Remarks on the inviscid limit for the Navier-Stokes equations for uniformly bounded velocity fields

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1 Remarks on the inviscid it for the Navier-Stokes equations for uniformly bounded velocity fields Peter Constantin, Tarek Elgindi, Mihaela Ignatova, and Vlad Vicol ABSTRACT. We consider the vanishing viscosity it of the Navier-Stokes equations in a half sace, with Dirichlet boundary conditions. We rove that the inviscid it holds in the energy norm if the roduct of the comonents of the Navier-Stokes solutions are equicontinuous at x 2 =. A sufficient condition for this to hold is that the tangential Navier-Stokes velocity remains uniformly bounded and has a uniformly integrable tangential gradient near the boundary. June 16, 216 Consider the 2D Navier-Stokes equations 1. Introduction t u NS + u NS u NS + NS = ν u NS (1.1) u NS = (1.2) u NS 1 = u NS 2 = (1.3) with kinematic viscosity ν, in the half sace = {(x 1, x 2 ): x 2 > }, and the Euler equations with asymtotically matching initial conditions We denote by t u E + u E u E + E = (1.4) u E = (1.5) u E 2 = (1.6) ν uns u E L 2 () =. (1.7) u E 1 (x 1, t) = U E (x 1, t) the trace on of the Euler tangential flow. We omit ν in the notation for u NS. Throughout this aer we consider < ν ν, and t T, where ν is an arbitrary fixed kinematic viscosity, and T is an arbitrary fixed time. We assume that the Euler initial datum is smooth, u E s () for some s > 2, so that there exists an unique s smooth solution u E of (1.4) (1.6) on [, T ]. This aer establishes sufficient conditions for the family of Navier-Stokes solutions {u NS } ν (,ν ] to ensure that the inviscid it holds in the energy norm: ν uns u E L (,T ;L 2 ()) =. (1.8) Our main results are given in Theorems 1.1, 1.3, and 1.5. The main assumtions are either on the equicontinuity of the roduct u NS 1 uns 2 at x 2 =, or on the uniform boundedness of u NS 1 and the 1

2 2 P. CONSTANTIN, T. ELGINDI, M. IGNATOVA, AND V. VICOL uniform integrability of 1 u NS 1 near the boundary of the half sace. These conditions do not assume a riori any articular scaling law of the boundary layer with resect to ν. The conditions imosed imly that the Lagrangian aths originating in a boundary layer, stay in a roortional boundary layer during the time interval considered. The hysical interretation of our result is that, as long as there is no searation of the boundary layer, the inviscid it is ossible Known finite time, inviscid it results. The question of whether (1.8) holds in the case of Dirichlet boundary conditions has a rich history. Kato roved in [Kat84] that the inviscid it holds in the energy norm if and only if T ν ν x 2 Cν u NS (x 1, x 2, t) 2 dx 1 dx 2 dt =, (1.9) i.e. that the energy dissiation rate is vanishing in a thin, O(ν), layer near the boundary. Kato s criterion was revisited and sharened by many authors. For instance, in [TW97] and [Wan1] it is shown that the condition on the full gradient matrix u NS may be relaced by a condition on the tangential gradient of the Navier-Stokes solution alone, at the cost of considering a thicker boundary layer, of size δ(ν), where ν δ(ν)/ν =. In [Kel7] it is shown that ν u NS 2 L 2 ( x 2 Cν) may be relaced by ν 1 u NS 2 L 2 ( x 2 Cν) which has the same scaling in the Kato layer. In [Kel8] it is shown that (1.8) is equivalent to the weak convergence of vorticities ω NS ω E u E 1 µ in ( 1 ()) (1.1) where µ is the Dirac measure on, and ( 1 ) is the dual sace to 1 (not 1 ). In fact, it is shown in [BT7] that the weak convergence of vorticity on the boundary νω NS in D ([, T ] ) (1.11) is equivalent