas also depicted from [27].
In addition, for 2r + s < 2 − 1/p, we have and A useful technical lemma will now be presented. The proof of this lemma will be done in Appendix A.
Lemma 2.10 (L ∞ control of the spatial derivative) Let p > 3 and let 0 < T ≤ 1/4 (this condition is taken for simplification). Then for every u ∈ W 2,1 in the trace sense (see Lemma (2.9)), there exists a constant c(T, p) > 0 such that

BMO theory for parabolic equation
A very useful tool in this paper is the limit case of the L p theory, 1 < p < ∞, for parabolic equations, which is the BM O theory. Roughly speaking, if the function f appearing in (2.1) is in L p for some 1 < p < ∞, then we expect our solution u to have u t and u xx also in L p . This is no longer valid in the limit case, i.e. when p = ∞. In this case, it is shown that the solution u of the parabolic equation have u t and u xx in the parabolic/anisotropic BM O space (bounded mean oscillation) that is convenient to give its definition here.
Here the supremum is taken over all parabolic lower cylinders Q r (see the beginning of Section 2 for the notation).

Remark 2.12
The parabolic BM O(I T ) space, which will be refereed, for simplicity, as the BM O(I T ) space, and sometimes, where there is no confusion, as BM O space, is a Banach space equipped with the norm, We move now to the two main theorems of this subsection; the BM O theory for parabolic equations, and the Kozono-Taniuchi parabolic type inequality. To be more precise, we have the following: Theorem 2.13 (BM O theory for parabolic equations) Consider the following Cauchy problem: If f ∈ L ∞ (R × (0, T )) and f is a 2I-periodic function in space, i.e.
then there exists a unique solution u ∈ BM O(R × (0, T )) of (2.29) with Moreover, there exists c > 0 independent of T such that: The proof of this theorem will be presented in Appendix B. The next theorem shows an estimate concerning parabolic BM O spaces. This estimate, which will play an essential role in our later analysis, is a sort of control of the L ∞ norm of a given function by its BM O norm and the logarithm of its norm in a certain Sobolev space. It can also be considered as the parabolic version on a bounded domain I T of the Kozono-Taniuchi inequality (see [26]) that we recall here.
Theorem 2.14 (The Kozono-Taniuchi inequality in the elliptic case, [26, Theorem 1]) Let 1 < p < ∞ and let s > n/p. There is a constant C = C(n, p, s) such that the estimate holds for all f ∈ W s p (R n ).

Remark 2.15
It is worth mentioning that the BM O norm appearing in (2.31) is the elliptic BM O norm, i.e. the one where the supremum is taken over ordinary balls The original type of the logarithmic Sobolev inequality was found in [5,6] (see also [18]), where the authors investigated the relation between L ∞ , W k r and W s p and proved that there holds the embedding , sp > n provided u W k r ≤ 1 for kr = n. This estimate was applied to prove existence of global solutions to the nonlinear Schrödinger equation (see [5,23]). Similar embedding for vector functions u with div u = 0 was investigated in [3], with sp > n, where they made use of this estimate to give a blow-up criterion of solutions to the Euler equations. Estimate (2.31) is an improvement of (2.32) where a sharp version of (2.31) can be found in [34].
In our work, we need to have an estimate similar to (2.31), but for the parabolic BM O space and on the bounded domain I T . This will be essential, on one hand, to show a suitable positive lower bound of κ x (κ given by Theorem 1.1), and on the other hand, to show the long time existence of our solution. Indeed, there is a similar inequality and this is what will be illustrated by the next theorem.
This inequality is first shown over R x × R t , then it is deduced over I T (for a sketch of the proof, see Appendix B). for some constant c > 0. Suppose furthermore that:
Proof. Throughout the proof, we will extensively use the following notation: G a (y) = a 2 + y 2 a, y ∈ R. Define the quantity M by:

5)
γ(t) > 0 is a function to be determined. The proof could be divided into five steps.
Step 1. (Partial differential inequality satisfied by M ) We do the following computations in I T : and from (1.1) we deduce that (3.8) 13 We set From (3.6), (3.7) and (3.8), we get: where we have used in the last line that G ′ γ (y)G γ (y) = y. Define the function F γ by: we note that F ′ γ = (G ′ γ ) 2 and hence we have: We notice that Using Young's inequality 2ab ≤ a 2 + b 2 , we have: Plugging (3.10) into (3.9), we get: (3.11) Step 2. (The boundary conditions for M ) The boundary conditions (1.3), and the PDEs of system (1.1) imply the following equalities on the boundary (using the smoothness of the solution up to the boundary), (3.12) In particular (3.12) implies To deal with the boundary condition (3.13), we now introduce the following change of unknown function: (3.14) We calculate M on the boundary of I to get: We claim that it is impossible for M to have a positive minimum at the boundary of I.

