The van Dommelen and Shen singularity in the Prandtl equations

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1 The van Dommelen and Shen singularit in the Prandtl equations Igor Kukavica, Vlad Vicol, and Fei Wang ABSTRACT. In 198, van Dommelen and Shen provided a numerical simulation that predicts the spontaneous generation of a singularit in the Prandtl boundar laer equations from a smooth initial datum, for a nontrivial Euler background. In this paper we provide a proof of this numerical conjecture b rigorousl establishing the finite time blowup of the boundar laer thickness. March 1, Introduction We consider the 2D Prandtl boundar laer equations for the unknown velocit field (u, v) = (u(t, x, ), v(t, x, )): t u u + u x u + v u = x P E (1.1) x u + v = (1.2) u = = v = = (1.3) u = U E. (1.4) The domain we consider is T R + = {(x, ) T R: }, with corresponding periodic boundar conditions in x for all functions. The function U E = U E (t, x) is the trace at = of the tangential component of the underling Euler velocit field (u E, v E ) = (u E (t, x, ), v E (t, x, )), and P E = P E (t, x) is the trace at = of the Euler pressure p E = p E (t, x, ). The obe the Bernoulli equation t U E + U E x U E = x P E (1.5) for x T and t, with periodic boundar conditions. The goal of this paper is to prove the formation of finite time singularities in the Prandtl boundar laer equations when the underling Euler flow is not trivial, i.e., when U E. For this purpose, we consider the Euler trace U E = κ sin x (1.6) x P E = κ2 2 sin(2x) (1.7) proposed b van Dommelen and Shen in [vds8], where κ is a fixed parameter. These are stationar solutions of the Bernoulli equation (1.5). Moreover, the functions U E and P E above 1

2 2 I. KUKAVICA, V. VICOL, AND F. WANG arise as traces at = of the stationar 2D Euler solution u E (x, ) = κ sin x cos v E (x, ) = κ cos x sin p E (x, ) = κ2 (cos 2x + cos 2). 4 It is clear that the function (u E, v E ) described above is divergence free, obes the boundar condition v(x, ) =, and ields a stationar solution of the Euler equations in T R +. REMARK 1.1 (Numerical blowup is observed). The Lagrangian computation of van Dommelen and Shen [vds8] was revisited and improved b man groups in the past decades [Cow83, CSW96, HH3, GSS9, GSSC14]. The consensus is that all the numerical experiments indi- SEPARATING BOUNDARY LAYERS 127 t-o t-4 FIG. I. The distortion of a tpical Lagrangian grid with time FIGURE 1. The distortion of a tpical Lagrangian grid with time. The figure is from in [vds8, p. 127]. do blow up, but in the Lagrangian description those balancing large terms are replaced b a single time derivative. Therefore, as will be substantiated b the numerical results to be presented, the solution is better behaved in Lagrangian coordinates than in Eulerian ones. The present work extends the earlier work b van Dommelen and Shen ] 131, in which the same case as presented here was calculated, but with different initial data, cate a singularit formation in finite time from a smooth initial datum. We refer to the recent paper [CGSS15, Section 4.2] for a detailed discussion of the numerical singularit formation in the Prandtl sstem. The following is the main result of this paper: u(x,, ) =f() sin x, (5) where f() is the Hiemenz velocit profile [ 11, instead of the step function initial data of Eq. (3). From [ 131 we borrow Fig. 1, as it gives a beautiful picture of how the Lagrangian grid distorts with time and consequentl, like a geometrical mapping, THEOREM 1.2 (Finite time blowup for Prandtl). Consider the Cauch problem for the Prandtl equations (1.1) (1.4), with boundar conditions at = matching (1.6) (1.7), with κ. There exists a large class of initial conditions (u, v ) which are real-analtic in x and, such that the unique real-analtic solution (u, v) to (1.1) (1.4) blows up in finite time.

