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1 SIAM J. MATH. ANAL. Vol. 45, No. 6, pp c 3 Society for Industrial and Applied Mathematics ALMOST SURE EXISTENCE OF GLOBAL WEAK SOLUTIONS FOR SUPERCRITICAL NAVIER STOKES EQUATIONS ANDREA R. NAHMOD, NATAŠA PAVLOVIĆ, AND GIGLIOLA STAFFILANI Abstract. In this paper we show that after suitable data randomization there exists a large set of supercritical periodic initial data, in H α T d )forsomeαd) >, for both two- and threedimensional Navier Stokes equations for which global energy bounds hold. As a consequence, we obtain almost sure large data supercritical global weak solutions. We also show that in two dimensions these global weak solutions are unique. Key words. Navier Stokes, almost sure existence, supercritical regime AMS subject classifications. 76D3, 35Q3 DOI..37/8884. Introduction. Consider the initial value problem for the incompressible Navier Stokes equations given by t u =Δ u P u u); x T d or R d,t>,.) u =, ux, ) = fx), where f is divergence free and P is the Leray projection into divergence-free vector fields given via.) P h = h Δ h). It is well known that global well-posedness of.) when the space dimension d = 3 is a long-standing open question. This is related to the fact that equations.) are so-called supercritical when d>. Indeed, recall that if the velocity vector field ux, t) solves the Navier Stokes equations.), then u λ x, t) with u λ x, t) =λ uλx, λ t) is also a solution to the system.), for the initial data.3) u λ = λ u λx). The spaces which are invariant under such a scaling are called critical spaces for the Navier Stokes equations. Examples of critical spaces for the Navier Stokes equations are.4) Ḣ d L d Ḃ + d p p, BMO, <p<. Received by the editors June, ; accepted for publication August, 3; published electronically November, 3. Department of Mathematics, University of Massachusetts, Amherst, MA 3 nahmod@math. umass.edu). This author was supported in part by NSF DMS 443. Department of Mathematics, University of Texas at Austin, Austin, TX 787 natasa@math. utexas.edu). This author was supported in part by NSF DMS 75847, NSF DMS 9, and an Alfred P. Sloan Research Fellowship. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 39 gigliola@math.mit.edu). This author was supported in part by NSF DMS

2 343 A. R. NAHMOD, N. PAVLOVIĆ, AND G. STAFFILANI In particular, for Sobolev spaces, u λ x, ) Ḣsc = ux, ) Ḣsc when s c = d. We recall that the exponents s are called critical if s = s c, subcritical if s>s c,and supercritical if s<s c. On the other hand, classical solutions to.) satisfy the decay of energy, which canbeexpressedas t.5) ux, t) L + ux, τ) L dτ = ux, ) L. Note that when d =, the energy ux, t) L, which is globally controlled thanks to.5), is exactly the scaling invariant = L -norm. In this case the equations are Ḣsc said to be critical. When d = 3, the energy ux, t) L is at the supercritical level with respect to the scaling invariant Ḣ -norm, and hence the Navier Stokes equations are said to be supercritical, and the lack of a known bound for the Ḣ contributes to keeping the global well-posedness question for the initial value problem.) still open. One way of studying the Navier Stokes initial value problem.) is via weak solutions introduced in the context of these equations by Leray [3, 4, 5] in the 93s. Leray [5] and Hopf [7] showed existence of global weak solutions of the Navier Stokes equations corresponding to initial data in L R d ). When d =classical global solutions were later obtained by Ladyzhenskaya []. Lemarié-Rieusset generalized Leray s construction to prove existence of uniformly locally square integrable weak solutions; for details see []. However, questions addressing uniqueness and regularity of these solutions when d = 3 have not been answered yet, although there are many important contributions in understanding partial regularity and conditional uniqueness of weak solutions; see, e.g., Caffarelli, Kohn, and Nirenberg [5], Lin [6], Escauriaza, Seregin, and Šverak [3], and Vasseur [9]. Another approach in studying existence of solutions to.) is to construct solutions to the corresponding integral equation via a fixed point theorem. In such a way one obtains so-called mild solutions. This approach was pioneered by Kato and Fujita; see, for example, [4]. However, the existence of mild solutions to the Navier Stokes equations in R d for d 3 has been obtained only locally in time and globally for small initial data in various subcritical or critical spaces see, e.g., Kato [8], Cannone [6, 7], Planchon [7], Koch and Tataru [], and Gallagher and Planchon [6]) or globally in time under conditions on uniform-in-time boundedness of certain scaling invariant norms see, e.g., Kenig and Koch [9]). In this context, Cannone and Meyer [8] proved that if f X, awell-suited Banach space for the study of the Navier Stokes, then the fluctuation w := u e tδ f is better in that it belongs to the Besov-type space ḂX,. In this paper we consider the periodic Navier Stokes problem in.) and in particular address the question of long time existence of weak solutions for supercritical initial data in both two and three dimensions; see also Tao [8]. For d = we address uniqueness as well. Our goal is to show that by randomizing in an appropriate way the initial data in H α T d ),d =, 3forsomeα = αd) > ), which is below the critical threshold space H sc T d ), as well as below the space L where one has available deterministic constructions of weak solutions, one can construct a global-in-time weak solution to.). Such a solution is unique when d =. Similar well-posedness results for randomized data were obtained for the supercritical nonlinear Schrödinger equation by Bourgain [] and for supercritical nonlinear wave equations by Burq and Tzvetkov in [, 3, 4]. The approach of Burq and Tzvetkov was applied in the context of Navier Stokes in order to obtain local-in-time solutions to the corresponding inte-