to (1.8) (see also [Kel8, CKV15] in the case of stronger convergence in (1.11)). The idea to introduce a boundary layer corrector like Kato s, which is not based on ower series exansions, and to treat the remainders with energy estimates has roven to be very fruitful. See for instance: [Mas98] in the case of anisotroic viscosity; [BSJW14, BTW12] in the context of weak-strong uniqueness; [GN14] for a steady flow on a moving late; [BN14] for the comressible Navier-Stokes equations; [LFNLTZ14] for the vanishing α it of the 2D Euler-α model. There are three classes of functions for which there exist unconditional inviscid it results, that is, theorems whereby conditions imosed solely on initial data guarantee that (1.8) is true for a time interval indeendent of viscosity (but ossibly deending on initial data). The first class is that of real analytic initial data in all sace variables [SC98b], the second is that of initial data with vorticity suorted at an O(1) distance from the boundary [Mae14], and the third class is data with certain symmetries or secial restrictions [LFMNL8, LFMNLT8, MT8, Kel9]). It is worth noting that in these three cases the Prandtl exansion of the Navier-Stokes equation is valid in a boundary layer of thickness ν. Moreover, in all these results, the Kato criteria also hold [BT7, Kel14]. owever, to date, there is no robust connection between the well-osedness of the Prandtl equations, and the vanishing viscosity it in the energy norm. It is known that for a class of initial conditions close to certain shear flows the Prandtl equations are ill-osed [GVD1, GN11, GVN12] and even that the Prandtl exansion is not valid [Gre, GGN14b, GGN14c, GGN14a]. These results do not imly that the inviscid it in the energy norm is invalid, but rather just that the Prandtl exansion does not describe the leading order behavior near the boundary. It would be natural to exect that working in a function sace for which the local existence of the Prandtl equations holds (see, e.g. [Ole66, MW14, AWXY14], [SC98a, LCS3, KV13], [KMVW14], [GVM13]), there is a greater chance for (1.8) to be true. An instance of such

3 INVISCID LIMIT OF NAVIER-STOKES FOR BOUNDED VELOCITY FIELDS 3 a result is given in [CKV15], where a one-sided Kato criterion in terms of the vorticity is obtained, connecting Oleinik s monotonicity assumtion and the inviscid it: if T ( ( ν U E (x 1, t) ω NS (x 1, x 2, t) + δ(νt) )) 2 νt dt = (1.12) L 2 ( x 2 νt/δ(νt)) holds, where T δ(νt)dt as ν, then (1.8) holds. In articular, if there is no back-flow in the underlying Euler flow, U E, and the Navier-Stokes vorticity ω NS is larger than δ(νt)/ν (for instance if it is non-negative as in Oleinik s setting) in a boundary layer that is slightly thicker than Kato s, then the inviscid it holds. In contrast to the works (1.9) (1.12) mentioned above, the goal of this aer is to establish sufficient conditions for (1.8) to hold, which do not rely on any assumtions concerning derivatives of the Navier-Stokes equations. Alternately, we establish conditions which require only L 1 uniform integrability of tangential derivatives near the boundary. Our roofs kee the idea of Kato of building an ad-hoc boundary layer corrector, but its scaling is dictated by the heat equation in x 2 (with Prandtl scaling). No exlicit convergence rates are obtained with our assumtions. The main results of this aer are: 1.2. Results. TEOREM 1.1. Assume that the family {u NS 1 u NS 2 } ν (,ν ] is equicontinuous at x 2 =. (1.13) Then (1.7) imlies that the inviscid it holds in the energy norm. Secifically, in view of the Dirichlet boundary condition (1.3), by