3 THE VAN DOMMELEN AND SHEN SINGULARITY FOR PRANDTL 3 REMARK 1.3 (Inviscid limit). For a real-analtic initial datum in the Navier-Stokes, Euler, and Prandtl equations, it was shown in [SC98b] that the inviscid limit of the Navier-Stokes equations is described to the leading order b the Euler solution outside of a boundar laer of thickness νt, and b the Prandtl solution inside the boundar laer (see also [Mae14] for initial vorticit supported awa from the boundar). Indeed, if an series expansion (in ν) of the Navier-Stokes solution holds, then the leading order term near the boundar must be given b the Prandtl solution. The result in [SC98b] states that the inviscid limit holds on a time interval on which the Prandtl solution does not lose real-analticit. Our result in Theorem 1.2 shows that this time interval cannot be extended to be arbitraril large, and thus the Prandtl expansion approach to the inviscid limit should onl be expected to hold on finite time intervals. REMARK 1.4 (The case κ = ). In the case of a trivial Euler flow U E and a trivial Euler pressure P E, i.e., for κ = in (1.6) (1.7), the emergence of a finite time singularit for the Prandtl equations was established in [EE97]. There, the initial datum is taken to have compact support in and be large in a certain nonlinear sense. It is shown that either the solution ceases to be smooth, or that the solution along with its derivatives does not deca sufficientl fast as. The proof given in [EE97] does not appear to handle the case κ treated in this paper. Indeed, here the initial datum does not need to have compact support in and the pressure gradient is not trivial. Moreover, it is shown in [GSS9, Appendix] that from the numerical point of view the structure of the singularit for κ = is different from the complex structure of the singularities in [vds8]. REMARK 1.5 (Boundar laer separation and the displacement thickness). The displacement thickness (cf. [Sch6, vds8, CM7]) is defined as ( ) δ u(t, x, ) ( ) u(t, x, ) (t, x) = 1 U E d = 1 d. (1.8) (t, x) κ sin(x) Phsicall, it measures the effect of the boundar laer on the inviscid flow [CM7]. As long as 138 VAN DOMMELEN AND SHEN 1-7-t x FIG. 11. The variation of the displacement thickness with x, for various instants, FIGURE 2. The variation of the displacement thickness δ (t, x) for various time TABLE II instants. The figure is from [vds8, p. 138]. Computed Values of Several Variables for Various Gridsizes Gridsize 19 x 9 37x x x 65 s at min(lgrad xi) for T= u at min(lgrad.ui) for T= p.274 T at separation T for zero wall shear at.y = R

4 4 I. KUKAVICA, V. VICOL, AND F. WANG δ (t, x) remains bounded, the Prandtl laer remains of thickness proportional to ν, i.e., it remains a Prandtl laer. In turn, if the displacement thickness develops a singularit in finite time, this signals that a boundar laer separation has occurred, and after this point, the Prandtl expansion is not expected to hold anmore (see also[gre, GGN14c, GGN14b, GGN14a]). For a proof of a boundar laer separation for the stationar Prandtl equations, we refer to the recent paper [DM15]. The proof of Theorem 1.2 consists of showing that a Lapunov functional G(t) blows up in finite time. The functional G is defined b G(t) = (κφ(t, ) x u(t,, )) w()d where w() is an L 1 weight function and φ(t, ) is the solution of a nonhomogenous heat equation (see (2.11) and (2.26) below for details). We note that the Lapunov functional was built to emulate a weighted in version of κδ (t, x) x=. Indeed, as long as u is smooth near x =, we have that κδ (t, ) = lim x κδ (t, x) = (κ x u(t,, )) d. REMARK 1.6 (More general Euler flows). The blowup result of Theorem 1.2 holds if the Euler flow defined (1.6) (1.7) is replaced b an smooth and odd function U E (x), upon defining P E to equal (U E (x)) 2 /2, which is in turn even in x. REMARK 1.7 (More general classes of initial conditions). We note that besides ielding the local existence of solutions (cf. [SC98a]), the analticit of the initial datum is not required for proving Theorem 1.2. One ma instead consider an initial datum that is merel Sobolev smooth with respect to, analtic with respect to x, and decas sufficientl fast as (cf. [CLS1, KV13, IV15]). Alternativel Gevre-class 7/4 regularit in x ma be considered [GVM13], whose vorticit decas sufficientl fast with respect to. On the other hand, in view of the oddness in x of the boundar condition (1.6), we cannot consider initial datum which is in the Oleinik class of monotone in flows (uniforml with respect to x), although the local existence holds in this class [Ole66, MW15, AWXY15]. The datum in [XZ15] is also not allowed in view of the boundar conditions (1.6). The mixed analticit near x =, and monotonicit awa from the -axis (of different signs) ma however be treated, using the local existence result in [KMVW14]. REMARK 1.8 (Ill-posedness for the Prandtl equations). The ill-posedness of the Prandtl equations was established at the linear level in [GVD1], and at the nonlinear level in [Gre, GN11, GVN12]. We note that these results do not impl the finite time blowup from a given initial datum. REMARK 1.9 (Finite time blowup for the hrdostatic Euler equations). Here we note certain ver interesting blowup results [CINT13, Won15] for the hdrostatic Euler equations (which has man analogies with the Prandtl equations). These equations are set in a finite strip Ω = R [, h], and have different boundar conditions (onl v = is imposed at the top and bottom boundaries). For these equations the local existence was established for convex [Bre99, MW12] or analtic [KTVZ11] data, as well as for the combination thereof [KMVW14]. These equations are at the same time severel unstable (i.e., ill-posed in Sobolev spaces) if convexit or analticit is absent [Ren9]. In [CINT13] and [Won15] the finite blowup of odd solutions is established, b observing the behavior of u x = w z. The main difference with [EE97] is the presence of the pressure, which is quadratic in u (cf. [CINT13] for further details). The paper is organized as follows. In Section 2 we give the proof of Theorem 1.2 assuming certain properties of a boundar condition lift φ = φ(t, ) and of a weight function w = w(). Sections 3 and 4 are devoted to establishing these properties of φ and w respectivel.