Indeed we have
M has a positive minimum at M has a positive minimum at Both cases violate the equation (3.15) in the case of the choice of β satisfying: and hence the minimum of M is attained inside the interval I. We make the following calculation inside I T .
Using the previous identities into (3.11), we obtain: Let Since the minimum is attained inside I, and since M is regular, there exists We remark that we have: and hence we write down the equation satisfied by m, we get (indeed in the viscosity sense): We turn our attention now to the term R from (3.18). By Young's inequality 2ab ≤ a 2 + b 2 , we have: therefore the term R satisfies: By the hypothesis (3.2), for all β ∈ R, there exists a unique η = η(β) > 0 satisfying: From (3.22), we know that m(0) = α 2 1 > 0, and the continuity of m preserves its positivity at least for short time. Then, as long as m is positive, we have By using (3.24), (3.1), and the basic identities where Step 5. (The choice of γ and conclusion) When γ ′ ≤ 0, we deduce from (3.18) and (3.25) that We remind the reader that ρ x appearing in the previous inequality have the following form: where Two cases can be considered: Assume first that γ is C 1 (which is not the case in general). Then we plug the function m = γ 2 in (3.26) to deduce when γ ′ ≤ 0: inequality (3.28) implies: In other terms This directly implies that m(t) ≥ m(0)e −2(e c+c * )t .
Case B: the general case.
Simply choose where c * is given by (3.29), and α 1 is given by (3.23). We claim that γ 2 is a sub-solution of (3.26). Indeed, the function γ given by (3.31) is constructed in such a way that γ 2 is a sub-solution of (3.26). To see this, we remark that γ solves the equality that corresponds to the inequality (3.30) and therefore it solves (3.30) with the reverse inequality. Hence, coming back from (3.30), we can see that γ 2 is a sub-solution of (3.26). Since Finally, remark that Inequality (3.32) implies in particular that we have

Short time existence, uniqueness, and regularity
In this section, we will prove a result of short time existence, uniqueness and regularity of a solution of problem (1.1), (1.2) and (1.3). This could be done in two steps. At the first step, we show a short time existence and uniqueness result of a truncated system of equations that will be specified later. At the second step, we show an improved regularity of this solution by a bootstrap argument.

Short-time existence and uniqueness of a truncated system
Fix a time T 0 > 0. Consider the following system defined on with M 0 > 0 and γ 0 > 0 are two positive constants. Here, the function T a (x), x ∈ R and a > 0, is called a truncation function and is given by: (4. 2) The initial conditions are: and the boundary conditions:

Remark 4.1 (The terms p and α)
In all what follows, and unless otherwise precised, the term p is a fixed positive real number such that p > 3, and the term 0 < α < 1 is a fixed real number that is related to p by the following relation We write down our next proposition: be two given functions such that: and where γ 0 > 0 and M 0 > 0 are two given positive real numbers. Then there exists Moreover, this solution satisfies and Remark that the regularity (4.5) of the initial conditions that we have considered is somehow strange and not natural for a result of existence in the Sobolev space W 2,1 p . In fact, the regularity (4.5), which is natural in connection with the main theorem of this paper (see Theorem 1.1), was just taken for the simplification of the forthcoming announcements of our results.

Remark 4.4
It is worth noticing that (4.6) justifies the compatibility of zero order with the boundary conditions (4.4) (see (2.9)).

