Global existence of weak solutions to the three-dimensional Prandtl equations with A special structure

The global existence of weak solutions to the three space dimensional Prandtl equations is studied under some constraint on its structure. This is a continuation of our recent study on the local existence of classical solutions with the same structure condition. It reveals the sufficiency of the monotonicity condition on one component of the tangential velocity field and the favorable condition on pressure in the same direction that leads to global existence of weak solutions. This generalizes the result obtained by Xin-Zhang on the two-dimensional Prandtl equations to the three-dimensional setting.


Introduction
Consider the initial boundary value problem for the Prandtl boundary layer equations in three space variables, x, y), V (t, x, y)), (u, v)| t=0 = u 0 (x, y, z), v 0 (x, y, z) , where Q = {t > 0, (x, y) ∈ D, z > 0} with a fixed D ⊂ R 2 , (U (t, x, y), V (t, x, y)) and p(t, x, y) are the traces of the tangential velocity field and the pressure of the Euler flow on the boundary {z = 0}. Note that the traces satisfy Despite of its importance in physics, there are very few mathematical results on the Prandtl equations in three space variables. In fact, except the recent work [5] about the classical solution with special structure and those in the analytic framework [12,3], most of the mathemtical studies on this foundamental system in boundary layer theory are limited to the problem in two space dimensions, cf. [1,4,7,9,10,13,14] and the references therein.
Recently in [5], we obtain the local well-posedness of classical solutions to the problem (1.1) under some constraint on the structure of the solution, in order to avoid the appearance of secondary flow ( [8]) in boundary layers. Precisely, assuming that for the Euler flow given in (1.2), U (t, x, y) > 0, in the class of boundary layers that the direction of tangential velocity field is invariant in the normal variable z, and the x−component of velocity u(t, x, y, z) is strictly increasing in z, ∂ z u > 0, in [5] we have constructed a classical solution to the problem (1.1), and it is linearly stable with respect to any three dimensional perturbation. In the class of this special structure, the solution of (1.1) takes the form: x, y, z), k(x, y)u(t, x, y, z), w(t, x, y, z)) , where the function k(x, y) satisfies the following condition (H): (H1) the function k depending only on (x, y) satisfies the inviscid Burgers equation in D, (1.4) k x + kk y = 0; (H2) the outer Euler flow (U (t, x, y), k(x, y)U (t, x, y), 0, p(t, x, y)) with U (t, x, y) > 0, satisfies that from the system (1.2), Moreover, the authors recently observed in [6] that for the shear flow (u s (t, z), v s (t, z), 0) of the three-dimensional Prandtl equations, the special solution structure (1.3) is the only stable case.
For this reduced problem under the assumption that in the class of ∂ z u > 0, we apply the method developed by Oleinik [10] for two dimensional Prandtl equations. Precisely, by the Crocco transformation, the problem (1.5) becomes the following initial boundary value problem, For the problem (1.7) of the degenerate parabolic equation, we established local existence of classical solutions in [5]. As a continuation of the paper [5], the purpose of this paper is to prove the global in time existence of a weak solution to the problem (1.7) for data satisfying (1.6) and the favourable pressure condition: For this, we will adopt the approach introduced by Xin-Zhang in [14] for the the two-dimensional Prandtl equations to the three-dimensional setting. As observed in [11], the main motivation of introducing the favourate condition on pressure is to avoid the separation of boundary layers. For completeness, the definition of weak solutions to (1.7) is given as follows.
in t < T, if the following conditions hold: (i) There exists a positive constant C such that (ii) W satisfies the first equation and the initial boundary conditions of (1.7) in the weak sense: for any test function ψ(τ, ξ, η, ζ) ∈ C ∞ (Ω T ) satisfying ψ = 0, at t = T or (ξ, η) ∈ γ + ; ψ ζ = 0 at ζ = 0. Here, and the function k n = (1, k) · n with n being the unit outward normal vector of γ − .
