On the ill-posedness of active scalar equations with odd singular kernels

Transcription

1 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page On the ill-posedness of active scalar equations with odd singular kernels Igor Kukavica Vlad Vicol Fei Wang Department of Mathematics, University of Southern California, Los Angeles, CA Department of Mathematics, Princeton University, Princeton, NJ Department of Mathematics, University of Southern California, Los Angeles, CA kukavica@usc.edu vvicol@math.princeton.edu wang828@usc.edu We consider active scalar equations with constitutive laws that are odd and very singular, in the sense that the velocity field loses more than one derivative with respect to the active scalar. We provide an example of such a constitutive law for which the equation is ill-posed: Either Sobolev solutions do not exist, from Gevrey-class datum, or the solution map fails to be Lipschitz continuous in the topology of a Sobolev space, with respect to Gevrey class perturbations in the initial datum. Keywords: Active scalar equations, ill-posedness, stability, Gevrey-class.. Introduction In this paper we address the well-posedness of the Cauchy problem for active scalar equations t θ + u θ = 0 ) u = 0 2) θx, 0) = θ 0 x) 3) posed on T 2 [0, ) = [ π, π] 2 [0, ) with a certain constitutive law for the incompressible drift u = T θ,

2 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page Igor Kukavica, Vlad Vicol, Fei Wang to be specified precisely below cf. 9) below). The datum θ 0 and the solution θ are taken to have zero mean on T 2. The study of active scalar equations of type ) 2) is motivated by several important fluid models: When T = ), the system becomes the vorticity formulation of the 2D Euler equations; the case T = ) /2 corresponds to the surface quasi-geostrophic equation ; the constitutive law T = ) models flow in an incompressible porous medium with Darcy s law 2 ; while T = B, where B is a scalar bounded Fourier multiplier, appears in magneto-geostrophic dynamics 3. The well-posedness theory of active scalar equations either inviscid or with fractional dissipation) has attracted considerable attention in the last two decades. We refer the readers to 23 and the references therein. Recently, these equations have been considered with velocity fields determined by a singular constitutive law i.e., the map T : θ u is unbounded) which is also odd by which we mean the corresponding Fourier multiplier is an odd function of frequency). In particular, in 24 the authors considered the velocity field given by u = R Λ β θ, 4) where β 0, ], Λ =, R = Λ, and showed the local existence and uniqueness of solutions for ) 3) with 4), in the Sobolev space H 4. The two key ingredients in their proof were an estimate for the commutator [ i Λ s, g]f L 2 and the identity faf g = f[a, g]f 5) 2 which holds for smooth f, g, where A = i Λ β is an odd operator. This result was later sharpened in 2 where the local existence was established in H σ with σ > 2 + β, using an improved commutator estimate. In contrast when the constitutive law is even and singular there are some ill-posedness results available for these equations. In 25,26, the authors established ill-posedness for even singular constitutive laws, i.e., for operators T that have an even and unbounded Fourier multiplier symbol. The main ingredients in these works were a linear ill-posedness result severe linear instability) in the spirit of 27,28, and a classical linear implies nonlinear ill-posedness argument 29,30. On the other hand, we are not aware of any ill-posedness result for ) 3) with an odd constitutive law. This is due to the special cancelation property 5), which technically reduces the order of the constitutive law by

