A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALEKATOMAJDA ESTIMATE. 1.


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1 A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALEKATOMAJDA ESTIMATE THOMAS CHEN AND NATAŠA PAVLOVIĆ Abstract. We prove a BealeKatoMajda criterion for the loss of regularity for solutions of the incompressible Euler equations in H s R 3, for s >. Instead of double exponential estimates of BealeKatoMajda type, we obtain a single exponential bound on ut H s involving the dimensionless parameter introduced by P. Constantin in []. In particular, we derive lower bounds on the blowup rate of such solutions.. Introduction In this paper, we revisit the BealeKatoMajda criterion for the breakdown of smooth solutions to the 3D Euler equations. More precisely, we consider the incompressible Euler equations t u + u u + p =. u =. ux, = u.3 for an unknown velocity vector ux, t = u i x, t i 3 R 3 and pressure p = px, t R, for position x R 3 and time t [,. Existence and uniqueness of local in time solutions to..3 in the space C[, T ], H s C [, T ]; H s,.4 has long been known for s >, see for instance [6]. However, it is an open problem to determine whether such solutions can lose their regularity in finite time. An important result that addresses the question of a possible loss of regularity of solutions to Euler equations..3 is the criterion formulated by BealeKato Majda [] in terms of the L norm of the vorticity ω = u. More precisely, BealeKatoMajda in [] proved the following theorem: Theorem.. Let u be a solution to..3 in the class.4 for s 3 integer. Suppose that there exists a time T such that the solution cannot be continued in the class.4 to T = T. If T is the first such time, then T ω, t L dt =.. Date: June 7,.
2 T. CHEN AND N. PAVLOVIĆ The theorem is proved with a contradiction argument. Under the assumption T ω, t L dt <, the authors of [] show that u, t H s C, for all t < T contradicting the hypothesis that T is the first time such that the solution cannot be continued to T = T. In particular, BealeKatoMajda obtain a double exponential bound for u, t H s, which follows from the following estimates: Step An energytype bound on u H s in terms of Du L, where Du = [ i u j ] ij is a 3 3matrix valued function. More specifically, one applies the operator D α to equations.., where α is an integervalued multiindex with α s and uses a certain commutator estimate to derive d dt u, t H C Du s L u, t Hs,.6 which via Gronwall s inequality gives the bound: t u, t H s u H s exp C Du, τ L dτ..7 Step An estimate on Du, t L based on the quantities ω, t L, ω, t L, and log + u, t H 3, given by Du, t L C { + + log + } u, t H 3 ω, t L + ω, t L,.8 where C is a universal constant. Step 3 The bound on ω, t L in terms of ω, t L given by d dt ω, t L D ω, t L ω, t L, which follows from taking the L R 3 inner product of ω with the equation for vorticity. Then, Gronwall s inequality yields t ω, t L ω, L exp D ω, τ L dτ..9 Consequently, one obtains the double exponential bound t u, t H s u H s exp exp C ω, τ L dτ.. from combining.7,.8 and.9. It is an open question whether. is sharp. While we do not attempt to answer that question itself in this paper, we obtain a single exponential bound on the H s norm of solution to Euler equations. .3 in terms of the quantity l δ t = min { L, ωt C δ u L δ+ },. Single exponential bounds have been obtained in other solution spaces than those displayed above, see for instance [7] for such a result in BMO.