Proof of Proposition (4.2). Let
We will prove the existence and uniqueness for T small enough using a fixed point argument. Define the application Ψ by: where (ρ, κ) is a solution of the following system: with the same initial and boundary conditions given by (4.3) and (4.4) respectively.
Recall that ρ T 0 and κ T 0 verify (4.6). Hence we deduce from Theorem 2.3 (using on one hand, the fact that the source terms of both equations of (4.12) are in L p (I T 0 ,T ); the fact that this is a direct consequence of (4.5)", and on the other hand, the compatibility of the boundary conditions (see Remark 4.4)), the existence and uniqueness of the solution (ρ, κ) ∈ Y 2 of (4.12), (4.3) and (4.4). We claim that Ψ is a contraction map over some suitable closed subset of Y 2 for T small enough. Let us clarify that the constant c that will frequently appear in the proof may vary from line to line but always has the form: Assume we are searching for some T > 0 such that The proof is divided into three steps.
Step 1. (Defining the map Ψ over a suitable subset) Let λ be any fixed constant. Define D ρ λ and D κ λ as the two closed subsets of Y given by: and (4.14) We will prove that Ψ is a well defined map over D ρ λ ×D κ λ into itself, at least for sufficiently small time T . Let (ρ,κ) ∈ D ρ λ × D κ λ and let Ψ(ρ,κ) = (ρ, κ).
We use system (4.12) to write down some estimates. Takē From (4.12), the equations satisfied byρ andκ are: and respectively. We use equation (4.16) together with the estimate (2.17), we obtain and from (4.15), we deduce that Therefore, choosing T satisfying: In the same way, we use equation (4.17) with the estimate (2.17) to obtain where we have used again, passing from the first to the second line, the equation (4.16) together with the estimate (2.17). Precisely, we have used that: From (4.20) and (4.15), we deduce that In this case, choosing ensures that κ x p,I T 0 ,T ≤ λ and hence From (4.19) and (4.22), we deduce that for sufficiently small time T , the map Ψ is a well defined map from D ρ λ × D κ λ into itself.
Step 2. (Ψ is a contraction map) The couple (ρ − ρ ′ , κ − κ ′ ) is the solution of the following system: Step 2.1. From the second equation of (4.23), and (2.17), we have: By the boundary conditions (4.24) and the L p parabolic estimate (2.17), we deduce that for some c > 0, we have:

27)
Step 2.2. Let F be the function given by: (4.28) From the first equation of (4.23) and using (2.17), we get The function F can be rewritten as follows: We are going to use the system (4.23), (4.24) together with the inequality (2.17) in order to estimate each term of (4.30). First, from (4.27), we have: For the term A 2 , we proceed as follows. We apply the L ∞ control of the spatial derivative (see Lemma 2.10) to the functionρ −ρ ′ , we get: For the term ρ ′ xx , we first remark that if we letρ ′ = ρ ′ − ρ T 0 , this function satisfies (4.16) withκ x replaced byκ ′ x , and hence we deduce that From (4.32) and (4.33), we deduce that The term A 3 could be treated in a similar way as the term A 2 , and we obtain the following estimate: Step 2.3. From (4.27) in Step 2.1, and (4.29) in step 2.2, we finally get: and therefore, taking T satisfying: (4.19) and (4.22), we deduce that Ψ is a contraction from D ρ λ × D κ λ into itself.
Step 3. (Conclusion) In order to terminate the proof, it remains to show (4.9) and (4.10), again for sufficiently small time T . In fact, this will be done by controlling the modulus of continuity in time of ρ x and κ x uniformly with respect to T . The time T that we will use in Step 3 is that determined by (4.19), (4.22) and (4.36), ensuring existence and uniqueness. However, additional conditions will be imposed on T so that the inequalities (4.9) and (4.10) are valid onQ T .
where for the last line we have used estimate (2.17) for equation (4.16). Hence we have Call m 1 = c(M 0 + λ), and recall thatρ = ρ − ρ T 0 , we therefore obtain From (2.8), (4.8), and (4.37), we deduce that for any (x, t) ∈Q T , we have and then for Step 3.2. (Controlling the quantity κ x ) We argue in a similar manner in order to control κ x with Following the same arguments as above, we obtain that for we have