condition (1.13) we mean that there exists a function γ(x 1, t) L 1 t,x 1 ([, T ] R) (1.14) with the roerty that for any ε >, there exists ρ = ρ(ε) > such that u NS 2 (x 1, x 2, t) εγ(x 1, t), for all x 2 (, ρ], (1.15) and all (t, x 1 ) [, T ] R, uniformly in ν (, ν ]. REMARK 1.2. We note that condition (1.13) holds for the solution of the Stokes equation (the linearization of the Navier-Stokes equations around the trivial flow (, )) and for the solution of the Oseen equations (the linearization of the Navier-Stokes equations around a stationary shear flow (U(y), )), if one considers sufficiently smooth and comatible initial datum. Indeed, the equicontinuity of the family {u 1 u 2 } ν (,ν ] at {x 2 = } follows (with an exlicit rate) for instance from the bound x2 u 1 (x 1, x 2, t)u 2 (x 1, x 2, t) u 1 (x 1, x 2, t) 1 u 1 (x 1, y, t) dy and the fact that both u 1 and k 1 u 1 obey the same equation for any k 1. That is, since 1 is a tangential vector field to, so that it commutes with the Stokes oerator, for initial datum that is comatible one may obtain the same good bounds for k 1 u 1 for all k (one may use the Ukai formula directly for the Stokes equation, or the argument in [TW95] for Oseen). The integrability (uniform in ν) of 1 u NS 1 and the boundedness (uniform in ν) of u NS 1 is thus related to condition (1.13):

4 4 P. CONSTANTIN, T. ELGINDI, M. IGNATOVA, AND V. VICOL TEOREM 1.3. Let C NS > be a constant. Assume that for any ε > there exists ρ >, such that T su 1 u NS 1 (t) 1 <x2 <ρ 2 L 1 () dt ε, (1.16) ν (,ν ] meaning that the tangential derivative of the tangential comonent of the Navier-Stokes flow is uniformly integrable near {x 2 = }, and that su ν (,ν ] T u NS 1 (t) 1 <x2 <ρ 2 L () dt C NSν, (1.17) meaning that the tangential comonent of the Navier-Stokes flow is bounded near {x 2 = }. Then (1.8) holds. The quantity in condition (1.17) is natural to consider: it is scale invariant under the Navier- Stokes isotroic scaling, and it aears in three dimensions as well. A similar quantity was used in [BSJW14] to establish conditional weak-strong uniqueness of weak solutions in ölder classes. Condition (1.16) requires that the family of measures µ ν (dx 1 dx 2 ) = 1 u NS 1 (t, x 1, x 2 ) dx 1 dx 2 is uniformly absolutely continuous at x 2 = with values in L 2 (, T ), i.e. that they do not assign uniformly bounded from below mass to a boundary layer. Note that 1 u NS 1 vanishes identically on, which is not the case for the Navier-Stokes vorticity ω NS = 2 u NS 1 1u NS 2, which is exected to develo a measure suorted on the boundary of the domain in the inviscid it [Kel8]. Thus, the vorticity is not exected to be uniformly integrable in L 2 t L 1 x. Therefore, in (1.16) it is imortant that instead of a uniform integrability condition on ω NS or equivalently 2 u NS 1, we have only assumed a uniform integrability condition on 1 u NS 1. Also, note that (uniform in ν) higher integrability of the Navier-Stokes vorticity, such as L for > 2 cannot hold unless U E, as is shown in [Kel14]. REMARK 1.4. From the roof of Theorem 1.3 it follows that in (1.16) (1.17) we may relace (both) L 1 x 1,x 2 () by L 2 x 1 L 1 x 2 () and L x 1,x 2 () with L 2 x 1 L x 2 (). That is, the boundedness of u NS 1 and the uniform integrability of 1 u NS 1 is to be only checked with resect to the x 2 variable. With resect to x 1 we only need that these functions are square integrable. Again, either of these conditions may be checked directly for the Stokes and Oseen equations if the initial datum is sufficiently smooth (with resect to x 1 ) and comatible. A similar result to the one in Theorem 1.3, has