3 BLOWUP RATE FOR EULER EQUATION 3 where ωx ωy ω C δ = sup x y <L x y δ. denotes the δholder seminorm, for L > fixed, and δ >. More precisely, we prove the following theorem: Theorem.. Let u be a solution to. .3 in the class.4, for s = + δ. Assume that l δ t is defined as above, and that T l δ τ dτ <..3 Then, there exists a finite positive constant C δ = Oδ independent of u and t such that [ t ] u, t H s u H s exp C δ u L l δ τ dτ holds for t T. The quantity l δ t has the dimension of length, and was introduced by Constantin in [] see also the work of Constantin, Fefferman and Majda [4] where a criterion for loss of regularity in terms of the direction of vorticity was obtained, where it was observed that T l δ t dt =.4 is a necessary and sufficient condition for blowup of Euler equations. In particular, the necessity of the condition follows from the inequality obtained in [] ω, t L u, t L l δ t,. and Theorem. of BealeKatoMajda. This is so because Theorem. implies that if the solution cannot be continued to some time T, then T ω, t L dt =. As a consequence of., and conservation of energy u, t L = u L,.6 this in turn implies.4. However, by invoking the result of BealeKatoMajda in this argument, one again obtains a double exponential bound on u, t H s in terms of T l δt dt. We refer to [3, ] for recent developments in this and related areas. In this paper, we observe that one can actually obtain a single exponential bound on the H s norm of the solution ut in terms of T l δt dt, as stated in Theorem.. This is achieved by avoiding the use of the logarithmic inequality.8 from []. More precisely, we combine the energy bound.6 with a CalderonZygmund type bound on the symmetric and antisymmetric parts of Du. Also, we obtain a lower bound on the blowup rate of solutions in H +δ. Specifically, we prove:
4 4 T. CHEN AND N. PAVLOVIĆ Theorem.3. Let u be a solution to..3 in the class C[, T ]; H +δ C [, T ]; H 3 +δ..7 Suppose that there exists a time T such that the solution cannot be continued in the class.7 to T = T. If T is the first such time then there exists a finite, positive constant Cδ, u L such that + u, t H +δ Cδ, u L δ T,.8 t under the condition that t is sufficiently close to T see the conditions 3. and 3.3 below, with t = t. The proof of Theorem.3 can be outlined as follows. We assume that u is a solution in the class.7 that cannot be continued to T = T, and that T is the first such time. Invoking the local in time existence result, we derive a lower bound T loc,t > on the time of existence of solutions to Euler equations in.7 for initial data ut H +δ at an arbitrary time t < T. By definition of T, we thus have t + T loc,t < T..9 Based on an energy bound on the H +δ norm of the solution, we obtain in Section 3 an expression for T loc,t of the form C u,t, which together with.9 +δ H implies that u, t H +δ > CT t,. for all t < T. This is an a priori lower bound on the blowup rate. Subsequently, we improve. by a recursion argument in Theorem.3 for times t close to T, to yield the stronger bound.8.. Proof of theorem. First we recall that the full gradient of velocity Du can be decomposed into symmetric and antisymmetric parts, where Du + is called the deformation tensor. Du = Du + + Du. Du ± = Du ± Du T.. In the following lemma we recall important properties of Du + and Du. For the convenience of the reader, we give proofs of those properties, although some of them are available in the literature, see e.g. []. Lemma.. For both the symmetric and antisymmetric parts Du +, Du of Du, the L bound holds. Du ± L C ω L..3
5 BLOWUP RATE FOR EULER EQUATION The antisymmetric part Du satisfies Du v = ω v.4 for any vector v R 3. The vorticity ω satisfies the identity ωξ = 4π P.V. σŷ ωx + y dy y 3,. P.V. denotes principal value where σŷ = 3 ŷ ŷ, with ŷ = y y. Notably, S σŷ dµ S y =,.6 where dµ S denotes the standard measure on the sphere S. The matrix components of the symmetric part have the form Du + ij = l T l ijω l = l K l ij ω l,.7 where ω l are the vector components of ω, and where the integral kernels K l ij have the properties K l ijy = σ l ijŷ y 3.8 σ l ij C S C.9 S σ l ijŷ dµ S y =.. Thus in particular, Tij l is a CalderonZygmund operator, for every i, j, l {,, 3}. Proof. An explicit calculation shows that the Fourier transform of Du as a function of ω is given by where and Ĝξ := ξ Duξ = [ i ω j ξ] i,j = Ĝξ + Ĥξ. ξ ξ ω 3 ξ ξ 3 ω ξ ξ 3 ω ξ ξ 3 ω 3 ξ ξ 3 ω ξ ξ 3 ω ξ ξ ω 3 ξ ξ 3 ω 3 ξ ξ ω ξ ξ ω ξ ξ 3 ω ξ ξ 3 ω Ĥξ := ξ using the notation ω j ω j ξ for brevity. ξ ω 3 ξ3 ω ξ ω 3 ξ3 ω ξ ω ξ ω.,.3 Clearly, every component of G is given by a sum of Fourier multiplication operators with symbols of the form ξiξj ξ, i j, applied to a component of ω. For instance, G x = const. P.V. ŷ ŷ 3 ω x + y dy y 3.4 corresponds to the component G. It is easy to see that every component G ij is a sum of CalderonZygmund operators applied to components of ω, with kernel