The main result of this paper can be stated as follows.

The proof of the main result
Following the approach introduced in [14], a viscous splitting method is used to construct a sequence of approximate solutions to the problem (1.7). Precisely, divide the time interval [0, T ] into n equal sub-intervals: First, in the time step [0, t 1 ] or [t i , t i+1 ] for an even i, we construct the approximate solution by solving the following initial boundary value problem for a porous mediatype equation: and in the time step [t i , t i+1 ] for an odd i, we construct the approximate solution by solving the following problem for a transport equation: Here, the coefficient function b in (2.1) will be chosen later to satisfy the boundary condition W | Γ− = W 1 for the solution of (2.1). What is needed is to prove that the function W constructed in the time interval [0, T ] is uniformly bounded in n and has a uniform total variation with respect to the spatial variables ξ, η and ζ. This implies that as n → ∞, the limit function of the approximate solutions W constructed in (2.1)-(2.2) is a weak solution to the problem (1.7). The proof is divided into the following subsections.

Porous medium-type equation.
In this subsection, consider the following problem for a porous medium-type equation In order to match the boundary condition W | Γ− = W 1 , by observing that W 1 > 0 for 0 ≤ ζ < 1 and where f (ξ, η) is a non-negative smooth function defined on the closure of the domain D satisfying f (ξ, η)| γ− = 1. By the formulations in (1.8): and the assumption given in Theorem 1.1, there exists a positive constant M 0 depending on the parameters of (1.7), such that for the function b(t, ξ, η, ζ) given in (2.4), The problem (2.3) can be viewed as a one space dimensional problem by regarding variables ξ and η as parameters.
Note that the equation in (2.3) is degenerate on the boundary {ζ = 1}. As [14], consider the following uniformly approximated parabolic problem: , for a positive constant ǫ > 0. It is known that problem (2.6) has a unique smooth solution. After getting some uniform bounds of W ǫ with respect to ǫ, we can obtain a solution to the problem (2.3) by taking ǫ → 0 in (2.6). In fact, we have (2.8) (2) there exists a positive constant C 2 , depending on W L ∞ , ∂ ζ W 0 L ∞ and the C 1 -norm of the parameters of (2.3), such that (3) for any t > 0, Here, the positive constant C 3 depends on W L ∞ and the C 1 -norm of the parameters of (2.3). Also, W η and W t satisfy similar estimates as (2.11) by simply replacing the partial derivative with respect to ξ in (2.11) by the partial derivatives with respect to η and t, respectively.
Proof. To prove the inequality (2.7), we first show W | ζ=0 ≥ 0 by using the favorable condition on the pressure function. For this, we can assume that px U < 0 because we can replace it by px U − δ for some constant δ > 0 and then let δ → 0. Then, we want to show that holds for the problem (2.6). Otherwise, by using the continuity of W ǫ and W 0 | ζ=0 > 0, there exist ǫ 0 > 0 and a point P on {ζ = 0}, such that W ǫ0 | P = 0. That is, W 2 ǫ0 attains its minimum at P , which implies that ∂ ζ W 2 ǫ0 | P ≥ 0. But from the boundary condition of (2.6) on ζ = 0, we have which is a contradiction. Hence, we obtain (2.12), which implies that W | ζ=0 ≥ 0 for the problem (2.3) by letting ǫ → 0. Now, combining W | ζ=0 ≥ 0 with the boundary condition W W ζ | ζ=0 = px U ≤ 0, we obtain W ζ | ζ=0 ≤ 0. The rest of proof for (2.7) is similar to that of Lemma 4.2 in [14] so that we omit the details. Moreover, there is a constant m > 0, such that (2) We now turn to the estimate (2.9). Consider the problem (2.6) for W ǫ , and the corresponding problem for ∂ ζ W ǫ as follows, Here, note that ∂ 2 ζ W ǫ | ζ=1 = 0 because W ǫ | ζ=1 = 0 and A| ζ=1 = 0 in the first equation in (2.6).