been obtained indeendently in [GKLF + 15], where the authors rove that if u NS is uniformly in ν bounded in L (, T ; L 1 (Ω)), for a domain Ω such that the embedding W 1,1 (Ω) L 2 is comact, then the vanishing viscosity it holds in L (, T ; L 2 (Ω)). Note moreover, that in [Wan1], the author establishes a Kato-tye criterion for the inviscid it to hold, under the assumtions that ν T x 2 δ(ν) 1u NS 1 2 dt as ν and ν/δ(ν) as ν. The modification to condition (1.16) of this aer, as described in Remark 1.4, is weaker than the condition of [Wan1], as may be seen by taking ρ = ν and alying the ölder inequality to bound L 1 x 2 with L 2 x 2. owever, we also need to imose condition (1.17). We conclude the introduction by noting that a similar roof to that of Theorem 1.1 yields: TEOREM 1.5. Let δ be an increasing non-negative function such that ν δ(νt) = uniformly for t [, T ], such that (νt)δ (νt)/δ(νt) is uniformly bounded for ν (, ν ] and t [, T ],

5 and such that INVISCID LIMIT OF NAVIER-STOKES FOR BOUNDED VELOCITY FIELDS 5 T ν ν δ(νt) + δ(νt) dt = (1.18) t1+c holds for some c >. We may for instance take δ(νt) = (νt) a for any a (, 1). Assume that holds for a.e. (t, x 1, y) [, T ], and that Then (1.8) holds. T ν uns 1 (x 1, δ(νt) y, t)u NS 2 (x 1, δ(νt) y, t) = (1.19) su u NS 1 (t)u NS 2 (t) L ({x 2 δ(νt)(log(1/ν)) 1/2 }) <. (1.2) ν (,1] 1.3. Organization of the aer. In Section 2 we lay out the scheme of the roof for the above mentioned theorems, by identifying the rincial error terms in the energy estimate for the corrected u NS u E flow. In Section 3 we build a caloric lift of the Euler boundary conditions, augmented by an O(1) correction at unit scale. In Section 4 we conclude the roof of Theorem 1.1, in Section 5 we give the roof of Theorem 1.3, while in Section 6 we show why Theorem 1.5 holds. 2. Setu of the Proof of Theorem 1.1 We consider a boundary layer corrector u K (to be constructed recisely later) which for now obeys three roerties u K = (2.1) u K 1 = U E (2.2) u K 2 =. (2.3) The main difference between the corrector u K we consider, and the one considered in [Kat84], is its characteristic length scale: we let u K obey a Prandtl νt scaling (see also [TW95, Gie14]). Roughly seaking, u K 1 is a lift of the Euler boundary condition which obeys the heat equation ( t ν x2 x 2 )u K 1 = to leading order in ν. In view of (2.1) (2.3) we then obtain uk 2 from uk 1 as The function is divergence free u K 2(x 1, x 2, t) = and obeys Dirichlet boundary conditions The equation obeyed by v is x2 v = u NS u E u K v = v =. t v ν v + v u E + u NS v + q 1 u K 1(x 1, y, t)dy. (2.4) = ν u E ( t u K ν u K + u NS u K + u K u E) (2.5)

6 6 P. CONSTANTIN, T. ELGINDI, M. IGNATOVA, AND V. VICOL where q = NS E. Multilying (2.5) with v and integrating by arts, yields where we have denoted 1 d 2 dt v 2 L + ν v 2 2 L u E 2 L v 2 L + ν u E 2 L 2 v L 2 + T 1 + T 2 + T 3 + T 4 + T 5 + T 6 (2.6) T 1 = T 2 = T 3 = T 4 = T 5 = T 6 = ( t u K ν u K ) v (2.7) (u NS u E ) u K (2.8) (u K u E ) v (2.9) u NS 1 u NS 2 1 u K 2 (2.1) ( (u NS 1 ) 2 (u NS 2 ) 2) 1 u K 1 (2.11) u NS 1 u NS 2 2 u K 1 (2.12) The corrector u K is designed to einate the contribution from T 1 to leading order in ν. In turn, this leads to u K L u K L 2 as ν, so that the terms T 2, T 3, T 4, and T 5 are harmless. Such is the case if u K is localized in a layer near the boundary, which is vanishing as ν. Assumtion (1.13) only comes into lay in showing that T 6 is bounded conveniently. The next section is devoted to the construction of an u K with these roerties, and the conclusion of the roof is given in Section 4 below. Throughout the