On a one-dimensional \alpha-patch model with nonlocal drift and fractional dissipation

We consider a one-dimensional nonlocal nonlinear equation of the form: $\partial_t u = (\Lambda^{-\alpha} u)\partial_x u - \nu \Lambda^{\beta}u$ where $\Lambda =(-\partial_{xx})^{\frac 12}$ is the fractional Laplacian and $\nu\ge 0$ is the viscosity coefficient. We consider primarily the regime $0<\alpha<1$ and $0\le \beta \le 2$ for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D $\alpha$-patch models. In the critical and subcritical range $1-\alpha\le \beta \le 2$, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range $0 \le \beta<1-\alpha$, we prove formation of singularities in finite time for a class of smooth initial data. Our proof is based on a novel nonlocal weighted inequality which can be of independent interest.

Equation (1.1) becomes more singular as α decreases. It reduces to the inviscid Burger's equation when α = ν = 0, in which case it is well known that solutions may develop gradient blowup in finite time. When α = 0 and β ∈ (0, 2], (1.1) becomes the so-called fractal Burgers' equation, which is perhaps one of the simplest nonlinear equations with nonlocal terms. The fractal Burgers' equation was studied in detail recently; see, for instance, [2,1,16,13,20,5]. It is known that in the super-critical dissipative case β ∈ (0, 1), with very generic initial data the equation is locally well-posed and its solution may develops gradient blowup in finite time. In the critical and sub-critical dissipative case β ∈ [1,2], the equation is globally wellposed with arbitrary initial data in suitable Sobolev spaces. Another borderline situation is when α = 1 and ν = 0, in which case the equation is globally wellposed; see [21] for a proof in the periodic boundary condition case. As a matter of fact, in this case in the same spirit as the Beale-Kato-Majda criterion the solution is regular up to time T as long as T 0 u(t, ·) L ∞ dt < ∞.
This holds for any T since (1.1) is a transport equation and u(t, ·) L ∞ is nonincreasing.
In this paper, we consider primarily the regime 0 < α < 1 and 0 ≤ β ≤ 2 for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D α-patch models. We are interested in proving either the global regularity or finite-time blowup for (1.1). In the general case, the corresponding regularity criterion turns out to be which are not straightforward to verify. See Theorem 2.2 and Remark 2.5.
Let us describe the main results of the paper. In the critical and sub-critical range 1 − α ≤ β ≤ 2, we obtain global wellposedness for arbitrarily large initial data in suitable Sobolev spaces. In the full supercritical range 0 ≤ β < 1 − α, we prove formation of singularities in finite time for a family of smooth initial data. These results are in the same spirit of the results in [16,13] for the fractal Burgers' equation.
We state our main results of the paper more precisely in the following two theorems.
Theorem 1.2 (Global wellposedness in the critical and supercritical cases). Let 0 < α < 1, 1 − α ≤ β ≤ 2, and ν > 0. Assume 1 < p < 1 α and set k 0 = max{ For the proof of Theorem 1.1, we consider the evolution of a weighted integral of Λ −α u. The main difficulty is to show that such quantity satisfies some ordinary differential inequality, and would blow up in finite time under conditions a) and b) above. This then implies the blowup of the solution. Before, this type of approaches can be found, for instance, in [7,8,12,18,19,3]. We point out that, in these papers, weighted integral of solutions themselves are investigated, which unfortunately does not seem to work in our case. The key new observation in our case is that instead of working with the solution u itself, it is more natural to consider certain weighted integral of the nonlocal quantity Λ −α u and establish positive lower bounds expressed in terms of Λ −α u. More precisely we have the following nonlocal weighted inequality, which could be of independent interest. Proposition 1.3 (A nonlocal weighted inequality). Let 0 < α < 1 and 2α < δ < 2. There is a constant C α,δ > 0 such that for any odd function The parity assumption in Proposition 1.3 can probably be removed or relaxed by a more elaborate analysis but we will not do it here. The nontrivial point of Proposition 1.3 is to guess the correct positive lower bound such as the right-hand side of (1.3). We briefly explain the difficulty as follows. By a simple computation (see e.g. Lemma 3.1), we have for some constant C ′ α,δ > 0, From this and a scaling heuristic, one is led to conjecture the inequality The inequality (1.4) is close in spirit to the type of inequalities used in [7,8,12,18,19,3]. However a preliminary calculation shows that the inequality (1.4) is probably false unless some negative terms are added on the right-hand side of (1.4). To circumvent this difficulty we prove (1.3). Note that these two lower bounds are not equivalent although they obey the same scaling relations.