Regularity of the solution
This subsection is devoted to show that the solution of (4.1), (4.3) and (4.4) enjoys more regularity than the one indicated in Proposition 4.2. This will be done using a special bootstrap argument, together with the Hölder regularity of solutions of parabolic equations. and Then the unique solution (ρ, κ) ∈ Y 2 given by Proposition 4.2, satisfying (4.9) and (4.10), is in fact more regular.
To be more precise, it satisfies:
Step 3. (The C ∞ regularity) At this point, we will show how to obtain more regularity of the solution (ρ, κ) away from the initial data. Remark that if we want to follow similar arguments of what was done in the previous two steps, we might think of increasing the regularity of ρ by using the Hölder solvability, Theorem 2.1, and the fact that κ x ∈ C 2+α, 2+α 2 (I T ) (see (4.55) above). In fact, this requires higher order compatibilty conditions that are not satisfied having only (4.41) and (4.42). We send the reader to [27, Chapter 4, Section 5, page 319] for the details of these compatibility conditions. To overcome this difficulty, we introduce the following function. Let 0 < δ < T , define the test function ϕ δ ∈ C ∞ [0, T ] by: We can easily check that these quantities satisfy two parabolic equations with the higher order compatibility of the initial data are all satisfied. By the bootstrap argument (see Steps 1, 2 above), we get: (ρ, κ) ∈ C ∞ (I T ).

Exponential bounds
In this section, we will give some exponential bounds of the solution given by Proposition (4.2) and having the regularity shown by Proposition (4.6).
It is very important, throughout all this section, to precise our notation concerning the constants that may certainly vary from line to line. Let us mention that a constant depending on time will be denoted by c(T ). Those who do not depend on T will be simply denoted by c. In all other cases, we will follow the changing of the constants in a precise manner.
Proposition 5.1 (Exponential bound in time for (ρ x (., t), κ x (., t)) ∞,I ) Let be a long time solution of the following system: Suppose furthermore that Then we have (ρ x (., t), κ x (., t)) ∞,I ≤ ce ct ,  Proof of Proposition 5.1. We use the special coupling of the system (5.1) to find our a priori estimate. Roughly speaking, the fact that κ x appears as a source term in the second equation of system (5.1) permits, by the L p theory for parabolic equations, to have L p bounds, in terms of κ x p,I T , on ρ x and ρ xx which in their turn appear in the source terms of the first equation of (5.1) satisfied by κ. All this permits to deduce our estimates. To be more precise, let T > 0 an arbitrarily fixed time.
Step 1. (estimating κ x in the L p norm) Let κ ′ be the solution of the following equation: As a solution of a parabolic equation, we use the L p parabolic estimate (2.16) to the function κ ′ to deduce that: where the term 1 comes from the value of κ ′ = κ on S T . Takē then the system satisfied byκ reads: (5.8) Using the special version (2.17) of the parabolic L p estimate to the functionκ, we obtain: where we have plugged into the constant c the terms ε, τ , p and B ∞ . Combining (5.6), (5.7) and (5.9), we get: The term ρ W 2,1 p (I T ) appearing in the previous inequality is going to be estimated in the next step.
Step 2. (estimating ρ in the W 2,1 p norm) As in Step 1, let ρ ′ ,ρ be the two function defined similarly as κ ′ ,κ respectively (see (5.5) and (5.7)). ρ ′ satisfies an inequality similar to (5.6) that reads: (5.11) The term 1 disappered here because ρ ′ = ρ = 0 on S T . We write the system satisfied bȳ ρ, we obtain: hence the following estimate onρ, due to the special L p interior estimate (2.17), holds: Again, we have plugged ε, τ and p into the constant c, and we have assumed that T ≤ 1. Combining (5.11) and (5.13), we get in terms of ρ: (5.14) We will use this estimate in order to have a control on κ x p,I T for sufficently small time.
Step 3. (Estimate on a small time interval) From (5.10) and (5.14), we deduce that: Let us remind the reader that all constants c and c(T ) have been changing from line to line. In fact, the important thing is whether they depend on T or not. Let T * = 1 2c 2 , c is the constant appearing in (5.15), we deduce, from (5.15), that where c 3 = c 3 (T * ) > 0 is a positive constant which depends on T * . Recall the special coupling of system (5.1); the brief introduction in the beginning of the proof of this proposition, and the above estimate, we can deduce that: with c 4 = c 4 (T * ) > 0 is also a positive constant depending on T * but independent of the initial data.