text we shall denote by C E any constant that deends on u E L (,T ; s ()). Various other ositive constants shall be denoted by C; these constants do not deend on ν, but they are allowed to imlicitly deend on the fixed length of the time interval T, and on the largest kinematic viscosity ν. 3. A seudo-caloric lift of the boundary conditions The tye of corrector we construct here was also used for instance in [TW95, Gie14] (see also references therein) to address the vanishing viscosity it for the linear Stokes system with comatible, resectively non-comatible initial datum The tangential comonent of the lift u K. Let z = z(x 2, t) = x 2 be the self-similar variable for the heat equation in x 2, with viscosity ν. Let η be a non-negative bum function such that su(η) [1, 2] which in addition obeys that η L + η L C η, for some constant C η. and 2 1 η(r)dr = 1 π (3.1)

7 INVISCID LIMIT OF NAVIER-STOKES FOR BOUNDED VELOCITY FIELDS 7 We let u K 1 consist of a caloric lift of the Euler boundary conditions, augmented with a localization factor at large values of x 2. We define ( u K 1(x 1, x 2, t) = U E (x 1, t) erfc(z(x 2, t)) ) η(x 2 ) (3.2) where erfc(z) = 1 erf(z) = 2 ex( y 2 )dy. π The normalization of the mass of η was chosen recisely so that ( u K 1(x 1, x 2, t)dx 2 = U E (x 1, t) erfc(z(x 2, t)) ) η(x 2 ) dx 2 = U E (x 1, t) ( ) erfc(z)dz η(x 2 )dx 2 z =. (3.3) Proerty (3.3) of u K 1 allows the uk 2 defined in (2.4) (see also below) to decay sufficiently fast as x 2. This decay of u K 2 will be used essentially later on in the roof. Note that u K 1 is seudo-localized to scale x 2. Indeed, we have that and erfc(z(x 2, t)) L x2 (, ) = () 1/(2) 1 erf(z) L z (, ) C(νt) 1/(2) x2 erfc(z(x 2, t)) L x2 (, ) = () 1/(2) 1/2 z erfc(z) L z (, ) C(νt) 1/(2) 1/2 for all 1, where C > is a constant. The above bounds yield u K 1 L x1,x 2 () C U E (t) L x1 (() 1/(2) + C η () 1/2) C E (νt) 1/(2) (3.4) 1 u K 1 L x1,x 2 () C η 1 U E (t) L x1 (νt) 1/(2) C E (νt) 1/(2) (3.5) 2 u K 1 L x1,x 2 () C η U E (t) L x1 (νt) 1/(2) 1/2 C E (νt) 1/(2) 1/2 (3.6) 12 u K 1 L x1,x 2 () C η 1 U E (t) L x1 (νt) 1/(2) 1/2 C E (νt) 1/(2) 1/2 (3.7) for all 1, where C E > is a constant that deends on the Euler flow, on, the cutoff function η, through the constant C η, on ν and T. We emhasize however only the deendence on the Euler flow. We moreover have that t u K 1 ν u K 1 = ( t U E (x 1, t) ν 11 U E (x 1, t) ) ( erfc(z(x 2, t)) ) η(x 2 ) ( ) + U E (x 1, t)( t ν 22 ) η(x2 ) and thus t u K 1 ν u K 1 L 2 C η t U E L 2 + ν 11 U E L 2 (νt) 1/4 + C η U E L 2ν 1/2 t 1/2 C E ((νt) 1/4 + ν 1/2 t 1/2) (3.8) where as before the deendence of all constants on ν and T is ignored.

8 8 P. CONSTANTIN, T. ELGINDI, M. IGNATOVA, AND V. VICOL 3.2. The normal comonent of the lift u K. Combining (2.4) with (3.2), we arrive at ( x2 u K 2(x 1, x 2, t) = 1 U E (x 1, t) erfc(z(y, t))dy x2 ) η(y)dy = ( z(x2,t) ) x2 1 U E (x 1, t) erfc(z)dz η(y)dy An exlicit calculation shows that ( 1 x2 ) R(x 2, t) = π η(y)dy =: 1 U E (x 1, t)r(x 2, t). (3.9) 1 1 π ex ( z(x 2, t) 2) + z(x 2, t) erfc(z(x 2, t)). Moreover, note that in view of the choice of η in (3.1), the first term on the right side of the above is identically vanishing for all x 2 2. It is clear that R obeys R(, t) = = R(x 2, t), x 2 and thus we may hoe that R is integrable with resect to x 2, which is indeed the case. To see this, first we note that R(t) L x2 1. π Then, we have that 2 R(t) L 1 x2 1 x2 η(y)dy π dx ex( z(x 2, t) 2 ) + z(x 2, t) erfc(z(x 2, t))dx 2 1 π C η + ex( z 2 ) + z(1 erf(z))dz π C η where the deendence of all constants on ν and T is ignored. By interolation it then follows