6 6 T. CHEN AND N. PAVLOVIĆ satisfying the asserted properties.8.. The same is true for the symmetric part, G + = G + GT. The symmetric part of Ĥξ is given by Ĥ + ξ = ξ ξ ω 3 ξ ξ3 ω ξ ξ ξ ω 3 ξ3 ξ ω ξ ξ3 ω ξ3 ξ ω. so that each component defines a Fourier multiplication operator with symbol of the form ξ i ξ j ξ, i j, acting on a component of ω with associated kernel of the form x i x j x n+. That is, for instance, H + x = const P.V. ŷ ŷ ω 3 x + y dy y 3..6 The properties.8. follow immediately. The Fourier transforms of the integral kernels Kij l can be read off from the components Ĝ+ ij + Ĥ+ ij. In position space, one finds that σl ij ŷ is obtained from a sum of terms proportional to terms of the form ŷ i ŷ j and ŷi ŷj. For the antisymmetric part Du, one generally has Du v = u v for any v R 3, and from u = ω, we get Du v = ω v, using that u =. As a side remark, we note that while H does not by itself exhibit the properties.8., it combines with G in a suitable manner to yield the stated properties of Du, thanks to the condition ω =. Next, Lemma. below provides an upper bound in terms of the quantity l δ t on singular integral operators applied to ω of the type appearing in.7. We note that similar bounds were used in [] and [4] for the antisymmetric part Du. Here, we observe that they also hold for the symmetric part Du +. As shown in [4] for Du, the proof of such a bound follows standard steps based on decomposing the singular integral into an inner and outer contribution. The inner contribution can be bounded based on a certain mean zero property, while the outer part is controlled via integration by parts. Lemma.. For L > fixed, and δ >, let l δ t be defined as above. Moreover, let ω l, l =,, 3, denote the components of the vorticity vector ωt. Then, any singular integral operator with satisfies T ω l x = 4π P.V. σ T ŷ ω l x + y dy y 3.7 S σ T ŷdµ S y =, σ T C S < C,.8 T ω l L Cδ u L l δ t.9 for l {,, 3}, for a constant Cδ = Oδ independent of u and t.
7 BLOWUP RATE FOR EULER EQUATION 7 Proof. Let χ x be a smooth cutoff function which is identical to on [, ], and identically for x >. Moreover, let χ R x = χ x/r, and χ c R = χ R. We consider y >ɛ for ɛ > arbitrary, where I := and σ T ŷ ω l x + y dy = I + II. y 3 y >ɛ II := σ T ŷ ω l x + y χ lδ t y dy y 3. σ T ŷ ω l x + y χ c dy l δ t y y 3.. From the zero average property.8, we find I L = σ T ŷ ω l x + y ω l x χ lδ t y dy y >ɛ y 3 dy ω l C δ y 3 δ y <l δ t C δ l δt δ ω l C δ since from the definition of l δ t, C δ u L l δ t.3 ω l C δ u L l δ t δ.4 follows straightforwardly. We can send ɛ, since the estimates are uniform in ɛ. On the other hand, II = σ T ŷ yi u j yj u i x + y χ c dy l δ t y y 3.. It suffices to consider one of the terms in the difference, σ T ŷ yi u j x + y χ c dy l δ t y y 3 = dy u j x + y yi σ T ŷ χ c l δ t y y 3 C u j L yi σ T ŷ χ c l δ t y L y 3 C u L l δ t.6 where to obtain the last line we used the conservation of energy.6 and the following three bounds: i yi χ c σt ŷ R y y 3 C L R R< y <R dy y 6 C R,.7
8 8 T. CHEN AND N. PAVLOVIĆ ii iii for R = l δ t. σ T ŷ χ c dy R y yi y 3 C L y >R y 8 χ c R y where we used that y σ T ŷ = holds. y 3 y i σ T ŷ C L C R..8 y >R y 6 y dy C R,.9 y zσ T z, z, z 3 z=ŷ y σ T C S.3 Summarizing, we arrive at T ω l L Cδ u L l δ t.3 for Cδ = Oδ, which is the asserted bound. The form of the singular integral operator that appears in the statement of Lemma. is suitable for application to Du + and Du, as we shall see in the following corollary. Corollary.3. There exists a finite, positive constant C δ = O δ independent of u and t such that the estimate holds. Du + L + Du L C δ u L l δ t.3 Proof. According to Lemma., the matrix components of both Du + and Du have the form.7. Accordingly, Lemma. immediately implies the assertion. Now we are ready to give a proof of Theorem., which is based on combining an energy estimate for Euler equations with Corollary.3. For s 3 integervalued, the energy bound.6 t ut H Dut s L ut H.33 s was proven in []. For fractional s >, we recall the definitions of the homogenous and inhomogenous Besov norms for p, q, u Ḃs = p,q j Z jqs u j q L p q,.34