3 SUPERCRITICAL NS AND ALMOST SURE GLOBAL EXISTENCE 3433 gral equation for randomized initial data in L T 3 ), as well as global-in-time solutions to the corresponding integral equation for randomized initial data that are small in L T 3 ) by Zhang and Fang [3] and by Deng and Cui []. Also in [], Deng and Cui obtained local-in-time solutions to the corresponding integral equation for randomized initial data in H s T d )ford =, 3 with <s<. The paper at hand is the first to offer a construction of a global-in-time weak solution to.) for randomized initial data without any smallness assumption) in negative Sobolev spaces H α T d ),d=, 3, for some α = αd) >. Roughly speaking the idea of the proof is the following: We start with a divergence-free and mean zero initial data f H α T d )) d,d =, 3, and suitably randomize it to obtain f ω see Definition. for details), which in particular preserves the divergence-free condition. Then we seek a solution to the initial value problem.) in the form u = e tδ f ω + w. In this way, the linear evolution e tδ f ω is singled out and the difference equation that w satisfies is identified. At this point it becomes convenient to state the equivalence lemma, Lemma 4., between the initial value problem for the difference equation and the integral formulation of it. This equivalence is similar to Theorem. in []; see also [5]. We will use the integral equation formulation near time zero and the other one away from zero see section 5 for more details). The key point of this approach is the fact that although the initial data are in H α for some α>, the heat flow of the randomized data gives almost surely improved L p bounds see section 3). These bounds in turn yield improved nonlinear estimates arising in the analysis of the difference equation for w almost surely see section 5 for details), and consequently a construction of a global weak solution to the difference equation is possible see section 6 for details). It is important to note that, almost surely in ω, the randomized initial data f ω belongs to W α,p for any p, and hence it is in B + d p p,q for p large enough so that d p >α,andanyq. In particular, f ω belongs to the critical Besov spaces for which Gallagher and Planchon [6] proved, when d =, global existence, uniqueness, and suitable bounds. Since f ω also belongs to BMO, small data almost sure) well-posedness follows when d =, 3 from Koch and Tataru s result []. The goal of this paper, however, is to show that there exists a large set of supercritical periodic initial data of arbitrary size in H α T d ), d =, 3, that evolve to global solutions for which once the linear evolution is removed we directly obtain energy bounds... Organization of the paper. In section we introduce appropriate notation and state the main results. In section 3 we prove some useful bounds for the heat flow on randomized data. In section 4 we introduce the difference equation for w and establish two equivalent formulations for the equation that w solves. In section 5 we prove energy estimates for w, and in section 6 we construct weak solutions to the difference equation via a Galerkin method. In section 7 we prove uniqueness of weak solutions when d =. Finally in section 8 we combine all the ingredients to establish the main theorems.. Notation and the statement of the main result... Notation. In this subsection we list some notation that will be frequently used throughout the paper. Let B be a Banach space of functions. The space C weak,t),b) denotes the subspace of L,T),B) consisting of functions which are weakly continuous,

4 3434 A. R. NAHMOD, N. PAVLOVIĆ, AND G. STAFFILANI i.e., v C weak,t),b) if and only if φvt)) is a continuous function of t for any φ B. If XT d )) d denotes a space of vector fields on T d, we simply denote its norm by X. We introduce notation analogous to that of Constantin and Foias in [9]. In particular we write H = the closure of { f C T d )) d f =} in L T d )) d, V = the closure of { f C T d )) d f =} in H T d )) d, V = the dual of V. We finally introduce some notation for the inner products in some of the spaces introduced above. The notation is similar to that used in [9]. Given two vectors u and v in R d we use the notation.) u, v = u v. In L T d )) d we use the inner product notation.) u, v) = ux) vx) dx. In Ḣ T d )) d we use the inner product notation.3) u, v)) = d D i u, D i v). Finally we introduce the trilinear expression.4) b u, v, w) = u j D j v i w i dx = u v, w dx. i= Also we note that when u is divergence free, we have.5) b u, v, w) = v u), w dx. Finally, as it is now customary, we use A CB to denote the estimate A CB for an absolute positive constant C... A general randomization setup. Before stating the main theorem we recall a large deviation bound from [] that we will use below in order to analyze the heat flow on randomized data in section 3. Lemma. see [, Lemma 3.]). Let l r ω)) r= be a sequence of real, -mean, independent random variables on a probability space Ω,A,P) with associated sequence of distributions μ r ) r=. Assume that μ r satisfy the property.6) c >: γ R, r, e γx dμ r x) ecγ. Then there exists α> such that for every λ>, every sequence c r ) r= l of real numbers, ) P ω : c r l r ω) >λ e αλ r c r. r=

5 SUPERCRITICAL NS AND ALMOST SURE GLOBAL EXISTENCE 3435 As a consequence there exists C>such that for every q and every c r ) r= l, c r l r ω) C ) q c r. r= L q Ω) Burq and Tzvetkov showed in [] that the standard real Gaussian as well as standard Bernoulli variables satisfy the assumption.6)..3. Our randomization setup. We now introduce the diagonal randomization of elements of H s T d )) d, which we will apply to our initial data. Definition. diagonal randomization of elements in H s T d )) d ). Let l n ω)) n Z d be a sequence of real, independent, random variables on a probability space Ω,A,P) as in Lemma.. For f H s T d )) d,let a i n ),i=,,...,d, be its Fourier coefficients. We introduce the map from Ω,A) to H s T d )) d equipped with the Borel sigma algebra, defined by.7) ω f ω, f ω x) = l n ω) a n e nx),..., l n ω) a d n e nx), n Z d n Z d where e n x) =e in x, and we call such a map randomization. By the conditions in Lemma., the map.7) is measurable and f ω L Ω; H s T d )) d )isanh s T d )) d -valued random variable. Also we remark that such a randomization does not introduce any H s regularization see Lemma B. in [] for a proof of this fact); indeed f ω H s f H s. However, randomization gives improved L p estimates almost surely see Proposition 3. below). Remark. Since the Leray projection.) can be written via its coordinates.8) P h) j = h j + R j R k h k, k=,...,d where R j φ)n) = inj ˆφn), n n Z d, one can easily see that the diagonal randomization defined in.7) commutes with the Leray projection P. Having defined diagonal randomization, we can state the main results of this paper..4. Main results. Using the notation from subsection. we introduce the following definition. Definition.3. Let f H α T d )) d,α >, f =, and be of mean zero. A weak solution of the Navier Stokes initial value problem.) on [,T] is a function u L loc,t); V ) L loc,t); H) C weak,t); H α T d )) d ) satisfying d u dt L loc,t),v ) and d u.9) dt, v + u, v)) + b u, u, v) = for a.e. t and v V,.) lim ut) = f weakly in the H α T d )) d topology. t + This is assumed without loss of generality. Since the mean is conserved, if it is not zero, one can replace the solution with the solution minus the mean. This new function will satisfy an equation with a first order linear modification which does not affect the estimates. r=