that R(t) L x2 C η (3.1) for all 1. In view of (3.1) and (3.2), we have that the bounds u K 2 L x1,x 2 () C η 1 U E L x1 C E (νt) 1/2 (3.11) 1 u K 2 L x1,x 2 () C η 11 U E L x1 C E (νt) 1/2 (3.12) hold for 1, where we have as before suressed the deendence on C η and of the constant C E. Lastly, we obtain from (2.4) and (3.9) that ( t ν )u K 2(x 1, x 2, t) = ν 12 u K 1(x 1, x 2, t) ν 111 U E (x 1, t)r(x 2, t) + ν 1/2 t 1/2 1 U E (x 1, t)r(x 2, t) + 1 U E (x 1, t) t R(x 2, t) = ν 12 u K 1(x 1, x 2, t) ν 111 U E (x 1, t)r(x 2, t) + ν 1/2 t 1/2 1 U E (x 1, t)r(x 2, t) ν 1/2 t 1/2 1 U E (x 1, t)z(x 2, t) erfc(z(x 2, t))

9 INVISCID LIMIT OF NAVIER-STOKES FOR BOUNDED VELOCITY FIELDS 9 where we have used that t R(x 2, t) = 1 2t z(x 2, t) erfc(z(x 2, t)). Using (3.7) and (3.1) we conclude that ( t ν )u K 2 L 2 x1,x 2 () C η ν 1/2 t 1/2 (νt) 1/4 1 U E (t) L 2 x1 holds. + C η ν(νt) 1/2 111 U E L 2 x1 + C η ν 1/2 t 1/2 1 U E L 2 x1 C E (ν 1/2 t 1/2 + (νt) 1/2) (3.13) 4. Conclusion of the Proof of Theorem 1.1 aving constructed the corrector function u K, we estimate the terms on the right side of (2.6) Bounds for T 1, T 2, T 3, T 4, and T 5. Using (3.8) and (3.13) we arrive at In order to bound T 2 we first estimate T 1 v L 2 ( t ν )u K L 2 ( C E v L 2 ν 1/2 t 1/2 + (νt) 1/4). (4.1) T 2 u E L u K L 2 u NS L 2 u E L u K L 2 u NS L 2 where we have used the L 2 energy inequality for the Navier-Stokes solution. Combining the above with (3.4) and (3.11) we arrive at T 2 C E (νt) 1/4 (4.2) since u NS L 2 C( ue L 2 + 1), for all ν ν, as we assume u NS ue L2 as ν. Similarly to T 2, we may estimate T 3 u E L u K L 2 v L 2 C E (νt) 1/2 v L 2. (4.3) Then, similarly to T 2 we estimate T 4. We aeal to the energy inequality for the Navier-Stokes solution and estimate (3.12), which is valid also for =, to conclude that T 4 u NS 2 L 2 1 u K 2 L u NS 2 L 2 1 u K 2 L C E (νt) 1/2. (4.4) Finally, we estimate T 5 by writing u NS = v + (u E + u K ) so that by the triangle inequality T 5 C v 2 L 2 1 u K 1 L + C v L 2 u E + u K L 1 u K 1 L 2 + C u E + u K 2 L 1u K 1 L 1 C E v 2 L 2 + C E v L 2(νt) 1/4 + C E (νt) 1/2 C E v 2 L 2 + C E (νt) 1/2. (4.5) In the above estimate we have imlicitly used the bounds (3.4) (3.5) and (3.11) (3.12).

10 1 P. CONSTANTIN, T. ELGINDI, M. IGNATOVA, AND V. VICOL 4.2. Bound for T 6. First we note that by the definition of u K 1 in (3.2) we have T 6 () 1/2 u NS 2 (x 1, x 2, t)u E (x 1, t)η (x 2 )dx 1 dx 2 + u NS 2 (x 1, x 2, t)u E (x 1, t) x2 erfc(z(x 2, t))dx 1 dx 2 C E (νt) 1/2 u NS 2 L 2 + T 6,ν C E (νt) 1/2 + T 6,ν (4.6) where we have used the energy inequality u NS L 2 u NS L 2 C(1+ ue L2), and have denoted T 6,ν = u NS 2 (x 1, x 2, t)u E (x 1, t) x2 erfc(z(x 2, t))dx 1 dx 2 = 1 u NS 2 (x 1, x 2, t)u E (x 1, t) ex x2 2 dx 1 dx 2 πνt = 2 π u NS 1 (x 1, y, t)u NS 2 (x 1, y, t)u E (x 1, t) ex( y 2 )dx 1 dy. (4.7) The goal is now to show that assumtions (1.17) (1.13) imly T ν T 6,ν (t) dt = (4.8) which yields the desired T 6 estimate. In order to rove (4.8), we fix an ε >, arbitrary, which in turn fixes a ρ = ρ(ε) > such that (1.15) holds. We then have T T T 6,ν (t) dt C E u NS 2 (x 1, x 2, t) ex x 2 2 dx 1 dx 2 dt x 2 ρ πνt T + C E u NS 2 (x 1, x 2, t) ex x2 2 dx 1 dx 2 dt x 2 ρ πνt T ex ρ 2 T x 2 ex 2 C E u NS (t) 2 πνt L dt + εc 2 E γ(x 1, t) dx 1 dx 2 dt x 2 ρ T ex ρ 2 C E u NS 2 4νT L + εc 2 E γ L1 (,T ;L πνt 1 x 1 ) y ρ/ ex( y 2 )dy ex ρ 2 4νT C E + εc E γ L πνt 1 (,T ;L 1 x 1 ). (4.9) By assing ν in (4.9), for ρ and T are fixed, we arrive at T ν T 6,ν (t) dt εc E γ L 1 (,T ;L 1 x 1 ). (4.1) Since γ is indeendent of ε, and ε > is arbitrary, (4.1) imlies (4.8) as desired.