Since
(ρ x (., t), κ x (., t)) ∞,I ≤ h(t), the result easily follows.   I×(t,t+T * ) ) We remark that from the Sobolev embedding in Hölder spaces (see Lemma 2.9): the previous result could be improved to an exponential bound of |ρ x |     Proof. Throughout the proof, we will omit, without loss of generality, the dependence on B ∞ . The ideas of the proof are somehow contained in the proof of the previous proposition. In fact, we will not only show the exponential bound for the L ∞ norm of ρ xx , but also for the C α norm. The proof is done in two steps.
Step 1. (Estimating ρ in the C 2+α, 2+α 2 norm) We start by writing down the Hölder estimate (2.11) for the second equation of (5.1). Indeed, since κ x ∈ C α,α/2 (I T ), and since the compatibility conditions of order 1 are satisfied, we have that: We aim to control |κ x | , c > 0 independent of T, and hence, from (5.24) and the definition (5.7) ofκ, we obtain: For the term where it interferes the κ ′ , we have used the following: Having in mind that the term κ x p,I T satisfies: I T , inequality (5.27) can be written: and hence for T * small enough, namely Plugging (5.28) into (5.23), we deduce that: (5.29) Here we consider c 11 ≥ 1 for technical reasons.

Step 2. (The exponential estimate by iteration)
This is similar to Step 4 of Proposition 5.1. We first notice that the arguments presented in that step can be adapted to get an exponential bound on the function g given by (5.17). Indeed, we use (5.18) and the estimate of the traces of functions in Sobolev spaces (see Lemma 2.9, estimate (2.23)), to deduce that, for every t ≥ 0: with c 12 ≥ 1 is a fixed positive constant independent of the initial conditions. Also here c 11 ≥ 1 is taken for technical reasons. Let I×(t,t+T * ) , T * is given in Step 1.
we start with the first lemma.
where γ := γ(T ), with d ≥ 1 is a positive constant depending on the initial conditions but independent of T , and will be given at the end of the proof.