9 BLOWUP RATE FOR EULER EQUATION 9 respectively, u B s p,q = u q L p + u q Ḃ s p,q q,.3 where u j = P j u is the PaleyLittlewood projection of u of scale j. In analogy to.6, we obtain the bound on the B s Besov norm of ut given by t ut B s Dut L ut B s,.36 from a straightforward application of estimates obtained in [8]; details are given in the Appendix. Accordingly, since the left hand side yields we get However, Corollary.3 implies that t ut B s = ut B s t ut B s,.37 t ut B s Dut L ut B s..38 Dut L Du + t L + Du t L Therefore, by combining.38 and.39 we obtain which implies that C δ u L l δ t..39 t ut B s C δ u L l δ t ut B s, ut H s ut B s u B s exp [ C δ u L [ u H s exp C δ u L t t for s, where we recall from.3 that C δ = Oδ. l δ s ds ] l δ s ds ], This completes the proof of Theorem.. 3. Lower bounds on the blowup rate In this section, we prove Theorem.3. Recalling the energy bound.38, t ut B s Dut L ut B s, 3. we invoke the Sobolev embedding Du L Du L dξ ξ 3 δ Du H 3 +δ C δ u H +δ C δ u B +δ, 3.
10 T. CHEN AND N. PAVLOVIĆ with C δ = Oδ, to get, for s = + δ, t ut B s C δ ut B s. 3.3 Straightforward integration implies C δ t t. 3.4 ut B s ut B s Hence, ut H s ut B s ut B s t t C δ ut B s ut H s, 3. t t C δ ut H s where a possible trivial modification of C δ is implicit in passing to the last line. This implies that the solution ut is locally wellposed in H s, with s = + δ, for t t < t C δ ut H s In particular, this infers that if T is the first time beyond which the solution u cannot be continued, one necessarily has that T > t C δ ut H s This in turn implies an a priori lower bound on the blowup rate given by ut H s > C δ T t 3.8 for all t < T. The lower bound on the blowup rate stated in Theorem.3 is stronger than this estimate, and we shall prove it in the sequel. To begin with, we note that ωt C δ C δ ωt H 3 +δ C δ ut H +δ C δ ut H +δ t t C δ ut H +δ. 3.9 That is, local wellposedness of u in H +δ implies δholder continuity of the vorticity. The parameter L in the definition. of l δ t is arbitrary. Thus, in view of 3.9, we may now let L for convenience. Then, l δ t = ωt C δ δ+ u L C δ ut H +δ δ u L Cδ δ ut H s δ, u L t t C δ ut H s 3.
11 BLOWUP RATE FOR EULER EQUATION where δ := δ + δ and s = + δ. 3. We note that while the right hand side of 3. diverges as t approaches the integral t t := t + t l δ t dt Cδ u L, 3. C δ ut H s δ t t ut H s t t C δ ut H s δdt =: B δ 3.3 converges for δ > δ >. This implies that the solution ut for t [t, t can be extended to t > t. In particular, we obtain that ut H +δ ut H +δ exp t C δ u L ut H +δ exp C δ u L B δ l δ t dt t 3.4 from Theorem.. We may now repeat the above estimates with initial data ut in H +δ, thus obtaining a local wellposedness interval [t, t ]. Accordingly, we may set t to be given by t := t + More generally, we define the discrete times t j by We then have ut j+ H s where B j δ is defined by C δ u L B j δ := C δ u L t j+ := t j + Cδ. 3. C δ ut H s. 3.6 C δ ut j H s exp C δ u L B j δ ut j H s, 3.7 u L = δ C δ u L δ δ ut j H s δ tj+ t j ut j H s t t j C δ ut j H s δdt Letting u =: b L δ δ. 3.8 ut j H s ρ j u δ := exp b L δ, 3.9 ut j H s
12 T. CHEN AND N. PAVLOVIĆ we have ut j H s ρ j ut j H s, 3. and we remark that ρ j j satisfy the recursive estimates u ρ j exp b L δ ρ j ut j H s = ρ j ρ δ j = exp ρ δ j ln ρ j We note that from its definition, ρ j > for all j. δ. 3. We shall now assume that T > is the first time beyond which the solution ut cannot be continued. Thus, by choosing t close enough to T, 3.8 implies that ut H s can be made sufficiently large that the following hold: The quantity is small. u b L δ δ 3. ut H s There is a positive, finite constant C independent of j such that ut j H s C ut H s 3.3 holds for all j N. Without any loss of generality by a redefinition of the constant b δ if necessary, we can assume that C =. Accordingly, 3.3 with C = implies that ρ j ρ for all j. Then, for any N N, T t N t j+ t j j= = + + C δ ut H s ut N H s = + ut H s + + ut H s C δ ut H s ut H s ut N H s C δ ut H s ρ ρ ρ N C δ ut H s ρ ρ N 3.4 from ρ j ρ for all j, and the fact that ρ > since the argument in the exponent 3.9 is positive.