3 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page 2 On the ill-posedness of active scalar equations with odd singular kernels 2 one. The main purpose of this paper is to establish an ill-posedness result for ) 3) with an odd constitutive law u = T θ = R Λ β Mθ 6) where < β 2 and M is a zero-order, scalar, even Fourier multiplier operator with symbol m. That is, Mθk) = mk)ˆθk), where the specific Fourier multiplier symbol m we consider is given by for all k Z 2, where we define mk) = k 2 k 2 φk) 7) φk) = {, k2 = ±,, otherwise. 8) For simplicity of notation, we denote by φi ) the Fourier multiplier operator with symbol φk). We may then rewrite 6) 8) concisely as u = R R 2 φi )Λ β θ 9) where we recall that β, 2] throughout this paper. Our main result Theorem 2.2 below) is to prove that the active scalar equation ) 3) with constitutive law 9) is ill-posed in any Gevrey space G s with s > 4 β)/β )3 β)). Here, by ill-posedness we mean that: either, given an initial datum θ 0 in G s, there is no local in time Sobolev solution θ L t Hx β/2++δ, where δ > 0 is arbitrary; or, the solution map θ 0 θ is not Lipschitz continuous in H β/2++δ, with respect to G s perturbations in the initial datum see Definition 2.2 below). We note that the condition β > is strictly necessary for our result to hold. Indeed, for β 0, ], by the results in 2,24, we know that for the system ) 3) and 9) we have local existence and uniqueness in the Sobolev space H σ with σ > 2 + β. Additionally, using the identity 5) and the aforementioned commutator estimates, one may show that for β 0, ] the equations are locally Lipschitz H β/2+, H β+3 ) well-posed, in the sense of Definition 2.2 below. Thus, the requirement β, 2] in this paper is necessary and sufficient. The proof of our main result is in the spirit of the earlier Lipschitz ill-posedness works 25,26 with the main difference being that as opposed to these works the eigenfunctions are constructed as sums of sines and cosines; this interplay leads to an eigenvalue problem involving a continued fraction

4 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page Igor Kukavica, Vlad Vicol, Fei Wang with all positive signs cf. 29) below). Such continued fractions require a different treatment than 25,26 due to non-monotonicity of the corresponding approximation sequence. Finally, we need a way to bound the Sobolev and Gevrey norms ensuring that the L 2 norm of the solutions grow arbitrarily fast in any short period of time. The paper is organized as follows. In Section 2, we linearize the active scalar equation around a steady solution and state our main linear and nonlinear results. In Section 3, we give the proof of the ill-posedness in Sobolev spaces and Gevrey classes for the linear equation. Finally, in Section 4, we conclude the ill-posedness for the non-linear problem in the corresponding Sobolev and Gevrey spaces using a perturbation argument. 2. The linearized problem and main results Consider a scalar function of the form θx, t) = θx 2 ), where θx 2 ) is a real function in T with zero mean. According to 9) we have that ū = T θ = 0, in view of the differentiation with respect to x inherent in T. Thus, any such θ is a steady state of ). We choose a particular function θx 2 ) = cosx 2 ) 0) and consider the linearization of the nonlinear term in ) around it. Denote the corresponding linear operator by Lθ = ū θ u θ = R 3 φi )Λ β θ 2 θ = sinx 2 ) R 3 φi )Λ β θ. ) We shall use the method of continued fractions see e.g ) to prove that the operator L has a sequence of eigenvalues whose positive real parts diverge to infinity cf. Theorem 2.3 below). In order to state our main results, we recall here the definition of the Gevrey-space G s cf. 3 ). Definition 2. Gevrey-space). A function θ C R 2 ) belongs to the Gevrey class G s, where s, if there exists a positive constant τ > 0, called the Gevrey-class radius, such that the G s τ -norm is finite, i.e., θ 2 G s τ = k Z 2 ˆθk) 2 k 4 e 2τ k /s <. 2) With this definition, G s τ = {θ C T 2 ): θ G s τ < } is an algebra for s, τ > 0, and we have G s = τ>0 Gs τ.