6 3436 A. R. NAHMOD, N. PAVLOVIĆ, AND G. STAFFILANI Theorem.4 existence and uniqueness in two dimensions). Fix T >, < α< and let f H α T )), f =and be of mean zero. Then there exists asetσ Ω of probability such that for any ω Σ the initial value problem.) with datum f ω has a unique global weak solution u in the sense of Definition.3 of the form.) u = u f ω + w, where u f ω = e tδ f ω and w L [,T]; L T )) ) L [,T]; Ḣ T )) ). Theorem.5 existence in three dimensions). Fix T >, <α< 4 and let f H α T 3 )) 3, f =, and be of mean zero. Then there exists a set Σ Ω of probability such that for any ω Σ the initial value problem.) with datum f ω has a global weak solution u in the sense of Definition.3 of the form.) u = u f ω + w, where u f ω = e tδ f ω and w L [,T]; L T 3 )) 3 ) L [,T]; Ḣ T 3 )) 3 ). 3. The heat flow on randomized data. In this section we obtain certain a priori estimates on the free evolution of the randomized data. These bounds will play an important role in the proof of existence d =, 3) and uniqueness d =)in the later sections. 3.. Deterministic estimates. We have the following estimates for the solutions to the linear evolution. Lemma 3.. Let <α<, letk be a nonnegative integer, and let u f ω = e tδ f ω. If f ω H α T d )) d, then we have 3.) 3.) k u f ω,t) L x k u f ω L x C + t α+k ) f H α, ) C max{t,t k+α+ d ) } f H α. Proof. To prove 3.) we write u f ωn, t) =e n t f ω n). Then we have that k u f ω,t) L x e n t n k f ω n) l n C + t α+k ) f H α, where to obtain the last line we used that e n t t α+k n α+k C. To prove 3.) using the Fourier representation k u f ωx, t) wehave where k u f ωx, t) n k e n t n α n α f ω n) n f H α n α n k e n t I := n I f H α, + ρ ) α ρ k e ρt ρ d dρ. )

7 SUPERCRITICAL NS AND ALMOST SURE GLOBAL EXISTENCE 3437 In order to calculate the integral I we perform the change of variables z = tρ and obtain where I = I = I = ) α ) k + z z e z z d t d dz = I + I, t t z t z > t ) α + z z t + z t We bound I from above as follows: 3.3) I Ct d ) α z d t t t t Ct e t ). On the other hand, we bound I from above as 3.4) I Ct d z > t = t k+α+ d ) Ct k+α+ d ). z α+k)+d ) k e z z d t d dz, ) k e z z d t d dz. e z zdz e z t) α+k) z > t By combining 3.3) and 3.4) we obtain that dz z α+k)+d e z dz I C max{t,t k+α+ d ) }, which thanks to 3.3) and 3.3) implies the claim 3.). 3.. Probabilistic estimates. We also have the following result for the randomized solution to the linear evolution. Proposition 3.. Let T>and α. Let r p q, letσ, andlet γ R be such that σ + α γ)q <. Then there exists C T > such that for every f H α T d )) d 3.5) t γ Δ) σ e tδ f ω Lr Ω;L q [,T ];L p x) C T f H α, where C T may depend also on p, q, r, σ, γ, andα. Moreover, if we set 3.6) E λ,t, f,σ,p = {ω Ω: t γ Δ) σ e tδ f ω Lq [,T ];L p x) λ}, then there exists c,c > such that for every λ> and for every f H α T d )) d [ ] λ 3.7) PE λ,t, f,σ,p ) c exp c C T f. H α

8 3438 A. R. NAHMOD, N. PAVLOVIĆ, AND G. STAFFILANI Proof. For t, using the notation hx) = Δ α fx) and recalling the notation defined in.7), we have 3.8) where for t γ Δ) σ e tδ f ω x) =t γ Δ) σ α Δ e tδ Δ α f ω x) = t γ n σ + n ) α e t n hω n)e n x) n Z d CJ σ + J σ+α, 3.9) β {σ, σ + α} we introduced J β as follows: J β = t γ β n Z d t β n β e t n hω n)e n x). In order to estimate J β,weobservethatt β n β e t n C, which together with two applications of Minkowski s inequality, followed by an application of Lemma., implies the first inequality in the estimate below: J β Lr Ω;L q [,T ];L p x)) C r β ) tγ hn)en x) 3.) as long as n = C r t hn)en γ β x) n C r,p h L T t β γ = C r,p,q h L T q + γ β, 3.) β γ) q <, ) q L q [,T ];L p x) L q p [,T ]; L ) q which is for our range 3.9) of β satisfied under the assumption that σ +α γ)q <. Now the estimate 3.5) follows from 3.8), 3.), and 3.9). To prove estimate 3.7) one uses the Bienaymé Tchebishev inequality as in Proposition 4.4 in [], which relies on Lemma.. 4. Difference equation and equivalent formulations. In this section we consider two formulations of the initial value problem for the difference equation in 4.) t w =Δ w P w w)+c [P w g)+p g w)] + c P g g), w =, wx, ) =. We start with the definition of weak solutions to 4.). Again using the notation from subsection. we have the following. x )