5 THE VAN DOMMELEN AND SHEN SINGULARITY FOR PRANDTL 5 2. Proof of Theorem Local existence. As initial datum for the Prandtl equation we consider ( u (x, ) = κerf sin x + ū (x, ) (2.1) 2) where ū (x, ) is a real-analtic function of x and which is also odd with respect to x, and Erf(z) = 2π 1/2 z exp( z2 )d z is the Gauss error function. We assume that ū decas sufficientl fast (at least at an integrable rates) as, and takes the value at =. Moreover, we assume that obes and that G = a () = ( x ū )(, ) a () > for all > a ()w()d C κ,w (2.2) where the weight w is as constructed in Section 4, and C κ,w 1 is a constant that depends on κ and on the L 1 (R + ) weight function w. For instance, we ma consider ū (x, ) = A 2 exp( 2 ) sin x (2.3) where A = A(κ, w) > is a sufficientl large constant. This choice for ū (x, ) ields that a () is a large constant multiple of ϕ() = 2 exp( 2 ). The local in time existence of a unique realanaltic solution of the Prandtl sstem (1.1) (1.7) with initial datum given b (2.1) (2.3) follows from [SC98a]. We note though that the real-analticit is not needed in the blowup proof. It is onl used to ensure that we have the local in time existence and uniqueness of smooth solutions. Much more general classes of initial conditions ū ma be considered as long as the are odd with respect to x, the Cauch problem is locall well-posed (cf. Remark 1.7 above), and (2.2) holds Restriction of Prandtl dnamics on the -axis. Consider an initial datum u (x, ) for the Prandtl equations that is odd in x (such as the one defined in (2.1)). Note that the boundar condition at = is homogenous and thus automaticall odd in x, the boundar condition at = given b the Euler trace in (1.6) is also odd in x, and the derivative of the Euler pressure trace (1.7) is odd in x as well. Therefore, the unique classical solution u(t, x, ) of (1.1) (1.7) is also odd in x. Hence, as long as the solution remains smooth we have u(t,, ) = ( u)(t,, ) = ( 2 xu)(t,, ) =. (2.4) Phsicall, this smmetr freezes the Lagrangian paths emanating from the -axis, introducing a stable stagnation point in the flow. As in [EE97] (see also [CINT13, Won15]) this allows one to consider the dnamics obeed b the tangential derivative of u at x =, i.e., b(t, ) = ( x u)(t, x, ) x=.

6 6 I. KUKAVICA, V. VICOL, AND F. WANG As long as the solution remains smooth (so that we ma take traces at x = ), using (2.4) one derives that the equation obeed b b(t, ) is t b b b b b = κ 2 (2.5) b = = (2.6) b = κ. (2.7) In (2.5) and throughout the paper, we denote the integration with respect to the vertical variable as 1 ϕ(t, ) = ϕ(t, )d for an function ϕ(t, ) which is integrable in. In order to obtain (2.5), one applies x to (1.1) and then evaluates the resulting equation on the -axis. Similarl, (2.7) follows upon taking a derivative with respect to x of (1.6) and setting x = A shift of the boundar conditions. In order to homogenize the boundar condition at = when t =, we add to b a lift φ(t, ) defined as the solution of the nonhomogenous heat equation t φ φ = κ 2 (2.8) φ = = (2.9) φ = κ + κ 2 t (2.1) with an initial datum that we ma choose, as long as it obes compatible boundar conditions. We consider ( φ () = κerf, 2) so that the solution of (2.8) (2.1) is explicit ( ) φ(t, ) = κerf 4(t + 1) ( ( ( ) ) ( ( ) + κ 2 2 t Erf 1 + Erf + ))) exp ( 2. (2.11) 2t 4t 4t πt 4t In Section 3 we prove a number of properties (such as φ and that φ ) of the function φ defined in (2.11). Letting the sstem (2.5) (2.7) becomes a(t, ) = b(t, ) + φ(t, ) (2.12) t a a = (a φ) 2 1 (a φ) (a φ) (2.13) a = = (2.14) a = κ 2 t. (2.15) The equation (2.13) is similar to the one obtained in [EE97] for κ =, except for two additional terms on the right side: a forcing term F (t, ) = φ(t, ) 2 1 φ(t, ) φ(t, ) (2.16)