5 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page 23 On the ill-posedness of active scalar equations with odd singular kernels 23 Next we recall the definition of the Lipschitz well-posedness cf. 26,32 ). Definition 2.2 Locally Lipschitz X, Y ) well-posedness). Let Y X H β/2++δ be Sobolev spaces, where δ > 0 is arbitrary. We say that the Cauchy problem for the active scalar equation ) 3) with 6) is locally Lipschitz X, Y ) well-posed, if there exist continuous functions T, K : [0, ) 2 0, ), so that for every pair of initial data θ ) 0, ), θ 2) 0, ) Y there exist unique solutions θ ), θ 2) L 0, T ; X) of the initial value problem associated to ) 3) with 6), such that θ ) t, ) θ 2) t, ) X K θ ) 0, ) θ 2) 0, ) Y 3) for every t [0, T ], where T = T θ ) 0, ) Y, θ 2) 0, ) Y ) and K = K θ ) 0, ) Y, θ 2) 0, ) Y ). The role of the space H β/2++δ in the above definition is to ensure that the ranges of the linear operator L defined in ), and that of the nonlinear operator in ), defined as N[θ] = T θ θ, lie in L 2. Here, recall that u = T θ is given by 9). More precisely, by the Sobolev embedding we have and Lθ L 2 C θ H β C θ H β/2++δ 4) N[θ] L 2 = T θ θ L 2 C θ 2 H β/2++δ 5) for a sufficiently large constant C > 0. Note that we can set δ = 0 if β < 2. The role of the two-spaces X and Y in Definition 2.2 is to allow the solution to lose regularity, as it is expected due to the derivative losses of order β from T ) and from ) present in the nonlinearity. The first main result in this paper asserts that singular odd active scalar equations are locally Lipschitz ill-posed in Sobolev spaces. Theorem 2.. Assume β, 2], let δ > 0, and define r = β/2 + + δ. Then the system ) 3) with 6) is locally Lipschitz H r, H s ) ill-posed for any s > r. In fact, in view of the bound 8), one may obtain a stronger illposedness result: The solution is not well-posed in a class of Gevrey spaces. Theorem 2.2. Let β, 2], r = β/2 + + δ for some δ > 0, and let s > 4 β)/β )3 β). Then the system ) 3) with 6) is locally Lipschitz H r, G s τ ) ill-posed for τ > 0.

6 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page Igor Kukavica, Vlad Vicol, Fei Wang Both theorems above hold with δ = 0 in the case β < 2. The main ingredient in the proofs of Theorems 2. and 2.2 is the following bound on the eigenvalues of the linearized operator L defined in ). Theorem 2.3. The linearized operator L defined in ) has a sequence of entire real-analytic eigenfunctions {θ k } k with corresponding eigenvalues {λ k } k, such that λ k C 0 kβ 6) for all k, where C 0 0 is a universal constant. Moreover, we may normalize θ k such that either one of the following statements holds: a) Given s 0 θ k H s = and θ k L 2 C s k s4 β)/3 β) 7) for all k, where C s is a constant that depends only on s; b) given s and τ > 0 θ k G s τ = and θ k L 2 Cs,τ exp C s,τ k 4 β)/s3 β)) 8) for all k, where C s,τ is a constant that depends only on s, τ. A standard perturbation argument see e.g. 29,30 ) then implies the illposedness ) 3) with the constitutive law 9). 3. Proof of linear ill-posedness Proof. [Proof of Theorem 2.3] Fix an integer k. We seek an eigenvalueeigenfunction pair λ, θ) = λ k, θ k ) for L, with θ oscillating at frequency k with respect to x, i.e., we are looking for θ of the form θx, x 2 ) = c sinkx ) cosx 2 ) + sinkx ) c n cosnx 2 ) + coskx ) even n 2 odd n 3 c n sinnx 2 ) 9) where the real coefficients {c n } n are to be determined. Using φi )θx, x 2 ) = sinkx ) odd n c n cosnx 2 ) + coskx ) even n 2 c n sinnx 2 )