Set V = ∂ ζ W ǫ − αζ with α being a constant to be chosen later. It follows that From (2.15) and (2.14), we have with the following initial and boundary conditions Firstly, note that for an arbitrarily fixed constant α > 0, from (2.17), we have ∂ ζ V 1 | ζ=1 < 0. Also, from (2.13) and the relation in (2.17) it implies that V 1 does not attain its positive maximum on ζ = 1 or ζ = 0. Then, if V 1 attains its positive maximum in the interior of Ω T or when t = T , we have from If V 1 achieves its positive maximum when t = 0, it follows that Therefore, we conclude that Secondly, for an arbitrarily fixed constant α < 0, by considering the possible negative minimal points of V 1 on Ω T , similar to the above argument, we have Hence, combining (2.18) with (2.19) and letting α → 0 yield that Thus, we obtain (2.9) as ǫ → 0.
(3) The proofs of (2.10) and (2.11) are similar to those given in Lemmas 4.6 and 4.7 of [14], respectively. And the proof for the uniqueness of solution to the problem (2.3) is similar to that of Theorem 4.1 in [14]. Thus, we omit the detail for brevity and this completes the proof of the theorem.
To study the estimate of the solution to problem (2.20), we first give the following proposition for the representation of the solution.
with positive constants C 1 and M 1 being given in (2.8). Moreover, there existsβ depending only on W 0 L ∞ , W 1 C 2 and the parameters of problem (1.7), such that Proof. When 0 ≤ t ≤ t 1 , the estimates (2.35) and (2.36) follow from (2.7) in Theorem 2.1 immediately. Assume that (2.35) holds for 0 ≤ t ≤ t i with i ≥ 1, and consider the case for t i ≤ t ≤ t i+1 . If i is even, from (2.7), we have by using the induction hypothesis. If i is odd, from (2.28) and (2.21) it follows by using the induction hypothesis again, or Thus, we conclude the estimate (2.35). Next, suppose that (2.36) holds for 0 ≤ t ≤ t i with i ≥ 1. Then if i is odd, from (2.7) in Theorem 2.1,we have that there existsβ depending on C 1 , b L ∞ and the parameters in the problem (1.7) such that  7), and satisfying C 0 ≥ C 0 with the constant C 0 being given in (1.10), such that ζ). Now, we study the L 1 estimate of the first order derivatives of the approximate solution with respect to the spatial variables for obtaining the uniform estimate on the total variation of the solution. Before it, we give the following two propositions for the problem (2.2) of transport equation. Proposition 2.5. For the problem (2.2), there exists a constant C 4 depending on the domain D, the constant C 0 given in (2.37), W 1 C 1 and the C 1 estimates of the parameter in the problem (2.2), such that for all t ∈ [t i , t i+1 ] and ζ ∈ (0, 1), (2.38) Proof. For the problem (2.2), we know that ( 1 W ) ξ satisfies (2.39) Taking (2.37) into account, we multiply the above equation , and integrate the resulting equation over D with respect to (ξ, η), (2.40) From (2.2), we obtain that on the boundary γ − , Obviously, there exist two bounded functions a 1 (ξ, η) and a 2 (ξ, η) defined on the boundary γ − such that Thus, from (2.41) one has Hence, it follows that by virtue of (2.42), where l(γ − ) is the length of γ − and the positive constant C 0 is given in (1.10).
By using (2.37), it follows where S(D) is the area of the domain D.