Remark 6.2 (The constant E depending on time)
Let us stress on the fact that, througout the proof, the term E = de dT of Lemma 6.1 might vary from line to line. In other words, the term d in the expression of E might certainly vary from line to line, but always satisfying the fact of just being dependent on the initial data of the problem. The different E's appearing in different estimates can be made the same by simply taking the maximum between them. Therefore they will all be denoted by the same letter E.
Proof. Define the functions u and v by: We write down the equations satisfied by u and v respectively: 4) and with B = ρx κx : The proof could be divided into three steps. As a first step, we will estimate the L ∞ (D) norm of the term v x = κ tx . In the second step, we will control the W 2,1 2 (D) norm of v = κ t . Finally, in the third step, we will show how to deduce a similar control on the W 2,1 2 (D) norm of κ xx .
Step 1. (Estimating v x ∞,D ) Since v x = κ tx , it is worth recalling the equation satisfied by κ: In Step 3 of Proposition 4.6, we have shown that κ ∈ C 3+α, 3+α 2 . Therefore, writing the parabolic Hölder estimate (see (2.11)), we obtain: where the term 1 comes from the boundary conditions, and c H > 0 is the positive constant given by (2.12) that can be estimated as c H ≤ E. We use the elementary identity |f g| to the term ρxρxx κx (1+α) D with f = ρx κx and g = ρ xx , we get: where we have used the fact that κ x ≥ γ and κ x ≥ |ρ x |. We plug (6.8) in (6.7), we obtain: , (6.9) where we have used used the fact that the term |ρ| (2+α) D has an exponential bound (see Remark 5.5) of the form |ρ| (2+α) D ≤ E. It is worth noticing that the term E appearing in (6.9) is the maximum between different E's that might exist as different bounds. This will be frequently used for the sake of simplicity.
Step 1.2. Estimating |ρ| We recall the equation satisfied by ρ: As for the term κ at the beginning of this Step 1, we have the following estimate for ρ: Having a second look at the equation (6.6) of κ, we can use again the parabolic Hölder estimate but for a lower order. In fact, we have: Similar computations to those in Step 1.1 yield: and hence from (6.17), we also get a similar estimate for |ρ| Step 1.3. (The estimate for κ tx ∞,D ) By combining (6.9), (6.15), (6.18) , (6.19), and by using the fact that |κ x | (α) D has an exponential estimate (see estimate (5.21) of Remark 5.3), we deduce that: where we have frequently used that γ ≤ 1, and we have always taken the maximum of all the exponential bounds of the E = de dT form.
Step 2. (Estimating v W 2,1 2 (D) ) We turn our attention to the equation (6.4) satisfied by u. We will show that we are in the good framework for applying the L 2 theory of parabolic equations. In fact, note first that u = ρ t ∈ C(D), and hence the compatibility condition of order 0 is satisfied. Moreover, since v x = κ tx ∈ C(D) then v x ∈ L 2 (D). Finally, the initial data satisfies u 0 ∈ C 1+α (Ī), hence u 0 ∈ W 1 2 (I). The above arguments show that the L 2 theory for parabolic equations (see Theorem 2.3) can be applied to the function u, therefore we get: , with the following estimate: Here the term E of the previous inequality hides in it all the constant c of the Sobolev estimate (see (2.16) and (2.17)), where this constant c behaves like T or √ T . Also the term 1 in (6.21) comes from the initial data. Since v x = κ tx , we plug the estimate (6.20) obtained in Step 1.3 into (6.21), we get Using some elementary identities, we finally obtain: Let us remind the reader that the term E is changing from line to line. We now consider equation (6.5) satisfied by v. In fact, for the same reasons as above with the new fact that u ∈ W 2,1 2 (D), we can easily deduce that we are in the good framework for the L 2 theory applied to v. Indeed, we have: v ∈ W 2,1 2 (D) =⇒ κ t , κ tt , κ tx , κ txx ∈ L 2 (D), with the following estimate: hence from (6.20), (6.22), and some repeated computations, we deduce from (6.23) that: As a byproduct of this last inequality, we can also get, using the Sobolev embedding Lemma (see Lemma 2.8-(ii)), that: Remark that we can even get a better control by simply integrating (6.20) with respect to x, hence we obtain: Step 3. (Estimating κ xx W 2,1 2 (D) ) The estimate of κ xx W 2,1 2 (D) requires a special attention. We will mainly use the equations (6.16) and (6.6) satisfied by ρ and κ respectivly. The four parts κ xx 2,D , κ xxt 2,D , κ xxx 2,D and κ xxxx 2,D of the above norm will be estimated separately.
Step 3.1. (Estimate of κ xx 2,D ) This can be easily deduced from the equation (6.6) of κ. Indeed, this equation gives: where for the last line, we have used estimate (6.25), and the exponential bounds on ρ x ∞,D and ρ xx ∞,D . Indeed, by the same way, we can even get, from the L ∞ bound (6.25) on κ t , that Step 3.2. (Estimate of κ xxt 2,D ) As an immediate consequence of (6.24), we get Step 3.3. (Estimate of κ xxx 2,D ) Using (6.19), we deduce that therefore, the fact that |κ x | (α) D ≤ E (see Remark 5.3) gives: (6.28) and hence (6.18) implies that: This will be used in estimating κ xxt 2,D . In fact, we derive the equation (6.6) satisfied by κ, with respect to x, we obtain: and hence, using (6.28), we get: Step 3.4. (Estimate of κ xxxx 2,D ) We first derive (6.16) two times in x, we deduce (using (6.22)) that ρ xxxx 2,D has the same upper bound as κ xxx 2,D , i.e.
We derive the equation (6.29) once more with respect to x: and we use (6.31) and our controls obtained in the previous steps, in order to deduce that: In fact, the highest power comes from estimating the following term: where we have used the L ∞ estimate of κ xx ∞,D . All other estimates are easily deduced. Let us just state how to estimate the other term were κ xx ∞,D interferes. In fact, we have: Step 3.