9 SUPERCRITICAL NS AND ALMOST SURE GLOBAL EXISTENCE 3439 Definition 4.. Assume that g =. A weak solution to the initial value problem 4.) on [,T] is a function w L,T); V ) L,T); H) satisfying d w dt L,T); V ) and such that for almost every t and for all v V, d w 4.) dt, v + w, v)) + b w, w, v) +b w, g, v) +b g, w, v) +b g, g, v) = and 4.3) lim wt) = t + weakly in the H topology. Now we state and prove the equivalence lemma, which is similar to the version for the Navier Stokes equations themselves; see, e.g., [5] and Theorem. in []. Lemma 4. the equivalence lemma). Let T>, and assume that g =and 4.4) Furthermore, assume that gx, t) L C + t α 4.5) { g L 4 [,T ],L 4 x ) C if d =, Δ) 4 g L [,T ],L 6 x ) + Δ) 4 g g L 3 [,T ],L 3x L ) 8 [,T ],L 8 x ) C if d =3 for some C>. Then the following statements are equivalent. S) w is a weak solution to the initial value problem 4.). S) The function w L,T); H) L,T),V) solves 4.6) wt) = where t ). e t s)δ F x, s) ds, 4.7) F x, s) = P w w)+c [P w g)+p g w)] + c P g g). Proof. We first show that S) implies S). Assume that w solves the integral equation 4.6). Define W w)x, t) = t e t s)δ F x, s) ds. Using the assumption 4.4) on g and the fact that w L,T); H) L,T); V ), it follows that F x, s) L,T); L x). Hence F x, s) L,T); D x)and 4.8) e t s)δ F x, s) L,T); C x ). We can now take the time derivative of W w)t, x) and easily show that t W w)x, t) =Δ W w)x, t) F x, t), where the equality holds in V.Since w = W w), by 4.6) and in particular lim t + wx, t) =,wenowhavethat w solves 4.) weakly.

10 344 A. R. NAHMOD, N. PAVLOVIĆ, AND G. STAFFILANI Next we show that S) implies S). Let Ψx, t) := t e t s)δ F x, s) ds, where F x, s) isasabove. Undertheassumptionson g and w and arguing as in the proof of the energy estimates Theorem 5. near time zero, we have that Ψ L,T); H) L,T); V )and dψ dt L,T); V ). We then have t Ψ w) =Δ Ψ w), where the equality holds in V and lim t + wt) Ψt) ) = weakly in H. A standard uniqueness result for the heat flow finally gives Ψ= w. 5. Energy estimates for the difference equation. In this section we establish energy estimates for the difference equation in 4.). These a priori estimates for w will be used in section 6, where we construct weak solutions; see also [8]. At this point it is useful to give a name to the left-hand side of.5). We define in fact the energy functional for w, t 5.) E w)t) = wt) L + c w dx ds, T d and prove the following theorem. Theorem 5.. Let T>, λ>, γ<, andα> be given. Let g be a function such that g =and 5.) 5.3) k gx, t) L gx, t) L C + ), t α C max{t,t k+α+ d ) } ) for k =,, and 5.4) { t γ g L4 [,T ];L 4 x ) λ if d =, t γ Δ) 4 g L [,T ];L 6 x ) + t γ Δ) 4 g L 3 [,T ];L 3x ) tγ g L8 [,T ];L 8 x ) λ if d =3. Let w L,T); L T d )) d ) L,T); Ḣ T d )) d ) be a solution to 4.). Then 5.5) 5.6) E w)t) CT,λ,α) t [,T], d dt w CT,λ,α), L p t H x with { if d =, p = 4 3 if d =3. Remark. In the course of the proof below we will rely on the equivalence lemma, Lemma 4., and use the integral equation formulation 4.6) for w near time zero and the other one 4.) away from zero. Note that 5.), 5.3), and 5.4) ensure that 4.4) and 4.5) hold.

11 SUPERCRITICAL NS AND ALMOST SURE GLOBAL EXISTENCE 344 Proof. In order to prove 5.5), we consider two cases: t near zero and t away from zero. Case t near zero. First, thanks to the equivalence lemma, Lemma 4., we can write w using formulation 4.6), 5.7) wt) = t e t s)δ F x, s) ds, where F x, s) is as in 4.7). Then we use a continuity argument as follows. Assume τ δ,whereδ is to be determined later. We need to proceed in different ways depending on whether d =ord =3. Ford =andτ [,δ ]by applying Lemma 4. from [] we have 5.8) wt) L x C F L [,τ];l x ) t [,τ]. Also by applying the maximal regularity see, e.g., Theorem 7.3 in []), we obtain 5.9) w L [,τ];h x ) C F L [,τ];l x ). Hence it suffices to analyze F L [,τ];l x ).Wehave 5.) F L [,τ];l x ) C w w L [,τ];l x ) + w g L [,τ];l x ) + g w L [,τ];l x ) + g g L [,τ];l x ). We proceed by estimating the terms on the right-hand side RHS) of 5.). We observe that 5.) w w L [,τ];l x ) = w L 4 [,τ];l 4 x ) CE w)τ), where in the last step we used that the norm L 4 t L4 x can be bounded via interpolating the two spaces L t L x and L t Ḣ x that appear in E w)t). For the next two terms by Hölder s inequality we have 5.) 5.3) w g L [,τ];l x ) + g w L [,τ];l x ) C g L p [,τ];l p x) w L p p p p [,τ];lx ) C g L p [,τ];l p x) D d p w L p p [,τ];l x ) C t γ g L p [,τ];l p x)δ ) γ D d p w L p ) p d p, p [,τ];l x )δ where to obtain 5.) we used Sobolev embedding, and to obtain 5.3) we used Hölder s inequality in t under the assumption that p 4. By letting p = 4 in 5.3), it follows from the assumptions 5.4) on g, in conjunction with interpolation between the spaces that appear in E w)t), that 5.4) w g L [,τ];l x ) + g w L [,τ];l x ) Cλ δ ) γ E w)τ). Finally, thanks again to 5.4), the last term can be estimated as 5.5) g g L [,τ];l x ) = g L 4 [,τ];l 4 x ) λδ ) γ).