For the proof of Theorem 1.2, when β < 2 we adapt an idea of the non-local maximum principle for a suitably chosen modulus of continuity. This method was first used by Kiselev, Nazarov, and Volberg in [15], where they established the global regularity for the 2D critical dissipative quasi-geostrophic equations with periodic C ∞ data. We also refer the reader to [17] and the references therein for further applications and development of this method. In the case β = 2, this method does not seem to be applicable, and we use a different approach.
The remaining part of the paper is organized as follows. In the next section, we prove the local wellposedness, continuation criteria for (1.1), and Theorem 1.2. In section 3 we present several auxiliary lemmas, which will be used in the proofs of Proposition 1.3 and Theorem 1.1 in the last section.
We close this introduction by setting up some Notations. For any two quantities A and B, we use A B (resp. A B ) to denote the inequality A ≤ CB (resp. A ≥ CB) for a generic positive constant C.
The dependence of C on other parameters or constants are usually clear from the context and we will often suppress this dependence. The value of C may change from line to line. For any function f : R → R, we use f L p or sometimes f p to denote the usual Lebesgue L p norm of a function for 1 ≤ p ≤ ∞.
The fractional Laplacian operator Λ s , s ∈ R is defined via Fourier transform as The homogeneous Sobolev normḢ s for any s ≥ 0 is defined as f Ḣs = |∇| s f 2 or more explicitly: We will need to use the Littlewood-Paley (LP) frequency projection operators. For simplicity we shall fix the notations on R, but it is straightforward to define everything in R d for any d ≥ 1. To fix the notation, let φ ∈ C ∞ 0 (R) and satisfy For two real positive numbers α < β, define the frequency localized (LP) projection operator P α<·<β by Here F and F −1 denote the Fourier transform and its inverse transform, respectively. Similarly, the operators P <α and P >β are defined by and We recall the following Bernstein estimates: for any 1 ≤ p ≤ q ≤ ∞ and dyadic N > 0,

Local and global regularity
We first state the following positivity lemma which is a simple variant of Lemma 2.5 in [6]. We include the proof here for the sake of completeness.
Lemma 2.1 (Positivity lemma). Let F : R → R be a nondecreasing function. Assume 0 ≤ β ≤ 2. Then for any θ : R → R such that Λ β θ is well-defined and In particular for any 1 ≤ p < ∞, provided that the integral is well-defined.
Proof. Without loss of generality, we assume F is a smooth function. In the general case one can mollify F and deduce the result by a limiting argument. Consider first β = 2, in this case we just integrate by parts and obtain since F is non-decreasing. Now we assume 0 < β < 2. Recall that for 0 < β < 1, we have for some constant C β > 0. For β = 1 one can use Λg = H∂ x g (H is the Hilbert transform) and integration parts to show that the formula (2.2) still holds. For smooth g, an equivalent formula (without ǫ-limit) is given by the expression Similarly for 1 < β < 2 one can use Λ β g = −Λ β−2 ∂ xx g, fractional representation of the Riesz potential Λ −(2−β) and integration by parts (twice) to show that (2.2) also holds in this case. In this case a formula equivalent to (2.2) is given by In all cases we shall just use fractional representation of Λ β as in (2.2). Clearly Symmetrizing the above integral in x and y, we obtain which is clearly non-negative since F is a non-decreasing function.
Next we state and prove the local wellposedness and a continuation criterion for Equation (1.1). The main issue is the nonlocal drift term Λ −α u which induces some integrability constraints on u, see Remark 2.3 and Remark 2.4 below.
Remark 2.3. In Theorem 2.2, to have local wellposedness, we require the initial data u 0 to lie in L p for some 1 < p≤ 1 α . For 0 < α ≤ 1 2 , one can just work with pure H k spaces. However for 1 2 < α < 1, it is essential that u 0 belongs L p for some small p in order to control the low frequency part. The regularity condition k ≥ k 0 comes from bounding the quantity ∂ x u 1 α when we perform contraction estimates in C 0 t L p x (see e.g. (2.8)). Remark 2.4. Instead of the space L p ∩ H k , one can also choose the space W k,p for 1 < p < 1 α when 1 2 < α < 1. Another possibility is the space H −δ ∩ H k for some δ > 0. Of course, the choice of k in Theorem 2.2 is not optimal. We shall not dwell on these issues here. In any case these spaces provide natural L ∞ bounds on the drift term Λ −α u and produce classical solutions.