7 and a linear term THE VAN DOMMELEN AND SHEN SINGULARITY FOR PRANDTL 7 L[a](t, ) = 2a(t, )φ(t, ) + 1 φ(t, ) a(t, ) + 1 a(t, ) φ(t, ). (2.17) The forcing term is explicit in view of (2.11), while the linear operator L[a] has nice coefficients given in terms of φ. With the notation (2.16) (2.17), the evolution equation for a becomes t a a = a 2 1 a a + L[a] + F (2.18) a = = (2.19) a = κ 2 t. (2.2) In order to prove Theorem 1.2, we show that the solution of (2.18) (2.2) blows up in finite time from a ver large class of smooth initial data a Minimum principle. The main purpose of shifting the function b up b φ is so that the resulting function a obes a positivit principle. LEMMA 2.1. Assume that a = a(, ) is such that a () > for all (, ). Consider a smooth solution a(t, ) of the initial value problem associated with (2.18) (2.2) and initial condition a, on a time interval [, T ]. Then we have that a(t, ) for all and t [, T ]. Before proving Lemma 2.1, we need to establish certain positivit properties concerning the function φ. LEMMA 2.2. Let φ(t, ) be as defined in (2.11) with κ >. Then we have that φ(t, ) (2.21) φ(t, ) C κ (1 + t) (2.22) φ(t, ) (2.23) φ(t, ) (2.24) for all t and all, where C κ > is a constant that depends onl on κ. Moreover, the inequalit in (2.21) is strict for >. The proof of Lemma 2.2 is given in Section 3 below. PROOF OF LEMMA 2.1. We argue b contradiction. Since the solution is classical on [, T ] and decas sufficientl fast as, in order to reach a strictl negative value in [, T ] (, ) there must exist a first time t and an interior point >, such that a(t, ) = a(t, ) for all R + ( t a)(t, ). As is classical for the heat equation the contradiction arises b computing the time derivative of a at the point (t, ) and showing that it is strictl positive, contradicting the minimalit of t. In order to bound ( t a)(t, ) from below we use (2.18). Since a(t, ) has a global minimum at the interior point, we have ( a)(t, ) = ( a)(t, ). Since b assumption a(t, ) =, it follows that ( a + a 2 1 a a)(t, ). Moreover, since φ is a non-negative non-decreasing function, we immediatel obtain from (2.17) that

8 8 I. KUKAVICA, V. VICOL, AND F. WANG L[a](t, ). We conclude the proof b showing that F (t, ) >. Indeed, b (2.21) and (2.24) we have that and thus F (t, ) = F = φ φ 1 φ φ φ φ = 1 2 (φ 2 ) F (t, ȳ)dȳ 1 2 (φ 2 )(t, ȳ)dȳ = 1 2 (φ(t, ))2 > (2.25) whenever >, in view of Lemma 2.2. In order to full justif this argument, we appl the proof to ã(t, ) = a(t, ) + ε and show that ã(t, ) remains non-negative for ever ε >. The latter requires the additional observation that L[ε](t, ) = 2εφ(t, ) + ε φ(t, ) = ε (ȳ φ(t, ȳ) φ(t, ȳ)) dȳ in view of Lemma 2.2. This concludes the proof of the minimum principle Blowup of a Lapunov functional. Motivated b the displacement thickness (cf. (1.8)) we consider the evolution of the weighted average of a(t, ) on R +. For a suitable weight w() to be defined below and a non-negative solution a of (2.18) (2.2), we define the Lapunov functional G(t) = a(t, )w()d. (2.26) Note that since a as long as a remains smooth (cf. Lemma 2.1) we have that G(t) for all t. Our goal is to establish an inequalit of the tpe dg dt 1 C G2 C(1 + t)(1 + G) (2.27) for a constant C 1. Choosing a suitable initial datum, we then conclude that G blows up in finite time. The first step is to present the properties of the weight w in (2.26) which are needed in the proof of (2.27) Properties of the weight function w. We consider a weight function w() such that: w W 2, (R + ) w w = = w = w L 1 (R + ). The weight is given b glueing two functions f and g, i.e., { f(), Q, w() = g(), Q,