7 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page 25 On the ill-posedness of active scalar equations with odd singular kernels 25 we get Lθ = k 3 coskx ) k 3 sinkx ) = coskx ) odd n odd n + sinkx ) even n 2 n 2 + k 2 ) β 3)/2 c n cosnx 2 ) sinx 2 ) n 2 + k 2 ) β 3)/2 c n sinnx 2 ) sinx 2 ) k 3 n 2 + k 2 ) β 3)/2 c n even n 2 We introduce the positive coefficients sinn + )x 2) sinn )x 2 ) 2 k 3 n 2 + k 2 ) β 3)/2 c n cosn + )x 2) cosn )x 2 ). 20) 2 p n = p n,k = 2n 2 + k 2 ) 3 β)/2 k 3 2) omitting the k dependence for ease of notation. Then for all n we have even n 3 p n 2n 3 β k 3, 22) where by assumption 3 β. With 2) we simplify 20) as cn Lθ = coskx ) c ) n+ sinnx 2 ) c 2 sinkx ) cosx 2 ) p n p n+ p 2 even n 2 cn + sinkx ) c ) n+ cosnx 2 ). 23) p n p n+ The equation λθ = Lθ thus becomes λc sinkx ) cosx 2 ) + sinkx ) + coskx ) = coskx ) + sinkx ) even n 2 cn even n 2 even n 3 λc n sinnx 2 ) odd n 3 c n+ p n p n+ cn c n+ p n p n+ λc n cosnx 2 ) ) sinnx 2 ) c 2 sinkx ) cosx 2 ) p 2 ) cosnx 2 ). 24)

8 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page Igor Kukavica, Vlad Vicol, Fei Wang From 24) we obtain the recurrence relationship Denoting c n c n+ = λc n, n 2, 25) p n p n+ c 2 = λc, n =. 26) p 2 η n = cn p n ) ) cn, n 2 p n the recurrence relation 25) 26) becomes η n η n+ = λp n, n 2 27) η 2 = λp. 28) A real number λ yields a solution of 27) 28) if and only if it solves the continued fraction equation λp =. 29) λp 2 + λp 3 + Note that here the coefficients {p n } n are given by 2), and that λ is the only unknown. Next we show the existence of a solution λ > 0 of 29). For this purpose, we need to prove first that the continued fraction on the right side of 29) converges note that since the associated sequence is not monotone, this is not immediately clear), and then establish that it defines a continuous function of λ on 0, ). For n {2, 3, } and m {0,, 2, 3, }, define F n,m λ) = λp n + + λp n+m It is clear that F n,m λ) > 0 for λ 0, ). From the identity F n,m λ) = λp n + F n+,m λ).

9 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page 27 On the ill-posedness of active scalar equations with odd singular kernels 27 valid for n {2, 3, 4, } and m {, 2, 3, }, we deduce F n,m+ λ) F n,m λ) = λp n + F n+,m λ) λp n + F n+,m λ) F n+,m λ) F n+,m λ) = λp n + F n+,m λ))λp n + F n+,m λ)) from where we obtain F n,m+ λ) F n,m λ) λp n ) 2 F n+,mλ) F n+,m λ). Using induction on m and the bound F l, λ) F l,0 λ) = λp l + λp l+ λp l λ 3 p 2 l p l+ which is valid for all l 2, we obtain F n,m+ λ) F n,m λ) λp n ) 2 λp n+ ) 2 λp n+m ) 2 λ 3 p 2 n+mp n+m+ = λ 2m+3 p n+m+ p n p n+... p n+m ) 2 k 6m+9 n )! 2 ) 3 β 2 2m+3 λ 2m+3 n + m)!n + m + )! where used 22) in the last inequality. Hence, F 2,m+l λ) F 2,m λ) l j=0 k 6m+6j+9 2 2m+2j+3 λ 2m+2j+3 ) 3 β. 2 + m + j)!3 + m + j)! Since 3 β, the sequence F 2,m λ) is uniformly Cauchy on every compact subset of 0, ). Therefore, as m the sequence F 2,m λ) converges uniformly on compact subsets to a continuous function Since F 2 λ) =. λp 2 + λp 3 + we moreover have lim λ F 2 λ) = 0. The function F 2 λ) λp 2 30) Gλ) = F 2 λ) λp 3)