(Conclusion)
From the above estimates (6.26), (6.30) and (6.32), we finally deduce that: This terminates the proof. 2 We move now to the main result of this section. bound for ρ xxx ) Under the same hypothesis of Lemma 6.1, we have:
Combining the above two equations, we obtain: We write down, after doing some computations, the equation satisfied byw: Let us show that the framework of the L 2 theory for parabolic equations with Neumann boundary conditions (see Theorem 2.3 and Remark 2.4 that follows) is well satisfied.
First, from Step 2 of Lemma 6.1, we know that κ tt ∈ L 2 (D). Moreover, since we have supposed (ρ 0 , κ 0 ) ∈ (C ∞ (Ī)) 2 , then we eventually havew 0 ∈ W 1 2 (I). We note that the compatibility conditions are not necessary in this case because the singular index in the Neumann framework is 3 (see Remark 2.4). These arguments permit to use the L 2 theory of parabolic equations with Neumann boundary conditions, hence we get: Sincew = w −κ, we deduce, from (6.36), that: and eventually (6.37) with Lemma 6.1 gives immediately the result.  We start with the following lemma that reflects a useful relation between the BM O norm of f sym and f asym . The proof of this lemma will be presented in Appendix B. The next lemma gives a control of the BM O norm of (κ xx ) asym . where c > 0 is a constant depending on the initial conditions (but independent of T ). The function (κ xx ) asym is given via Definition 7.2.
Proof. The proof is based on the following simple observation on the boundary S T . In fact, recall that the hölder regularity C 3+α, 3+α 2 , up to the boundary, for the solution (ρ, κ) permits using to conclude that: hence a simple computation yields that: we write down the equation satisfied byκ: we also write the equation satisfied by v: the equation satisfied byv reads: We can assume, without loss of generality, that the initial conditionv 0 = 0. This is because being non-zero just adds a constant depending on the initial conditions in the final estimate that we are looking for. From the fact thatv x | S T = 0, we can easily deduce that the functionv sym satisfies: Recall the definition of the term E from Remark 6.2. At this stage, we write the following estimate: 12) which can be deduced using Lemma 7.3. The constant c > 0 appearing in (7.12) is independent of T . Finally, we deduce that: where we have used (7.9), (7.10), (7.11) and (7.12) for the first line, and Lemma 7.4 for the second line. For the last line, we have used that p > 3. From (H1) and (5.16), we know that: From the above two inequalities, and sincev xx = ρ xxx − τ κxx 1+ε , we easily arrive to our result. In this section, we use the results of Sections 5, 6 and 7, in order to give an L ∞ bound for ρ xxx via the Kozono-Taniuchi inequality. The next step is to improve some previously obtained results. Using (8.2) together with Lemmas 7.5 and 6.3, lead to the result. The only term left to control is ρ xxx 1,D . In fact, we know that: and since, by repeating the same arguments of the proof of Lemma 7.5, and of Lemma 2.7 (see Appendix B), using the L p estimates for parabolic equations instead of the BM O ones, we can conclude that: where from (5.16), we finally get: This inequality together with (8.3) terminates the proof. 2

Remark 8.2 (Improving the comparison principle)
The L ∞ bound on ρ xxx given by Proposition 8.1 shows that we can improve our choice of the function γ of Proposition 3.1. Although the function γ was essentially used, on one hand, to ensure the positivity of κ x for all time t ≥ 0, and on the other hand, for the boundedness of the ratio ρx κx , it was insufficient for showing the long time existence of (ρ, κ) given by Propositions 4.2 and 4.6; this lies from the fact that γ strongly depends, and in a dangerous way, on ρ xxx (see inequality 3.30). The remedy of this inconvenience is to revisit the comparison principle "Proposition 3.1" with the new information given by Proposition 8.1, namely estimate (8.1). Now, we show that we can even improve estimate (8.1) by eliminating the restrictive hypothesis (H1) and changing somehow the constant E appearing in (8.1). To be more precise, we write down our next corollary.
Proof. We know, from Propositions 4.2, 4.6, used for T 0 = 0, that there exists some small δ 1 > 0 only depending on the initial conditions, with: where c 13 > 0 is a constant only depending on the initial conditions. We now apply Proposition 8.1 with where it is important to indicate that the term E = E(δ 1 ) appearing in (8.6) depends on T 1 = δ 1 (see for instance the end of the proof of Lemma 7.5). Combining (8.5) and (8.6), we deduce that ∀T > 0: and hence the result follows. 2 The following proposition reflects how to improve Proposition 3.1.
where γ > 0 is a positive decreasing function depending on the initial conditions, and will be given in the proof.
Proof. In Proposition 3.1, we have that c (recall (3.1)) is a bound on ρ xxx ∞,I T . From the a priori estimate (8.4) we can choose for any T > 0. We assume that γ(t) is a continuous decreasing function, and that the solution (ρ, κ) satisfies: Therefore, from the proof of Proposition 3.1, we have that satisfies (3.26) on (0, T ). Here β satisfies (3.16) with I = (−1, 1). Therefore, using (8.8), we obtain with c 1 = β 2 4 + τ 2 8ε + εβ 2 , and c 2 = τ 2 cosh β 4ε . Since (8.9) is true for any T > 0, we deduce that: , t ∈ (0, T ), (8.10) (recall the definition of x 0 by (3.27)). Following the same reasoning of the proof of Proposition 3.1, in particular Step 5, Case A, we know that, as long as m = γ 2 is a solution of (8.10) with γ ∈ C 1 , γ ′ < 0 (which is not the case in general), we have (see (3.30)): γ(t), c * given by (3.29), t ∈ (0, T ). (8.11) Inequality (8.11) gives inspiration to the choice of γ as a solution of the following ODE: where α 2 is given by (3.33). It is easy to check that γ 2 is a subsolution of (8.10), hence In this case, as long as, we deduce that (8.14) Finally, from (8.13), (8.14) and the short-time existence result, Proposition 4.2, we easily deduce that κ x > 0 for all time and then the result follows. 2 In fact, Proposition 8.4, can be used to improve our L ∞ exponential bounds found in Propositions 5.1 and 5.4. This will be the result of the next proposition. Here b > 0 is a positive constant depending on the initial conditions and the fixed terms of the problem, but independent of time.
Proof. The proof of this proposition could be divided into three steps.