12 344 A. R. NAHMOD, N. PAVLOVIĆ, AND G. STAFFILANI To conclude, we combine the estimates 5.8) 5.) with 5.), 5.4), and 5.5) and obtain 5.6) E w)τ) C E w)τ)+c λδ ) γ E w)τ)+c3 λδ ) γ). Hence if we denote E w)τ) =X, we obtain the inequality X C X + C λδ ) γ X + C 3 λδ ) γ). If we choose δ small enough so that C 3 λδ ) γ) <ɛwhere ɛ is a small absolute constant depending only on C,C and C 3 ), then by a continuity argument X is bounded for all τ [,δ ]. We thus obtain 5.7) E w)τ) C for all τ [,δ ]. For d = 3 we cannot directly close the estimates in terms of the energy E w). Instead we need to work with E w) +E Δ) 4 w) =:E w). Now, for τ [,δ ], applying once again Lemma 4. from [], we have 5.8) Δ) 4 + I) wt) L x C Δ) 4 + I) F L [,τ];l x ) t [,τ]. Also by applying the maximal regularity see, e.g., Theorem 7.3 in []), we obtain 5.9) Δ) 4 + I) w L [,τ];h x ) C Δ) 4 + I) F L [,τ];l x ). Hence it suffices to analyze Δ) 4 + I) F L [,τ];l x ).Wehave 5.) Δ) 4 + I) F L [,τ];l x ) C{ [ Δ) 4 + I) w] w L [,τ];l x ) + [ Δ) 4 + I) w] g L [,τ];l x ) + [ Δ) 4 + I) g] w L [,τ];l x ) + [ Δ) 4 + I) g] g L [,τ];l x ) }. We proceed by estimating the terms on the RHS of 5.). By using the same argument as in the proof of Theorem 5. in [], we have that 5.) [ Δ) 4 + I) w] w L [,τ];l x ) CE w)τ). Next, by Hölder s and Sobolev s inequalities we have [ Δ) 4 + I) w] g L [,τ];l x ) C g L4 [,τ];l 6 x ) Δ) 4 + I) w L4 [,τ];l 3 x ) C g L 4 [,τ];l 6 x ) w L 4 [,τ];h x ). Now we note that w L4 [,τ];h x ) C[E w)] τ) by interpolation. Hence 5.) For the next term we have [ Δ) 4 + I) w] g L [,τ];l x ) C t γ g L 8 [,τ];l 8 x ) δ ) γ+ 8 [E w)]. w [ Δ) 4 + I) g] L [,τ];l x ) C Δ) 4 + I) g L [,τ];l 6 x ) w L [,τ];l 3 x ) Cδ ) γ t γ Δ) 4 + I) g L [,τ];l 6 x ) w L [,τ];h x ).

13 SUPERCRITICAL NS AND ALMOST SURE GLOBAL EXISTENCE 3443 Hence 5.3) w [ Δ) 4 + I)g] L [,τ];l x ) Cδ ) γ t γ Δ) 4 + I) g L [,τ];l 6 x ) [E w)]. Finally, thanks again to 5.4), the last term can be estimated as 5.4) [ Δ) 4 +I) g] g L [,τ];l x ) C Δ) 4 +I) g L 8 8 g 3 [,τ];l 3x L ) 8 [,τ];l 8 x ) λδ ) γ). To conclude, we combine the estimates 5.8) 5.) with 5.), 5.), 5.3), and 5.4) to obtain 5.5) [E w)τ)] C E w)τ)+c λδ ) γ+ 8 [E w)τ)] + C3 λδ ) γ). Henceifwedenote[E w)τ)] = X, we obtain the inequality X C X + C λδ ) γ+ 8 X + C3 λδ ) γ). If we choose δ small enough so that C 3 λδ ) γ) <ɛwhere ɛ is a small absolute constant depending only on C,C,andC 3 ), then by a continuity argument X is bounded for all τ [,δ ]. We thus obtain 5.6) E w)τ) CE w)τ) C for all τ [,δ ]. Case t [δ,t]. By the standard energy argument for 4.) we have d E w)t) = wx, t) w t x, t) dx + w x, t) dx dt T d T d 5.7) = w Δ wdx w P w w) dx + w dx T d T d T d ) 5.8) +c w P w g) dx + w P g w) dx T d 5.9) +c T d w P g g) dx. Now we observe that the expression in 5.7) equals zero as in the case of solutions to the Navier Stokes equations themselves. It remains to estimate 5.8) and 5.9). In order to do that we first note that since g is divergence free, w P w g) dx = T d T w P g ) wdxand the last expression equals zero thanks to the skew-symmetry d property. Also since w is divergence free too, we observe that 5.3) w P g w) dx = w P w g) dx C w L g L. T d T d x x On the other hand, by Hölder s inequality, 5.3) w P g g) dx w L x g L x g L x. T d T d