11 INVISCID LIMIT OF NAVIER-STOKES FOR BOUNDED VELOCITY FIELDS Proof of Theorem 1.1. From (2.6) and (4.1) (4.6) we conclude that d dt v 2 L 2 C E v 2 L 2 + C E ν 1/2 t 1/2 v L 2 + C E (νt) 1/4 + T 6,ν (4.11) where as usual C E imlicitly deends on ν and T. Uon integrating (4.11) in time, using (1.7) and (4.8) we arrive at v L ν (,T ;L 2 ()) =. The above yields the roof of (1.8) once we recall that u NS u E = v + u K, and that cf. (3.4) and (3.11) we have ν u K L (,T ;L 2 ()) =. 5. Proof of Theorem 1.3 The roof follows from the roof of Theorem 1.1, as soon as we manage to establish the it (4.8) for the T 6 term. Recall that π 2 T ex x2 2 6,ν(t) = u NS 2 (x 1, x 2, t)u E (x 1, t) dx 1 dx 2. Fix ε > and fix ρ(ε) > such that (1.16) holds. Then, as in (4.9), the contribution to T T 6,ν(t) dt from {x 2 ρ} may be estimated as ex ρ2 4νT C E as ν, (5.1) πνt for ε, and thus ρ, fixed. On the other hand, by aealing to both (1.17) and (1.16), the divergence free condition and the fact that u 2 (x 1,, t) =, the contribution to T T 6,ν(t) dt from {x 2 ρ} may be bounded by T x2 ex x2 2 C E u NS 1 (t) 1 <x2 <ρ L 1 u NS 1 (x 1, y, t)dy dx 1 dx 2 dt T ρ C E u NS 1 (t) 1 <x2 <ρ L 1 u NS 1 (x 1, y, t) ex x2 2 dydx 1 dx 2 dt C E u NS 1 1 <x2 <ρ L 2 (,T ;L ) 1 u NS 1 1 <x2 <ρ L 2 (,T ;L 1 ) εc E C NS ν. Therefore, by adding the estimates (5.1) and (5.2), we obtain T T 6,ν (t) dt Cε ν for any ε >, as desired. Note that Remark 1.4 follows once we distribute the x 1 integrability in (5.2), equally between u NS 1 and 1u NS 1 via the Cauchy-Schwartz inequality. (5.2)

12 12 P. CONSTANTIN, T. ELGINDI, M. IGNATOVA, AND V. VICOL 6. Proof of Theorem 1.5 The roof follows closely that of Theorem 1.1. To avoid redundancy, here we only oint out the main differences. Moreover, for the sake of simlicity, we first consider the case δ(νt) =, which clearly obeys condition (1.18) and the required conditions on δ(νt) stated in the theorem. We need to show that (1.19) and (1.2) imly that (4.8) holds. Once this is roven, the theorem follows with the same argument as Theorem 1.1. For this urose, we may decomose the integral defining T 6,ν into x 2 > ρ = (log(1/ν)) 1/2 and x 2 < ρ = (log(1/ν)) 1/2. The first integral, for x 2 away from is bounded as in (4.9) by ν C E u NS 2 L 2 T y<(log(1/ν)) 1/2 ex ρ2 dt = C E u NS 2 L 2 T ex ( log(1/ν)) dt C E ν 1/2 as ν. For the contribution to T 6,ν from x 2 < ρ, we change variables y = x 2 /, so that it remains to show that T u NS 2 (x 1, y, t)u NS 1 (x 1, y, t)u E (x 1, t) ex ( y 2) dx1 dydt =. Let M(t) be defined by su u NS 1 (t)u NS 2 (t) L ({x 2 δ(νt)(log(1/ν)) 1/2 }) = M 2 (t). ν (,1] By assumtion (1.2) we have that T M 2 (t)dt <, and thus the function is indeendent of ν, obeys A(x 1, y, t) = M 2 (t) U E (x 1, t) ex( y 2 ) A L 1 (dtdydx 2 ), since the Euler trace U E is bounded in L (, T ; L 1 x 1 (R)), and we have that u NS 2 (x 1, y, t)u NS 1 (x 1, y, t)u E (x 1, t) ex ( y 2) A(x1, y, t) for a.e. (x 1, y, t), and all ν (, ν ], in view of assumtion (1.2). Thus, in view of (1.19), which guarantees that u NS 1 (x 1, δ(νt) y, t)u NS 2 (x 1, δ(νt) y, t)u E (x 1, t) ex ( y 2) = ν we may aly the Dominated Convergence Theorem and conclude that (6.1) holds. This concludes the roof of the theorem when δ(νt) =. To treat the more general case δ(νt) which obeys (1.18), we need to define a corrector with a different length scale. For this urose, we either recall the construction in [CKV15], or note that we may consider the corrector u K constructed in (3.2) (3.9), in which we relace with δ(νt), and z(x 2, t) with x 2 /δ(νt). Similar estimates to those in Section 3 show that the bounds u K L () + t t u K L () + 1 u K L () + 11 u K L () C E δ(νt) 1/ 2 u K 1 L () C E δ(νt) 1+1/ 1 u K 2 L () C E δ(νt) (6.1)