Proof of Theorem 2.2. The proof is more or less a standard application of energy estimates and hence we only sketch the details here. Define u (0) = u 0 , and for n ≥ 0 inductively define the iterates u (n+1) such that Multiplying both sides by |w (n+2) | p−2 w (n+2) , integrating by parts and using again (2.1), we have By Hölder and Sobolev embedding, To bound ∂ x u (n+1) (t) 1 α we discuss two cases. If 0 < α ≤ 1 2 , then one can use Sobolev embedding to get (2.7) If 1 2 < α < 1, then we use the interpolation inequality where we have used the fact that k > k 0 ≥ 1 θ . Plugging the bounds (2.7)-(2.8) into (2.6) and using (2.5), we obtain By using the fact w (n+1) (0) = 0 for all n ≥ 0, and choosing T 1 smaller than T if necessary, we get for some 0 <θ < 1 that . By interpolation one also obtains strong convergence and the limit solution u in C([0, T 1 ], H k−1 ). By a standard argument (one has to discuss separately the case ν = 0 and the case ν > 0, and the fact that strong continuity in H k is equivalent to weak continuity together with norm continuity), one can show that u ∈ C([0, T 1 ], H k ). We omit the details. Now we turn to the proof of the continuation criterion (2.3).
and hence u(t) p is controlled by (2.3). It remains to control the H k norm. From (1.1), using integration by parts and Hölder, we have By using the Gagliardo-Nirenberg inequalities, we have for any 1 ≤ l ≤ k, Plugging the above estimates into (2.10), and we obtain Together with (2.9) this easily yields (2.3). Finally in the case ν > 0 and 0 < β ≤ 2, one can use the theory of mild solutions or simple energy estimates to gain additional regularity. See, for instance, [10]. We omit the details.

Remark 2.5. From the logarithmic type bound
it is easily seen that the condition (2.3) can be replaced by However, we shall not use this fact in the sequel.
For the proof of Theorem 1.2, when β < 2 we shall use the idea of the non-local maximum principle for a suitably chosen modulus of continuity. This method was first used by Kiselev, Nazarov, and Volberg in [15], where they proved the global regularity for the 2D critical dissipative quasi-geostrophic equations with periodic C ∞ data. In the borderline case when β = 2, this argument does not seem to work since (2.14) below no longer holds. Therefore, we will use a different approach in this case.
We say a function f has modulus of continuity where ω is an increasing continuous function ω : [0, +∞) → [0, +∞) and ω(0) = 0. We say f has strict modulus of continuity ω if the inequality is strict for x = y.
In what follows, we will choose a concave function ω satisfying Owing to Theorem 2.2 and the Sobolev imbedding theorem, we may assume θ 0 ∈ H 20 ∪ C ∞ . Because of the scaling property of (1.1), for any λ > 0, is also a solution of (1.1) with initial data λ α+β−1 u 0 (λ β x). Thus if we can show that u λ is a global solution, the same remains true for u. Note that for any ω satisfying (2.11) we can always find a constant λ > 0 such that ω(ξ) is a strict modulus of continuity of λ α+β−1 u 0 (λ β x) provided that α + β > 1. While in the critical case, i.e., α + β = 1, this still holds for any unbounded ω satisfying (2.11).
We shall show that for suitably chosen ω, the modulus of continuity is preserved for all the time. This gives ∂ x u(t) ∞ ≤ ω ′ (0), which together with the uniform boundedness of u(t) ∞ and Theorem 2.2 implies Theorem 1.2. Following the argument in [15] and [11], the strict modulus of continuity is preserved at least for a short time. Also it is clear that if u(t, ·) has strict modulus of continuity ω for all t ∈ [0, T ), then θ is smooth up to T and u(T, ·) has modulus of continuity ω by continuity. Therefore, to show that the modulus of continuity is preserved for all the time, it suffices to rule out the case that which in turn implies that there exist two different points x, y ∈ R satisfying due to (2.11) and the decay of u at the spatial infinity (see [11]). This possibility can be eventually ruled out if we are able to chose suitable ω such that under the conditions above we have ∂ ∂t (u(T, x) − u(T, y)) < 0. (2.12) To this end, we need the following two lemmas.