9 where Q > ; the function f is such that THE VAN DOMMELEN AND SHEN SINGULARITY FOR PRANDTL 9 f() = (2.28) f, on [, Q] (2.29) f, on [, Q] (2.3) f () c f f(), for all [, Q], where c f > (2.31) f () c f f(), for all [, Q], where c f > (2.32) and the function g, which we extend to be defined on [M, ) for some M (, Q) obes lim g() = = lim g () (2.33) g() >, for all M (2.34) g () <, for all M (2.35) g () >, for all M (2.36) g () 2 g()g β < 1, for all Q. (2.37) () Also let ψ(z) be a smooth non-decreasing cutoff function such that ψ(z) = for all z, ψ(z) = 1 for all z 1, and ψ (z) 2 for all (, 1). Define the cutoff function ( ) M η() = ψ. (2.38) Q M Note that η vanishes identicall on [, M] and equals 1 on [Q, ). Its derivative localizes to [M, Q] and obes η 2 () Q M for all (M, Q). Lastl, we require that the functions f and g obe the compatibilit conditions g () 2 η() f()g β < 1, for all (M, Q) (2.39) () 2η () g () η()g () f (), for all (M, Q). (2.4) The construction of two functions f and g that obe the properties (2.28) (2.4) listed above is provided in Section 4. A sketch of the graph of the resulting weight w() is given in Figure 3 below. Throughout paper, we shall denote derivatives of the functions w, f, g with primes, as the are onl functions of the variable Evolution of the Lapunov functional G. We use (2.18) (2.2) and the boundar values of w, given b (2.28) and (2.33), to deduce d dt G = = ( a + a 2 1 aw d + 2 aw d + 2 a a + L[a] + F ) wd a 2 wd + a 2 wd 1 2 a 1 a w d + ( 1 a) 2 w d + L[a]wd + L[a]wd F wd = I 1 + 2I I 3 + I 4. (2.41)

10 1 I. KUKAVICA, V. VICOL, AND F. WANG W().2 g.15.1 f.5 ε M Q FIGURE 3. Graph of the weight function w(). The weight is a linear function on [, ε], a quadratic function on [ε, Q], and an shifted negative power of on [Q, ). The above integrations b parts are justified b the fact that a and w are sufficientl smooth, w obes the Dirichlet boundar conditions, a and 1 a vanish at =, while w vanishes as. Here we have used that b (2.25) we have F (t, ), which combined with w() shows that the forcing is non-negative. We now bound each of the terms in (2.41) separatel Bound for I 1. To bound I 1 we use the convexit of w on [Q, ) (cf. (2.36)) and the fact that (2.31) holds. We deduce that I 1 Q Q af c f af c f G. (2.42) Bound for I 2. B the Cauch-Schwartz inequalit, and denoting c 1 = w L 1 (R + ) <, (2.43) we obtain and thus G = awd = a w wd a w L 2 w L 2 = c 1/2 1 I 1/2 2 I 2 1 c 1 G 2. (2.44)

11 THE VAN DOMMELEN AND SHEN SINGULARITY FOR PRANDTL Bound for I 3. We proceed to bound the difficult term I 3. Let η() be the cutoff function defined in (2.38). First we have I 3 = = = J + ( 1 a) 2 w η( 1 a) 2 w + η( 1 a) 2 g + Q M Q M (1 η)( 1 a) 2 f η( 1 a) 2 (f g ) + Q M (1 η)( 1 a) 2 f ( 1 a) 2 (f ηg ) (2.45) where we have used (2.3) in the second to last step to bound M (1 η)( 1 a) 2 f. The term J above is the leading term and it shall be bounded using integration b parts, which is justified since g vanishes as. Since 1 a =, we obtain from (2.33) and (2.35) that J = = 2 = 2 η( 1 a) 2 g ηa( 1 a)g ηa( 1 a) g + η( 1 a) 2 g η( 1 a) 2 g. Further, recalling that η is supported on [M, ), b appealing to (2.37) and (2.39) we get ( ) 1/2 ( J 2 η( 1 a) 2 g ηa 2 (g ) 2 ) 1/2 g + η ( 1 a) 2 g 2 ( ) 1/2 ( 1/2 β η( 1 a) 2 g a w) 2 + η ( 1 a) 2 g = 2 βj 1/2 I 1/2 2 + η ( 1 a) 2 g. Using the inequalit 2x x 2 / , we further estimate which in turn ields J J 2 + 2βI 2 + J 4βI Combining (2.45) and (2.46), we obtain I 3 4βI Q M η ( 1 a) 2 g + B appealing to (2.4) we then arrive at the bound η ( 1 a) 2 g η ( 1 a) 2 g. (2.46) Q M ( 1 a) 2 (f ηg ). (2.47) I 3 4βI 2 (2.48)