10 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page Igor Kukavica, Vlad Vicol, Fei Wang is continuous on 0, ) and satisfies On the other hand, we have the lower bound Gλ) = F 2 λ) λp lim Gλ) =. 32) λ λp 2 + λp 3 p p 3 λp = λ2p p 2 λp ) λp 3 Since p 3 > p, it follows that there exists λ 0 > 0, such that Gλ 0 ) > 0. 34) From 32), 34), and the intermediate value theorem, we conclude that there exists λ = λ k λ 0, ) such that Gλ k ) = 0 35) providing a solution of 29). Note that 33) and 35) imply providing a lower bound 0 λ 2 kp p 2 p p 3 36) λ k p /p 3 p p 2. Recalling 2), we obtain λ k k3 8/9 + k 2 )) 3 β)/ k 2 ) 3 β)/2 for all k. An explicit computation shows that lim k3 8/9 + k 2 )) 3 β)/2 k k β = 3 β 24 + k 2 ) 3 β)/2 for any β, 2]. Using x) a x for 0 a, x, we further obtain λ k C k β 37) for any k and any β, 2], where C is a universal constant. In particular, λ k as k at least as fast as the power law k β.

11 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page 29 On the ill-posedness of active scalar equations with odd singular kernels 29 Having found a root λ = λ k of 29), we next need to estimate the size of the coefficients c n defining θ in 9), and verify that they decay sufficiently fast so that θ is real-analytic. The function obeys the recurrence relationship or equivalently. Since also F n λ) =, λp n + λp n+ + F n λ) = λp n + F n+ λ), F n+ λ) = λp n + F n λ). 38) F 2 λ k ) = λ k p, 39) by the definition of λ k, we get, comparing the recurrence relations 27) 28) for η n and 38) 39) for F n λ k ), that F n λ k ) = η n. Using c n = p n /p n )c n η n for n 2 and we get F n λ k ) λ k p n, n 2 c n = η n η 2 p n c p = F n λ k ) F 2 λ k ) p n c λ k p n λ k p n λ k p 2 p n c p = p c λ n k p n... p showing rapid convergence of c n to 0. Without loss of generality we choose Then, by 22) and 37) we have c n C n k β from where c =. ) n k 3 ) n n )!) ) Cn k 4 β n 3 β n )!) 3 β 40) c n C n k 4 β) n n!) β 3, n. 4)

12 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page Igor Kukavica, Vlad Vicol, Fei Wang Furthermore, using n! n/c) n for a sufficiently large C, we obtain c n exp n log Ck 4 β n β 3)) 42) for all β, 2], n, and k. In particular, for any τ > 0, we see that for n nτ, k, β) := Ce 4τ k 4 β)/3 β), 43) where C is sufficiently large, it holds that c n exp 2τn 2 + k 2 ) /2). Using 9), we see that for l = l, l 2 ) Z 2, we have { c ˆθ k l) 2 2 = n, l = ±k, l 2 = ±n, 0, l k. Therefore, θ k 2 G = c 2 τ nk 2 + n 2 ) 2 exp 2τk 2 + n 2 ) /2) n nτ,k,β) n= + k 2 + n 2 ) 2 exp n>nτ,k,β) 50k 2τk 2 + n 2 ) /2 4 β + 2n log n 3 β n 2 + k 2 ) 2 exp 2τn 2 + k 2 ) /2) < which shows that the function θ k whose Fourier series n-th coefficients is c n, is in fact entire real-analytic. Next, we provide a proof of 7). For s 0, recalling 43) we estimate θ k 2 H s = n n 2 + k 2 ) s c 2 n )) n,k,β) 2 s n, k, β) 2s n= c 2 n + C s k 2s4 β)/3 β) θ k 2 L 2 + C s exp n 2 + k 2 ) s exp 2n 2 + k 2 ) /2) n>n,k,β) k 4 β)/3 β)) 2C s k 2s4 β)/3 β) θ k 2 L 2 44) where C s > 0 is a sufficiently large constant that depends only on s, and is independent of β, 2]; we have also used θ k 2 L, which follows 2 from c =. Since the equation Lθ k = λ k θ k is linear, we may renormalize