Plugging (2.43) and (2.44) into (2.40), we obtain that there exists a constant C 4 depending on D, C 0 , W 1 C 1 and the C 1 estimates of the parameter in the problem (2.2), such that Similarly, we can obtain another constant C 4 such that By letting C 4 = C 4 + C 4 , we have that from (2.45) and (2.46), Then, integrating the above inequality (2.47) over (t i , t) gives the estimate (2.38) immediately, and we complete the proof of the proposition. 2), such that for t ∈ [t i , t i+1 ] and ζ ∈ (0, 1), (2.48) Proof. From the problem (2.2), we know that W ζ satisfies Multiplying the above equation (2.49) by signW ζ or , and integrating over D with respect to (ξ, η), it follows that (2.50) As in (2.43), we have Obviously, from the bounded estimate (2.37) for W one has Plugging (2.51), (2.52) and (2.53) into (2.50), we obtain that there exists a constant C 5 depending on D, C 0 , W 1 C 1 and the C 1 estimates of the parameter in problem (2.2), such that Then, integrating the above inequality (2.54) over (t i , t), the estimate (2.48) in the proposition follows immediately. It is ready to give the L 1 estimate of the first order derivatives of the approximate solution W constructed in (2.1)-(2.2) with respect to the spatial variables.  Proof. The proof is divided into the following three steps.
(1) When t ∈ (t i , t i+1 ] for even i, W is determined by the initial boundary value problem (2.1) for a porous medium-type equation. From Theorem 2.1 and the boundedness (2.37) of W, we obtain that for t ∈ (t i , t i+1 ], (2.56) Moreover, there exists a constant C 3 depending on the domain D, the constant C 0 given in (2.37), W 1 C 3 and the C 1 estimates of the parameter in problem (1.7), such that which implies that by using the Gronwall inequality, (2.57) Combining (2.56) with (2.57), it follows that dξdη. (2.58) (2) When t ∈ (t i , t i+1 ] for odd i, we obtain W by the problem (2.2) for a transport equation. From Propositions 2.5, 2.6 and the estimate (2.37), it follows that there exists a constant C 6 , depending on the domain D, the constant C 0 given in (2.37), W 1 C 3 and the C 1 estimates of the parameter in problem (1.7), such that for t ∈ (t i , t i+1 ], which implies that by using the Gronwall inequality, (2.59) (3) On the other hand, when i is odd we have that from Proposition 2.2, By combining (2.58) , (2.59) and (2.60), and letting C 7 = max{ C 3 , C 6 }, we obtain that for any t ∈ (t i , t i+1 ], for even i, and G i (t) = 0 for odd i. Hence, Therefore, (2.61) implies that Note that W 0 | ζ=0 > 0, then where [ i 2 ] is the largest integer less than or equal to i 2 , and .

By choosing a constant
from which the estimate (2.55) follows immediately by using (2.37). Thus, we complete the proof of the lemma.
We are now ready to give the proof of the existence of weak solution as follows.
Proof of Theorem 1.1. Denote by W n (t, ξ, η, ζ) the approximate solution constructed in (2.1)-(2.2). Let Y be the dual space of H 2 0 (Ω). First, we claim that with a constant C 9 independent of n. Indeed, if i is even, we have that from (2.1), with a uniform constant C 10 depending only on D, A C 1 and b L ∞ . If i is odd, it follows that by using (2.2), with a uniform constant C 11 depending only in D, U C 1 and B − b L ∞ . Thus, by using (2.37), the estimate (2.66) follows immediately. Next, from (2.37) and Lemma 2.8, we know that where C 12 is a positive constant independent of n. Hence, by using the Lions-Aubin Lemma (see [2] for instance), we conclude that (1−ζ) 2 W n is compact in L 2 (0, T ) × Ω . Therefore, we may assume that and then W n → W, a.e. in (0, T ) × Ω.
Thus, for any ψ ∈ C ∞ 0 (0, T ) × Ω , we have that  (2.69) Therefore, from (2.69) we know that the function W satisfies the equation of problem (1.7) in the sense of distribution. Moreover, we can obtain that W satisfies the estimate (2.55) by letting n → ∞, which implies that W ∈ L ∞ 0, T ; BV (Ω) .
We can verify the other boundary conditions in (1.7) for W in the sense of distribution, respectively, through similar process as above to show that W satisfies the equation of (1.7) in the sense of distribution. For example, we have the following