13 INVISCID LIMIT OF NAVIER-STOKES FOR BOUNDED VELOCITY FIELDS 13 hold. The assumtion on νtδ (νt)/δ(νt) C(T ) is only needed to ensure that the t u K estimate holds. Although this corrector does not solve the heat equation we may aeal to the usual integration by arts and Poincaré inequality trick of [Kat84] to bound the T 1 term. It then follows that the terms T 1,..., T 5 defined in (2.7) (2.11) obey the estimates δ(νt) 1/2 T 1 C E v t L 2 + C E νδ(νt) 1/2 v L 2 + ν 2 2v 2 L + ν v 2 ν 2 L + C 2 E δ(νt) T 2 C E δ(νt) 1/2 (6.2) (6.3) T 3 C E δ(νt) 1/2 v L 2 (6.4) T 4 C E δ(νt) (6.5) T 5 C E v 2 L 2 + C E δ(νt) 1/2. (6.6) If there were no T 6 term, the roof is comleted uon integrating the above bounds on [, T ] and assing ν, since condition (1.18) ensures that the contribution from the first and last terms on the right side of (6.2) vanishes in the it. For the term T 6 we roceed as above, by aealing to (1.19) (1.19) and the Dominated Convergence Theorem. We omit further details. Acknowledgements The authors would like to thank Gung-Min Gie and Jim Kelliher for many useful comments about the aer, in articular for ointing out how to remove one unnecessary assumtion from an earlier version of Theorem 1.1. The work of PC was artially suorted by NSF grant DMS The work of VV was artially suorted by NSF grant DMS and by an Alfred P. Sloan Research Fellowshi. References [AWXY14] R. Alexandre, Y.-G. Wang, C.-J. Xu, and T. Yang. Well-osedness of the Prandtl equation in Sobolev saces. J. Amer. Math. Soc., 214. [BN14] C. Bardos and T.T. Nguyen. Remarks on the inviscid it for the comressible flows. arxiv rerint arxiv: , 214. [BSJW14] C. Bardos, L. Székelyhidi Jr, and E. Wiedemann. Non-uniqueness for the euler equations: the effect of the boundary. Russian Mathematical Surveys, 69(2):189, 214. [BT7] C. Bardos and E.S. Titi. Euler equations for an ideal incomressible fluid. Usekhi Mat. Nauk 62(3):5 46, 27. (Russian) Translation in Russian Math. Surveys 62(3):49 451, 27. [BTW12] C. Bardos, E.S. Titi, and E. Wiedemann. The vanishing viscosity as a selection rincile for the Euler equations: the case of 3d shear flow. C. R. Acad. Sci. Paris, 35(15 16):757 76, 212. [CKV15] P. Constantin, I. Kukavica, and V. Vicol. On the inviscid it of the Navier-Stokes equations. Proc. Amer. Math. Soc., 143(7):375 39, 215. [GGN14a] E. Grenier, Y. Guo, and T. Nguyen. Sectral instability of characteristic boundary layer flows. arxiv: , 214. [GGN14b] E. Grenier, Y. Guo, and T. Nguyen. Sectral instability of symmetric shear flows in a two-dimensional channel. arxiv: , 214. [GGN14c] E. Grenier, Y. Guo, and T. Nguyen. Sectral stability of Prandtl boundary layers: an overview. arxiv: , 214. [Gie14] G.-M. Gie. Asymtotic exansion of the Stokes solutions at small viscosity: the case of non-comatible initial data. Commun. Math. Sci., 12(2):383 4, 214. [GKLF + 15] G.-M. Gie, J.P. Kelliher, M.C. Loes Filho,.J. Nussenzveig Loes, and A.L. Mazzucato. The vanishing viscosity it for some symmetric flows. Prerint, 215. [GN11] Y. Guo and T. Nguyen. A note on Prandtl boundary layers. Comm. Pure Al. Math., 64(1): , 211.

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