12 12 I. KUKAVICA, V. VICOL, AND F. WANG which is a convenient estimate when β < Bound for I 4. Consider I 4 = L[a]w = 2 a φ w + a 1 φ w + φ 1 a w. (2.49) Using that (2.21) (2.23) hold (cf. Lemma 2.2), upon integrating b parts in the second term on the far right side of (2.49), we have that I 4 3 a φ w 3C κ (1 + t)g Q a 1 φ w a 1 φ (w ) + where (w ) + = max{w, }. In the last step we used that w is decreasing on [Q, ). Since the bounds (2.29), (2.32), and (2.22) hold on [, Q], we arrive at for [, Q]. We thus obtain 1 φ(w ) + φ(q)(w ) + C κ (1 + t) c f w I 4 (3 + c f )C κ (1 + t)g. (2.5) The lower bound for the growth of the Lapunov functional. Combining (2.41), (2.42), (2.44), (2.48), and (2.5), with the assumption that β < 1, we arrive at d dt G c f G + 2I 2 (1 β) (3 + c f )C κ (1 + t)g (c f + (3 + c f )C κ )(1 + t)g + 2(1 β) c 1 G 2 C κ,w (1 + t)g + 1 C κ,w G 2 (2.51) for some sufficientl large positive constant C κ,w 1, which onl depends on the choice of κ and the weight w. Note that β < 1 is essential here Conclusion of the proof of Theorem 1.2. Therefore, if we ensure that G = G t= is sufficientl large, the solution G(t) of (2.51) blows up in finite time. Quantitativel, it is sufficient to let G 4C 2 κ,w. (2.52) The condition (2.52) ma be achieved b a smooth initial datum. For instance we ma let a () = a(t, ) = be given b a large amplitude Gaussian bump, i.e., a () = Aϕ(), where ϕ is as in (2.3) above, and A > is sufficientl large. In view of Remark 1.7, more general classes of functions ϕ() ma be considered, including those with compact support in (cf. [CLS1, KV13]).

13 THE VAN DOMMELEN AND SHEN SINGULARITY FOR PRANDTL Properties of the boundar condition lift φ It is eas to verif that the function φ defined in (2.11), i.e., ( ) φ(t, ) = κerf 4(t + 1) ( + κ 2 2 t 2t ( = κerf ( ( ) ) ( ( ) Erf 1 + Erf + exp 4t 4t πt ) 4(t + 1) ))) ( 2 4t ( ( + κ 2 t 2z(t, ) 2 2z(t, ) (Erf (z(t, )) 1) + Erf (z(t, )) + exp ( z(t, ) 2))) π obes the non homogenous heat equation (2.8) (2.1), with initial value φ () = κerf(/2), where z(t, ) = / 4t is the heat self-similar variable. (3.1) 3.1. Proof of Lemma 2.2. PROOF OF (2.23). First note that for t > and >, the function φ obes the heat equation, i.e., ( t )( φ) =. Using the exact formula (2.11) we obtain the initial and boundar values for the quantit φ. Taking the derivative of φ gives φ = κ ( ( ( ) ) exp ( )+κ 2 2 Erf exp π t 4 t 4t 4πt ( 2 + t exp πt where t = t + 1. Sending and we obtain φ = = φ = = for all t >. Taking the limit t, we arrive at φ t= = κ + 2κ2 t > π t π κ ) exp ( 2. π 4 ( 2 ) ( 2 2 4t 2t πt exp ) 4t ( 2 4t )) ) The fact φ(t, ) for t, now follows from the parabolic maximum principle. For the sake of completeness we repeat this classical argument. We consider the nonnegative C 2 function { x 4, x ρ(x) =, x.