Lemma 2.6. Assume that u has modulus of continuity ω, which is a concave function. Then for any α ∈ (0, 1), Λ −α u has modulus of continuity Proof. For any r > 0, take x, y ∈ R such that |x−y| = r. Without loss of generality, we may assume x = −r/2 and y = r/2. Then Here and where in the last inequality we have used the mean value theorem. The lemma is proved.
i) Then we have ii) For any β ∈ (0, 2), it holds that
Due to Lemmas 2.6 and 2.7, the left-hand side of (2.12) is less than or equal to I 3 + I 4 + I 5 , where r 1+β ds, where Ω(r) is given in (2.13). With concave ω, both I 4 and I 5 are strictly negative. Now we are ready to prove Theorem 1.2.
Proof of Theorem 1.2. Let δ > 0 be a small number to be specified later. We consider three cases separately.
Remark 2.8. A slightly different proof for β = 2 is possible and we sketch it below for the sake of completeness. Multiplying both sides of (1.1) by −∂ xx u and integrating by parts, we have Since u(t) ∞ ≤ u 0 ∞ , a Gronwall in time argument then yields Now global wellposedness quickly follows from the continuation criterion (2.3) and (2.24), since Remark 2.9. The above arguments can be modified to prove to the global wellposedness of the following 1D model when β = 2: where ν > 0 is a constant. This equation has been studied recently in [7], and later in [10,18]. It is now known that when β ∈ [1, 2) the equation is globally wellposed. While in the range β ∈ [0, 1/4), evolving from a family of initial data solutions blow up in finite time. In [7], an additional positivity assumption is imposed on u 0 . On the other hand, the proof of the global wellposedness in [10] relies on the non-local maximum principle, which does not work when β = 2 by the same reasoning above.
To deal with this borderline case, we multiply both sides of (2.26) by u and integrate by parts to get where in the last inequality, we use the boundedness of the Hilbert transform in L 2 and the maximum principle. As before, by Young's inequality and Gronwall's inequality, we get u(t) 2 ≤ Ce Ct , which further implies that Hu(t) 2 ≤ Ce Ct . Therefore, by the classical Sobolev theory, u is globally regular.
Alternatively, we multiply both sides of (2.26) by −∂ xx u and integrate by parts to get Since u(t) ∞ ≤ u 0 ∞ , by Young's inequality and Gronwall's inequality, we get Note that (2.25) still holds when α = 0, which implies the global regularity of u by the Beale-Kato-Majda criterion.

Auxiliary lemmas
This section is devoted to several auxiliary lemmas, which will be used in the proofs of Proposition 1.3 and Theorem 1.1 in the following section.
Proof of Lemma 3.1. We first note that for any odd function f = f (y), where C α > 0 is a constant depending only on α. By a simple scaling argument, we then have for x > 0, where C ′ α,δ > 0 is a constant. It is easy to check that the integral in the last equality converges for 0 < α < 1, 0 < δ < 2. The identity (3.1) follows easily.
Next we prove (3.2). Without loss of generality, we can assume 0 < β < 1. Note that for any odd function f , we have Again by a scaling argument, we have for x > 0, Since 0 < δ < 2 and 0 < β < 1, it is not difficult to check the last integral converges. The lemma is proved.

Proof of Lemma 3.2. By using the Littlewood-Paley projectors and noting that
where in the second inequality we have used the Bernstein inequality. In the third inequality we used the fact that α < α 1 to make the summation over dyadic N < 1 converge.
The following lemma is crucial for the proof of Proposition 1.3.
Proof of Lemma 3.4. Throughout this proof we will denote by the letter C any constant which depends only on (α, θ) but may vary from line to line. Note that (3.8) is a simple consequence of (3.4) and the fact that zΓ(z) = Γ(z + 1). We first establish the weaker bound To show this we first differentiate g and write Then using the fractional representation of Λ α and a simple scaling argument, we have for any x > 0, By using the binomial expansion and a simple computation, we have Using the asymptotics it is not difficult to check the series representation of G(λ) converges. Clearly then Owing to our assumption 0 < θ < 1 − α, 0 < α < 1, it is easy to check directly from the above expression that (3.11) holds. Now by (3.9) and (3.11), we obtain It remains to prove the bound (3.10) for λ sufficiently large. For this we need to use the Stirling's formula for the Gamma function. It states that for any complex z with |arg(z)| < π − ǫ (here arg(z) takes values in [−π, π)) and |z| ≫ 1, we have Using (3.9) and a tedious computation, we arrive at where O(λ −2 ) denotes the remainder term (complex-valued) whose absolute value is less than λ −2 . It follows easily that where C 1 > 0 is a constant depending only on (α, θ). Clearly where C 2 > 0 is another constant. The sharp bound (3.10) follows.