Step 2. (Explicit minoration of γ)
It is clear that the decreasing function is the solution of (8.18), and hence , ∀t ≥ 0, (8.20) for some constant b > 0 depending on the initial conditions and some other fixed terms, but independent of t. Inequality (8.15) directly follows from (8.7) and (8.20).

and conclusion)
From the proof of Lemma 6.1, we can easily deduce that the estimate of κ tx ∞,D (see (6.20)) is also true replacing κ tx ∞,D by |κ| .
The proof could be divided into two steps.
Step 1. (B is a non-empty set) The inequality (1.8) ensures the existence of γ(0) = α 2 > 0 given by (3.23), such that From the boundary conditions of the initial data (1.7), we deduce, using Proposition 4.6, that this solution from W 2,1 p (I T 1 ) is in fact C 3+α, 3+α 2 (I T 1 ) and therefore The above identities (9.3) and (9.5) show that and hence B = ∅.

Set
T ∞ = sup B, our next step is to prove that T ∞ = ∞.
Step 2. (T ∞ = ∞) We will argue by contradiction. Suppose T 1 ≤ T ∞ < ∞. In this case, let δ > 0 be an arbitrary small positive constant, then there exist some T ∈ B such that and κ are constants. This argument together with (9.11) permit, using first, Proposition 4.6 on the regularity C 3+α, 3+α 2 , and then Proposition 8.4 on the minoration of κ x , to increase the regularity of this solution and then show that κ x > 0 and ρ x κ x < 1 onĪ × [T ∞ − δ, T ∞ − δ + T * ]. (9.12) From (9.12) and the above arguments, we find that with T * > 0 independent of δ. By choosing 0 < δ < T * , we deduce that which contradicts the definition of T ∞ = sup B. Therefore T ∞ = ∞. To complete the proof, we have to indicate that the C ∞ regularity (1.10) is automatically satisfied (see Step 3 of Proposition 4.6). As a first step, we will prove the result in the case where ε = 1, and in a second step, we will move to the case ε > 0. It is worth noticing that the term c may take several values only depending on p.
Step 1. (The estimate: case ε = 1) Suppose ε = 1. Recall that u ∈ W 2,1 p (I T ), p > 1 is the unique solution of (2.1) with f ∈ L p (I T ) and φ = Φ = 0. Letũ be a special extension of the function u defined over R × (0, T ) by: u(x + 2, t) =ũ(x, t) otherwise. (10.1) In exactly the same way, we can definef out of the function f . It is easy to verify that u satisfies: u(x, 0) = 0, on R.
Let u =ṽψ, we apply Theorem 11.1 to the function u, we get v ∞,I T ≤ c ṽψ BM O(R 2 ) 1 + log + ṽψ BM O(R 2 ) + log + ṽψ W 2,1 2 (R 2 ) . (11.18) The special extension of the function v permits to write: . (11.19) Moreover, repeating similar arguments as in the proof of Theorem 2.13, Steps 1 and 2, we can treat relatively small cubes Q s or relatively big cubes Q b for the BM O norm of ṽψ BM O(R 2 ) . As a final consequence, we get The only new case that we need to take care about is when the cube intersects the zone Z 2 \ Z 1 where ψ = 0, 1. In this case we use the fact that which return us to one of the above two cases considered above. Therefore, we obtain ṽψ BM O(R 2 ) ≤ c v BM O(I T ) . (11.20) From (11.18), (11.19) and (11.20), the result follows.