14 14 I. KUKAVICA, V. VICOL, AND F. WANG Taking the t derivative of the quantit ρ( φ(t, ))d, upon integrating b parts in and using the boundar conditions for φ we arrive at t ρ( φ) d = = = ρ ( φ) t φ d = ρ ( φ) 3 φ d ρ ( φ)( 2 φ) 2 d ρ ( φ = ) φ = ρ ( φ)( 2 φ) 2 d, from where we deduce that ρ( φ(t, )) d = for t since ρ( φ ()) d =. Therefore, we obtain φ(t, ), concluding the proof. PROOF OF (2.21). Since φ(t, ) =, for all t >, the non-negativit of φ follows from the fundamental theorem of calculus and the above established monotonicit propert φ. PROOF OF (2.22). The proof follows from (3.1) since 2z 2 (Erf (z) 1) + Erf(z) + 2 π z exp( z 2 ) C for all z, for some universal constant C >. We ma then take C κ = max{κ, C κ 2 }. PROOF OF (2.24). Taking the second derivative of φ defined in formula (2.11), we arrive at ( ) ( ( ) ) ) κ φ = exp 2 2 κ 2 t 4π t 3 4 t exp 2 ( 3 πt 3 4t 4 πt exp 2 5 4t + κ (Erf ) ) ) exp ( 2 ( 3 4t πt 4t 4 πt exp 2. (3.2) 3 4t From this expression we get the initial and boundar values for φ as φ = = κ 2, for t > φ =, for t >, φ t= = κ ) ( 2 π exp 2, for >. 4 An argument similar to the one above shows that b the parabolic maximum principle we have φ(t, ) for all t,. 4. Construction of a weight function w for the Lapunov functional We fix Q = 1 and let r > 1 be a free parameter, to be chosen below. In terms of this r we shall pick 1/2 < M = M(r) < 1, < β = β(r) < 1, and B = B(r) > 1 so that the conditions (2.28) (2.4) hold. Define the function f b Therefore, f() = and f() = 2B + r B r B + r B r 2 (4.1) f () = 2B + r B r 2(B + r) B r

15 with THE VAN DOMMELEN AND SHEN SINGULARITY FOR PRANDTL 15 f 2(B + r) () = B r. It thus follows that f W 2, ([, 1]), f, and f on (, 1), so that (2.28) (2.3) hold. Moreover, (2.32) holds with c f = 1. Note however that (2.31) does not hold in a neighborhood of the origin, since f() =. Instead, the function f defined in (4.1) needs to be modified in a small neighborhood near the origin so that it is linear there (see Remark 4.1 below). Next, we define g() = B ( + B 1) r set initiall for all 1, but which is a well-defined function on M as long as M + B > 1. Note that r > 1 implies that (2.43) holds. We have that and g rb () = ( + B 1) r+1 < g () = r(r + 1)B >. ( + B 1) r+2 Thus, the properties (2.33) (2.36) hold for this function g. Moreover, g () 2 g()g () = r r + 1 < 1 for all >. This verifies that condition (2.37) holds for an β [r/(r + 1), 1). Note that at = Q = 1 we have and f(1) = B 1 r = g(1) f (1) = rb r = g (1) and thus f and g ma be glued together at Q = 1 to ield a W 2, function on R. In order to assure that (2.39) holds, it is sufficient to verif that ( ) r M r + 1 ψ g() βf() (4.2) 1 M holds for all [M, 1], where r/(r + 1) < β < 1 is arbitrar. The condition (4.2) holds automaticall with if we ensure that β = 2r + 1 2r + 2 g() sup [M,1] f() 2r + 1. (4.3) 2r In view of the continuit of the above functions, (4.3) holds if we choose M (, 1) sufficientl close to 1. We need though to be more precise on this choice of M. Indeed, (4.3) holds for

16 16 I. KUKAVICA, V. VICOL, AND F. WANG [M, 1] if we impose that which is a consequence of B r+1 ( + B 1) r 2r + 1 ((2B + r) (B + r)) 2r B r+1 (M + B 1) r 2r + 1 MB. 2r Assuming that 1 M 1/(4r + 2), the above follows from which holds provided that The last condition ma be written as 1 (1 (1 M)/B) r 4r + 1 4r ( ) 4r 1/r 1 1 M 4r + 1 B. 1 M B ( 1 Therefore, (4.4) holds if we choose { ( 1 M = min 1 4r + 2, B 1 as long as 1 B (4r + 2) (1 ((4r)/(4r + 1)) ( ) ) 4r 1/r. (4.4) 4r + 1 ( ) )} 4r 1/r = 1 4r + 1 4r + 2 (4.5) 1/r). (4.6) This ensures the validit of the condition (4.2), and thus also (2.39) holds. We finall verif that (2.4) holds, or equivalentl ( ) ( ) 2 M rb M r(r + 1)B 1 M ψ 1 M ( + B 1) r+1 ψ 2(B + r) + 1 M ( + B 1) r+2 B r. (4.7) Since ψ 2 and φ, the above condition holds on [M, 1] once we ensure that which is implied b 4 rb 2(B + r) 1 M ( + B 1) r+1 B r B r+1 (1 M)(B + r). (M + B 1) r+1 2r The above condition holds if we take B sufficientl large, depending onl on r. More precisel, since M obes (4.5), letting B obe (4.6) and also ( ) 4r + 1 (r+1)/r B 2r(4r + 2), (4.8) 4r we complete the proof of (4.7).