14 3444 A. R. NAHMOD, N. PAVLOVIĆ, AND G. STAFFILANI Now by combining 5.8), 5.9), 5.3), and 5.3) and using the assumptions 5.) and 5.3) we obtain 5.3) with d dt E w)t) C{E w)t) g L + E x w)t) g L x g L x } C{ht)E w)t)+mt)e w)t)} ht) = max{t,t +α+ d ) } mt) = + t α ) ) ) max{t,t +α+ d ) }. Furthermore, using the above expressions for ht) andmt) weget 5.33) and 5.34) T T δ ht) dt = δ mt) dt = = = δ ht) dt + T δ t +α+ d ) dt + δ mt) dt + δ t +α+ d 4 T dt + ht) dt T t mt) dt T t +α, dt C δ ) α d 4 + T dt C {δ ) α d 4 Therefore by combining 5.3), 5.33), and 5.34) we obtain the bound 5.35) E w)t) CT,δ,α) for all t [δ,t]. Now 5.5) follows from 5.7) and 5.35). To prove 5.6) we let { if d =, p = 4 3 if d =3. + T α }. Since w satisfies 4.) we observe that d dt w 5.36) C{ Δ w L p [,T ];H L p [,T ];Hx x ) + w w) L p [,T ];Hx ) ) + w g) Lp [,T ];Hx ) + g w) L p [,T ];Hx ) + g g) Lp [,T ];Hx ) }. 5.37) We estimate the first term on the RHS of 5.36) as follows: Δ w Lp [,T ];Hx ) C { w L [,T ];L x ) if d =, w L 4 3 [,T ];L x ) CT 4 w L [,T ];L x ) if d =3.

15 SUPERCRITICAL NS AND ALMOST SURE GLOBAL EXISTENCE 3445 To estimate the second term on the RHS of 5.36) we notice that for d = 5.38) w w) Lp [,T ];Hx ) C w w L [,T ];L x ) w L 4 [,T ];L 4 x ) CE w)t ), while for d = 3 we use the Gagliardo Nirenberg inequality to obtain w w) L p [,T ];Hx ) C w L w x 3 L H 4 x 3 [,T ]) C w L [,T ];L ) w 3 x L [,T ];Hx ) CE w)t ). To estimate the third term on the RHS of 5.36), for d =, we proceed as follows: T T w g) = w g) ds = w g L p [,T ];Hx ) Hx L ds x g L 4 [,T ];L 4 ) w x L 4 [,T ];L 4 x ) 5.39) λt γ) w L4 [,T ];L 4 x ) CT,λ) E w)t ), where to obtain 5.39) we used the assumption 5.4) on g. The fourth term can be estimated analogously for d =. On the other hand to estimate the third term for d =3wehave w g) L p [,T ];H x ) w g L 4 3 [,T ];L x ) g L [,T ];L 6 x ) w L 4 [,T ];L 3 x ) C T λt γ ) t γ g L 8 [,T ];L 8 x ) w L 4 [,T ];H x ) CT,λ) E w)t ). Also the fourth term for d = 3 can be estimated analogously. Finally, in order to estimate the fifth term on the RHS of 5.36), for d =,we proceed similarly to when we estimated the third term for d =: T g g) = L p [,T ];Hx ) = T g g) H x ds g g L x ds g 4 L 4 [,T ];L 4 x ) λt γ) 4, where to obtain the last line we used the assumptions of the theorem. On the other hand, to estimate the fifth term for d =3wehave g g) T L p [,T ];Hx ) g g) T T L t H x g g L x ds T g L 4 [,T ];L 4 x ) C T g L 8 [,T ];L 8 x ) C T λt γ ). Collecting the above estimates, we obtain 5.6). 6. Construction of weak solutions to the difference equation. In this section we construct weak solutions to the initial value problem 4.). We denote the spatial Fourier transform of f by fk,t)= fx, t)e ik x dx, T d

16 3446 A. R. NAHMOD, N. PAVLOVIĆ, AND G. STAFFILANI with inverse transform fx, t) = fk,t)e ik x, k where k represents the discrete wavenumber, k = d πn j )e j, n j Z, j= and e j is the unit vector in the jth direction. By P M we denote the rectangular Fourier projection operator: P M f = fk)e ik x. {k : n j M for j d} Note that P M is a bounded operator in L p T d )for<p<. Theorem 6.. Let T>, λ>, γ<, andα> be given. Assume that the function g satisfies g =and 6.) 6.) k P M gx, t) L gx, t) L C + ), t α C max{t,t k+α+ d ) } ) for k =,. Furthermore, assume that we have 6.3) { t γ g L 4 [,T ];L 4 x ) λ if d =, t γ Δ) 4 g L [,T ];L 6 x ) + t γ Δ) 4 g L 3 [,T ];L 3x ) tγ g L8 [,T ];L 8 x ) λ if d =3. Then there exists a weak solution w for the initial value problem 4.) in the sense of Definition 4.. Remark 3. Since P M is a bounded operator in L p for <p<, P M g satisfies 6.) and 6.3) as g itself. So from this point on we will not make a distinction between P M g and g. Also, to keep the notation light, in the proof of Theorem 6. we shall consider the initial value problem 4.) with c = c =. Proof. In the construction of weak solutions, we follow in part the approach based on Galerkin approximations from Chapter 5 of Doering and Gibbon [] and from Chapter 8 of Constantin and Foias [9]. From now on we drop the vector notation to keep the notation light. The plan is to construct a global weak solution by finding its Fourier coefficients, which, in turn, will be achieved by solving finite-dimensional ODE systems for them. To determine the ODE systems we start by formally applying the Fourier transform to the difference equation 4.). d dtŵk,t)= k ŵk,t) + i k ŵk,t)=, I kkt k ) k +k =k ŵk,t) k ŵk,t)+ŵk,t) k ĝk,t) ) + ĝk,t) k ŵk,t)+ĝk,t) k ĝk,t),