13 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page 22 On the ill-posedness of active scalar equations with odd singular kernels 22 θ k to have the unit H s norm. Then, upon multiplying both sides of 44) with this constant, we obtain 7). The proof of 8) is similar. Given s and τ > 0 we have θ k 2 G = c 2 s τ nk 2 + n 2 ) 2 exp 2τk 2 + n 2 ) /2s) n nτ, k, β) 4 exp 4τnτ, k, β)) + n>nτ,k,β) 2C s,τ exp nτ,k,β) /s) c 2 n n= k 2 + n 2 ) 2 exp 2τk 2 + n 2 ) /2s) 2τn 2 + k 2 ) /2) C s,τ k 4 β)/s3 β)) θ k 2 L 2 45) where we have also used that c =. In order to conclude the proof we renormalize θ k to have unit G s τ norm, and then multiply both sides of 45) with this constant we obtain 8). 4. Proof of nonlinear Lipschitz ill-posedness Proof. [Proof for Theorem 2.] The proof of the above theorem follows directly from Theorem 2.3 and a classical perturbative argument see, e.g. 25,26,29,30,32 ), so we only give here a sketch of these details. Fix θ ) 0 x 2) = θ ) x 2, t) = cosx 2 ) a steady state of the system ) 3) with 6). Clearly θ ) 0 Hs =. For ɛ 0, ], let θ2,ɛ) 0 x) = θ ) 0 x 2) + ɛψ 0 x) where ψ 0 x) is a smooth function such that ψ 0 x) H s =, to be determined below. We have θ 2,ɛ) 0 H s 2 for all ɛ 0, ]. If the system ) 3) with 6) would be locally Lipschitz well posed, then there would exist T 0, K 0 > 0 uniform in ɛ), such that the family of solutions θ 2,ɛ) x, t) L 0, T ; H r ) of ) 3) with 6) with the initial datum θ 2,ɛ) 0 x) obey sup θ 2,ɛ) t) θ ) 0 H r K 0 θ 2,ɛ) 0 θ ) 0 H s = K 0ɛ 46) t [0,T 0] for all ɛ 0, ]. We define ψ ɛ x, t) = ɛ θ 2,ɛ) x, t) θ ) 0 to be the ɛ-perturbation of θ 2,ɛ) about θ ) 0. In view of 46) we have that ψ ɛ is uniformly bounded in L 0, T ; H r ), with the bound ), sup ψ ɛ t) H r K 0. 47) t [0,T 0]

14 April 26, :49 WSPC Proceedings - 9in x 6in Volum final page The equation obeyed by ψ ɛ is t ψ ɛ = Lψ ɛ ɛn[ψ ɛ ] so that by 4) 5) we have that t ψ ɛ is uniformly bounded in L 0, T ; L 2 ). Thus, Aubin Lions lemma ψ ɛ ψ in L 2 0, T ; L 2 ), where ψ L 0, T ; H r ) with a bound inherited from 47), solves the equation t ψ = Lψ, ψ0) = ψ 0. In order to conclude the proof, let ψ 0 x) = θ k x), where θ k is an eigenfunction of L as constructed in Theorem 2.3, with eigenvalue λ k, and we choose k large enough such that exp T 0 k β /2C 0 ) ) C s k s4 β)/3 β) 2K 0 where C 0 is the constant from 6), and C s is the constant from 7). Then using 7) we obtain that ψt 0 /2) L 2 = expλ k T 0 /2) ψ 0 L 2 2K 0 which contradicts 47) since r > 0, thereby concluding the proof. Proof. [Proof of Theorem 2.2] The proof is the same as the proof of Theorem 2., except that the eigenfunction ψ 0 = θ k is normalized to have a unit G s τ norm, and we pick k large enough so that expt 0 k β /2C 0 )) C s,τ expc s,τ k 4 β)/s3 β)) ) 2K 0, where C 0 is the constant from 6), and C s,τ is the constant from 8). Finding such a values of k uses the restriction on s. Using 8) we then obtain that ψt 0 /2) L 2 2K 0 which yields a contradiction. Acknowledgments IK and FW were supported in part by the NSF grant DMS-3943, while VV was supported in part by the NSF grant DMS-5477 and by an Alfred P. Sloan Research Fellowship. References. P. Constantin, A. Majda and E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity 7, ).

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