13 BLOWUP RATE FOR EULER EQUATION 3 Then, letting N, we obtain T C δ ut H s t ρ = C δ ut H s Next, we deduce a lower bound on the blowup rate. Invoking 3., we obtain u δ exp b L δ. 3. ut H s u δ T C δ ut H s exp b L δ t ut H s u C δ ut H sb L δ δ ut H s = C δ b δ u δ L ut δ H s. 3.6 This implies a lower bound on the blowup rate of the form ut H +δ Cδ, u L T t = Cδ, u L under the condition that 3. and 3.3 hold. δ δ+ T t, 3.7 This concludes our proof of Theorem.3. Appendix A. Proof of inequality.38 for s > In this Appendix, we prove.38 which follows from.36, t ut B s Dut L ut B s, A. for s >. We invoke Eq. 6 in the work [8] of F. Planchon, which is valid for s > + n in n dimensions thus, s > in our case of n = 3, for parameter values p = q = in the notation of that paper. It yields t js u j L js S j+ Du L u k L u j L k j + js j k k u k L u k L Du j L A. where u k = P k u is the PaleyLittlewood projection of u at scale k, and S j = j j P j is the PaleyLittlewood projection to scales j.
14 4 T. CHEN AND N. PAVLOVIĆ Summing over j, t js u j L sup S j+ Du L js u k L u j L j j j k j + sj k ks u k L k s u k L j k k j Du L js u j L To pass to the second inequality, we used that j + sj k ks u k L k j k Du L js u j L. A.3 j S j+ Du L = m j+ Du L Du L m j+ L, A.4 where m j is the symbol of the Fourier multiplication operator S j, and the fact that m j L uniformly in j. Accordingly, we get From t ut Ḃ s Dut L ut. A. Ḃ s ut B = s ut L + ut, A.6 Ḃ s and energy conservation, t ut L =, we obtain t ut B s = t ut Ḃ s Dut L ut Ḃ s Dut L ut B s. A.7 This proves A.. Acknowledgments. The work of T.C. was supported by NSF grant DMS The work of N.P. was supported NSF grant number DMS 7847 and an Alfred P. Sloan Research Fellowship. References [] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equations, Comm. Math. Phys. 94, 6 66, 984. [] P. Constantin, Geometric statistics in turbulence, SIAM Rev. 36, 73 98, 994. [3] P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc. N.S. 44 4, 63 6, 7. [4] P. Constantin, C. Fefferman, A. Majda, Geometric constraints on potentially singular solutions for the 3D Euler equations, Comm. Part. Diff. Eq., 34, 9 7 [] J. Deng, T.Y. Hou, X. Yu, Improved Geometric Conditions for NonBlowup of the 3D Incompressible Euler Equation, Commun. Part. Diff. Eq. 3, 93 36, 6.
15 BLOWUP RATE FOR EULER EQUATION [6] T. Kato, Quasilinear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations Proc. Sympos., Dundee, 974; dedicated to Konrad Jörgens, 7. Lecture Notes in Mathematics 448. Springer, Berlin, 97. [7] H. Kozono,Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Commun. Math. Phys. 4, 9,. [8] F. Planchon, An Extension of the BealeKatoMajda Criterion for the Euler Equations, Commun. Math. Phys. 3, 39 36, 3. T. Chen, Department of Mathematics, University of Texas at Austin. address: N. Pavlović, Department of Mathematics, University of Texas at Austin. address:
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