17 SUPERCRITICAL NS AND ALMOST SURE GLOBAL EXISTENCE 3447 where ŵ,t) =, and I kkt k is the projection onto the divergence-free vector fields in Fourier space. Here I is the unit tensor and k = k. As in [], we now introduce the Galerkin approximations as truncated Fourier expansions. More precisely, let M be a positive integer and consider 6.4) k = d πn j )e j, j= n j = ±, ±,...,±M. We look for the complex variables ŵm k,t), k as in 6.4), solving the following finite system of ODE: 6.5) 6.6) d ŵ dt M k,t)= k ŵm k,t) ) + i I kkt ŵm k k,t) k ŵm k,t) k +k M =k ) + ŵm k,t) k ĝk,t)+ĝk,t) k ŵm k,t)+ĝk,t) k ĝk,t), k ŵm k,t)=, where the sum over k extends over the range where both k and k are as in 6.4). The system 6.5) 6.6) is considered with zero initial data. Let T > be fixed. We now show that for any fixed M>, the ODE system 6.5) admits a unique solution in X T := C[,T],l ) L [,T], kl ). We proceed by a fixed point argument. Define t ΦŵM )k,t):= k ŵm k,s)ds t [ ) + i I kkt ŵm k k,s) k ŵm k,s)+ŵm k,s) k ĝk,s) k +k M =k + ĝk,s) k ŵm k,s)+ĝk,s) k ĝk,s)) ] ds. Assume t<t.letδ such that <δ<t be determined later. For t [,δ], we have ΦŵM ),t) l C{M δ ŵm L t l + δm+ d ŵ M L t l 6.7) + M + d + δ α ŵ M L t l + Mδhd) γ λ }, with { if d =, hd) = 3 4 if d =3. To obtain the second term on the RHS above we used Plancherel and Sobolev embedding for the L 4 x-norm. To obtain the third term, we used Plancherel, 6.), and the Sobolev embedding for the L x -norm of F ŵm ), the inverse Fourier transform

18 3448 A. R. NAHMOD, N. PAVLOVIĆ, AND G. STAFFILANI of ŵm. Finally for the fourth term we used Plancherel and 6.3). In a similar manner we can also show that kφŵm ),t) L [,δ],l ) C{M 3 δ 3 ŵ M L t l + δ 3 M + d + ŵm L t l 6.8) + M + d + δ γ+θd) ŵm L t l λ + M δ ρd) γ λ }, with { 5 4 if d =, θd) = 8 if d =3, ρd) = { if d =, 5 4 if d =3. If we let R =M and δ = δm,λ) small enough, we have that the Φ maps balls of radius R in X δ to themselves continuously. A similar argument shows Φ is also a contraction, and as a consequence the ODE system 6.5) has a unique solution ŵ M in X δ. Therefore by applying Plancherel s theorem we conclude that the function w M x, t), given by the inverse Fourier transform of {ŵm k,t)} k, belongs to L [,δ]; L T d )) d ) L [,δ]; Ḣ T d )) d ). We next note that in [,δ] the function w M x, t) is a solution to the following system: t w M = Δw M P M [P w M w M )+P w M P M g) ] 6.9) + P P M g w M )+P P M g P M g), w M =, w M x, ) =. Since P M g satisfies the same assumptions as g in section 5, we can repeat the proof of Theorem 5. to conclude that w M x, t) isinl [,δ]; H) L [,δ]; V ) and satisfies the energy bounds given by 5.5). As a consequence we can use an iteration argument to advance the solution of 6.5) up to time T. Thus the function w M x, t) given by the inverse Fourier transform of {ŵm k,t)} k is in L,T); H) L,T); V ), and it satisfies the energy bound given by 5.5) and 5.6) in Theorem 5.. Now by applying a standard compactness argument, together with the fact that P M g converges strongly to g in L p, one obtains a weak solution w to 4.) on [,T] see also Chapter 8 of [9]). Since T was arbitrarily large, we obtained a global weak solution. 7. Uniqueness in two dimensions. In this section we present a uniqueness result for solutions of the initial value problem 4.) in d =. The result can be stated as follows. Theorem 7.. Assume that g satisfies the same conditions as in Theorem 6.. Then in d =any two weak solutions to 4.) in L [,T]; V ) L,T); H) coincide. Proof. Our proof is inspired by the proof of Theorem. in Constantin and Foias [9], which establishes a related uniqueness result for solutions to the Navier Stokes equations. Let w j with j =, be two solutions of 4.) with g satisfying 6.) 6.3). Let v = w w.then v satisfies the equation 7.) t v =Δ v P w v) P w v)+c [P v g)+p g v)], v =, vx, ) =.

19 SUPERCRITICAL NS AND ALMOST SURE GLOBAL EXISTENCE 3449 By pairing in L T )) the first equation in 7.) with v we obtain ) d dt v, v =Δ v, v) + c P w v) vdx [P g v) v + P v g) v] dx, P w v) vdx which, thanks to the equality w j v) = v ) w j which is valid for j =, since each w j is divergence free and hence v is divergence free too), becomes 7.) ) d dt v, v =Δ v, v) + c P v ) w ) vdx [P g v) v + P v g) v] dx. P v ) w ) vdx Therefore after performing integration by parts in the last term of the above expression, and using Hölder s inequality to bound the last three terms on the RHS, we obtain d dt vt) L + v x L x w L x v L + w 4 x L x v L + c g 4 x L 4 x v L 4 x v L x ) 7.3) w L x v L + w 4 x L x v L + c 4 x μ g L 4 v x L + 4 μ v x L x 7.4) w j L x + c μ g L v 4 x L x v L x + c μ v L x j= 3 7.5) w j L ν + c x j μ g 4 L v ν 4 x L + c x μ + ν j v L, x 3 j= where to obtain 7.3) we used Young s inequality, and to obtain 7.4) we used the bound 7.6) v L 4 x v L x v L x, which follows from an interpolation followed by Sobolev embedding. On the other hand, we obtained 7.5) by applying Young s inequality three times. Now we choose μ and ν j s such that c μ + 3 j= ν j =. Then 7.5) implies that 7.7) d dt vt) L vt) x L x j= j= w j L ν + c x j μ g 4 L, ν 4 x 3 which after we apply Gronwall s lemma on [,ρ] [,T]gives ρ 7.8) vt) L C v) x L exp w x j L ν + c x j μ g 4 L dt ν 4 x 3. j=