17 THE VAN DOMMELEN AND SHEN SINGULARITY FOR PRANDTL 17 In summar, the conditions (2.28) (2.4), except for (2.31), are obeed once we set r = 2 β = 5 6 < 1 M = 9 1 < Q = 1 B = Condition (2.31). In order to ensure that (2.31) holds, we need to tweak the functions f and g defined above. Let < ε < 1/2 be a small parameter, to be determined. We then have f(ε) = 2B + r B r ε B + r B r ε 2 > f (ε) = 2B + r B r Therefore, we can extend f() b the linear function h ε () = f (ε)( ε) + f(ε) ( 2B + r 2(B + r) = B r B r ε on the interval [ ε, ε], where ) 2(B + r) B r ε >. ( 2B + r ( ε) + B r ε B + r ) B r ε 2 ε = f(ε) f (ε) ε. Note that h ε ( ε ) =, h ε (ε) = f(ε), and h ε(ε) = f (ε). Therefore, glueing h ε with f at = ε, and then f with g at = Q = 1, ields a W 2, function h ε (), [ ε, ε] w ε () = f(), [ε, 1] g(), 1 which obes all the properties (2.28) (2.4), but on the interval [ ε, ) instead of [, ). Here we used that ε is sufficientl small so that ε M/4. Also, (2.31) holds triviall on [ ε, ε) since on this interval h ε() = h ε (). Then in view of the continuit of f and f, and the fact that f onl vanishes at =, on the compact [ε, 1] we have that f () /f() c f for some constant c f 1. Therefore, to complete the construction of the weight function w, let w() = w ε ( + ε ) for a fixed < ε 1, where the values of Q and M are themselves shifted b ε. In view of the smoothness of all parameters on ε this is possible without affecting (2.37) (2.39), upon slightl increasing the value of β (so that it remains < 1). Acknowledgments IK and FW were supported in part b the NSF grant DMS , while VV was supported in part b the NSF grant DMS and b an Alfred P. Sloan Research Fellowship.

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19 THE VAN DOMMELEN AND SHEN SINGULARITY FOR PRANDTL 19 [MW12] [MW15] [Ole66] [Ren9] [SC98a] [SC98b] [Sch6] [vds8] [Won15] [XZ15] N. Masmoudi and T.K. Wong. On the H s theor of hdrostatic Euler equations. Arch. Ration. Mech. Anal., 24(1): , 212. N. Masmoudi and T.K. Wong. Local-in-time existence and uniqueness of solutions to the Prandtl equations b energ methods. Comm. Pure Appl. Math., 68 (215), O.A. Oleĭnik. On the mathematical theor of boundar laer for an unstead flow of incompressible fluid. J. Appl. Math. Mech., 3: (1967), M. Renard. Ill-posedness of the hdrostatic Euler and Navier-Stokes equations. Arch. Ration. Mech. Anal., 194(3): , 29. M. Sammartino and R.E. Caflisch. Zero viscosit limit for analtic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations. Comm. Math. Phs., 192(2): , M. Sammartino and R.E. Caflisch. Zero viscosit limit for analtic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution. Comm. Math. Phs., 192(2): , H. Schlichting. Boundar laer theor. Translated b J. Kestin. 4th ed. McGraw-Hill Series in Mechanical Engineering. McGraw-Hill Book Co., Inc., New York, 196. L.L. van Dommelen and S.F. Shen. The spontaneous generation of the singularit in a separating laminar boundar laer. J. Comput. Phs., 38(2):125 14, 198. T.K. Wong. Blowup of solutions of the hdrostatic Euler equations. Proc. Amer. Math. Soc., to appear., 143(3): , 215. C.-J. Xu and X. Zhang. Well-posedness of the Prandtl equation in Sobolev space without monotonicit. arxiv: , DEPARTMENT OF MATHEMATICS, UNIVERSITY OF SOUTHERN CALIFORNIA, LOS ANGELES, CA 989 address: kukavica@usc.edu DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NJ address: vvicol@math.princeton.edu DEPARTMENT OF MATHEMATICS, UNIVERSITY OF SOUTHERN CALIFORNIA, LOS ANGELES, CA 989 address: wang828@usc.edu