Finite time singularities
In this section, we complete the proofs of Proposition 1.3 and Theorem 1.1. The proof of Proposition 1.3 is inspired by an argument in [7] by using Mellin transforms and the corresponding Parseval identity. See also [8,12,18,19].
Proof of Proposition 1.3. Since by assumption u is odd, using Lemma 3.1, we get By using the Parseval identity for Mellin transforms, we have where By Lemma 3.4 and observing that ∂ x u is an even function on R, we obtain (note here θ = δ 2 − α and 0 < θ < 1 − α), where F α,θ (λ) was defined in (3.9). Substituting (4.3) into (4.1) and using (3.10), we have where the last step follows from (4.2) and the Parseval identity for Mellin transforms. This completes the proof of (1.3).
Lemma 4.1 (L 1 x norm is nonincreasing). Assume 0 < α < 1 and 0 ≤ β ≤ 2 in (1.1). Let the initial data u 0 ∈ L 1 x (R) ∩ H 1 x (R) be such that u 0 (x) ≥ 0 for any x ≥ 0 and u 0 is odd in x. Let u = u(t, x) be the corresponding solution to (1.1) with lifespan [0, T 0 ) where 0 < T 0 ≤ +∞. Then for any t ∈ [0, T 0 ), we have u(t, x) ≥ 0 for any x ≥ 0, and Proof of Lemma 4.1. We shall use the idea of time-splitting approximation. Namely the solution u = u(t, x) on any [0, T ′ ] with T ′ < T 0 can be approximated by interlacing the nonlinear evolution ∂ t f = Λ −α f ∂ x f with the linear evolution ∂ t g = −νΛ β g with small time step h = T ′ /N where N → ∞. Consider first the linear evolution ∂ t g = −νΛ β g on a time interval [0, h] with g 0 odd in x and g 0 (x) ≥ 0 for any x ≥ 0. Then clearly g = g(t, x) is also odd in x and has the representation where k(t, z) is the fundamental solution corresponding to the propagator ∂ t + νΛ β , which is nonnegative and radially decreasing. It is easy to check that g(t, x) ≥ 0 for any x ≥ 0. Furthermore where we used the fact that k(t, z) is nonnegative and ∞ −∞ k(t, z)dz = 1. We now check the nonlinear evolution ∂ t f = Λ −α f ∂ x f on the time interval [0, h]. Assume f 0 is odd in x and f 0 (x) ≥ 0 for any x ≥ 0. It is not difficult to check that f (t, x) is odd in x and f (t, x) ≥ 0 for any x ≥ 0. Then by using (3.3), integrating by parts, we compute (here for simplicity of presentation we omit any ǫ-regularization argument needed for the convergence of integrals): where in the last equality we have used a symmetrization in x and y to make the first integral corresponding to the kernel |x − y| −(2−α) sgn(x − y) vanish. Hence we obtain f (t, ·) L 1 x (R + ) ≤ f 0 L 1 x (R + ) . This completes the proof for the nonlinear evolution part.
We are now ready to complete the Proof of Theorem 1.1. We will argue by contradiction. Assume the solution corresponding to u 0 exists for all time, then by Lemma 3.1, we have ∞ 0 (Λ −α u)(t, x) x δ dx = C α,δ ∞ 0 u(t, x) x δ−α dx u(t, ·) L 1 x + 1 0 ∂ x u ∞ x δ−α−1 dx u(t, ·) L 1 x + ∂ x u L ∞ x < +∞. This shows that the integral ∞ 0 (Λ −α u)(t, x)x −δ dx is finite for any t ≥ 0. Next we show that the integral blows up at some finite T > 0 and obtain contradiction.
Substituting this last estimate into (4.4) and using again Lemma 3.2 and Cauchy-Schwartz, we obtain It is now clear that if u 0 satisfy (1.2) with sufficiently large constant C α,β,δ,ν , then we obtain the inequality of the form Clearly a(t) goes to infinity in finite time. We have obtained the desired contradiction and the proof is now completed.