20 345 A. R. NAHMOD, N. PAVLOVIĆ, AND G. STAFFILANI Since by the assumption each w j L [,T],H ) L,T),L ), we have 7.9) ρ w j L x dt < for every ρ T. On the other hand, by employing the assumptions 6.) 6.3), we obtain 7.) ρ g 4 L 4 x dt Cρ 4γ λ 4. Now we recall that v) = w ) w ) = and substitute that into 7.8), keeping in mind that estimates 7.9) and 7.) imply finiteness of the exponent on the RHS of 7.8). Hence we conclude that vt), which implies w t) = w t) for all t [,T]. 8. Proof of the main theorems. We find solutions u to.) by writing u = u ω f + w, where we recall that u ω f is the solution to the linear problem with initial datum f ω, and w is a solution to 4.) with g = u ω f. Note that u is a weak solution for.) in the sense of Definition.3 if and only if w is a weak solution for 4.) in the sense of Definition 4.. We also remark that uniqueness of weak solutions to 4.) is equivalent to uniqueness of weak solutions to.). From now on we work exclusively with w and the initial value problem 4.). The proof of the existence of weak solutions is the same for both d =andd = 3, and it is a consequence of Theorem 6.. For the uniqueness claimed in d = we invoke Theorem 7.. Now to the details. Let γ< be such that Given λ>, define the set <α< { +γ if d =, 4 +γ if d =3. 8.) E λ := E λ,α, f,γ,t = {ω Ω : t γ u ω f L 4 [,T ];L 4 x ) >λ}, when d =. Ford =3,set u ω f α, f,γ,t) := t γ Δ) 4 u ω f L [,T ];L 6 x ) + t γ Δ) 4 u ω f L 8 3 [,T ];L 8 3x ) + tγ u ω f L8 [,T ];L 8 x ) and define 8.) E λ := E λ,α, f,γ,t = {ω Ω : u ω f α, f,γ,t) >λ}. Then if we apply Proposition 3., we find that in either case d =ord =3)there exist C,C > such that ) λ 8.3) PE λ ) C exp C C T f. H α

21 SUPERCRITICAL NS AND ALMOST SURE GLOBAL EXISTENCE 345 Now, let λ j = j,j, and define E j = E λj.notethate j+ E j.thenifwe let Σ := Ej c Ω, we have that ) PΣ) = lim PE j) lim exp C j j j C T f =. H α Our goal is now to show that for a fixed divergence-free vector field f H α T d )) d and for any ω Σ, if we define g = u ω f, the initial value problem 4.) has a global weak solution. In fact given ω Σ, there exists j such that ω Ej c. In particular we then have 8.4) t γ g L4 [,T ];L 4 x ) λ j when d =and 8.5) t γ Δ) 4 g L [,T ];L 6 x ) + t γ Δ) 4 g L 8 3 [,T ];L 8 3x ) + tγ g L 8 [,T ];L 8 x ) λ j when d =3. Lemma 3. together with 8.4) and 8.5) implies that the assumptions on g in Theorems 5., 6., and 7. are satisfied. This concludes the proof. Acknowledgment. The authors thank Cheng Yu for noticing that an earlier version needed a clarification in the energy estimates for the d =3case. REFERENCES [] J. Bourgain, Invariant measures for the D defocusing nonlinear Schrödinger equation, Comm. Math. Phys., 76, 996), pp [] N. Burq and N. Tzvetkov, Random data Cauchy theory for super-critical wave equation I: Local theory, Invent. Math., 73 8), pp [3] N. Burq and N. Tzvetkov, Random data Cauchy theory for super-critical wave equation II: A global existence result, Invent. Math., 73 8), pp [4] N. Burq and N. Tzvetkov, Probabilistic Well-Posedness for the Cubic Wave Equation, preprint, arxiv:3.v,. [5] L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure. Appl. Math., 35 98), pp [6] M. Cannone, Ondelletes, paraproduits et Navier Stokes, Diderot, Paris, 995. [7] M. Cannone, Harmonic Analysis Tools for Solving the Incompressible Navier-Stokes Equations, in Handbook of Mathematical Fluid Dynamics, Vol. III, North-Holland, Amsterdam, 4, Chapter 3, pp [8] M. Cannone and Y. Meyer, Littlewood-Paley decomposition and Navier-Stokes equations, Methods Appl. Anal., 995), pp [9] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Math., The University of Chicago Press, 988. [] C. Deng and S. Cui, Random-data Cauchy problem for the Navier-Stokes equations on T 3, J. Differential Equations, 5 ), pp [] C. Deng and S. Cui, Random-data Cauchy Problem for the Periodic Navier-Stokes Equations with Initial Data in Negative-Order Sobolev Spaces, preprint, arxiv:3.67v.,. [] C. Doering and J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, UK, 4. [3] L. Escauriaza, G. Seregin, and V. Šverak, L3, -solutions of the Navier-Stokes equations and backward uniqueness, Russian Math. Surveys, 58 3), pp. 5. [4] H. Fujita and T. Kato, On the Navier-Stokes initial value problem, I, Arch. Rational Mech. Anal., 6 96), pp [5] G. Furioli, P. G. Lemarié-Rieusset, and E. Terraneo, Unicité dans L 3 R 3 ) et d autres espaces fonctionnels limites pour Navier-Stokes [Uniqueness in L 3 R 3 ) and other functional limit spaces for Navier-Stokes equations], Rev. Mat. Iberoamericana, 6 ), pp

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