On the local and global existence of solutions to 1D transport equations with nonlocal velocity

We consider the 1D transport equation with nonlocal velocity field: \begin{equation*}\label{intro eq} \begin{split}&\theta_t+u\theta_x+\nu \Lambda^{\gamma}\theta=0, \\&u=\mathcal{N}(\theta), \end{split} \end{equation*} where $\mathcal{N}$ is a nonlocal operator. In this paper, we show the existence of solutions of this model locally and globally in time for various types of nonlocal operators.

We here consider the 1D transport equations with nonlocal velocity field of the form θ t + uθ x + νΛ γ θ = 0, (1.1a) u = N (θ), (1.1b) where N is typically expressed by a Fourier multiplier. The study of (1.1) is mainly motivated by [11] where Córdoba, Córdoba, and Fontelos proposed the following 1D model (2) The existence of strong solution when the velocity u is more singular than θ. We intend to see the competitive relationship between nonlinear terms and viscous terms. More specifically, the topics covered in this paper can be summarized as follows.
• The model 1: N = −H and ν = 0. We first show the existence of local-in-time solution in a critical space under the scaling θ 0 (x) → θ 0 (λx). We then introduce the notion of a weak super-solution and obtain a global-in-time weak super-solution with θ 0 ∈ L 1 ∩ L ∞ and θ 0 ≥ 0.
• The model 2: N = −H(∂ xx ) −α , α > 0, ν = 1, and γ > 0. This is a regularized version of (1.2) which is also closely related to many equations as mentioned in [3]. In this case, we show the existence of weak solutions globally in time under weaker conditions on α and γ compared to [3].

Preliminaries
All constants will be denoted by C that is a generic constant. In a series of inequalities, the value of C can vary with each inequality. We use following notation: for a Banach space X, The Hilbert transform is defined as We will use the BMO space (see e.g. [4] for the definition) and its dual which is the Hardy space H 1 which consists of those f such that f and Hf are integrable. We will use the following formula 2H(f Hf ) = (Hf ) 2 − f 2 which implies that g = f Hf ∈ H 1 and for any f ∈ L 2 , (2.1) The differential operator Λ γ = ( √ −∆) γ is defined by the action of the following kernels [10]: where c γ > 0 is a normalized constant. Alternatively, we can define Λ γ = ( √ −∆) γ as a Fourier multiplier: Λ γ f (ξ) = |ξ| γ f (ξ). When γ = 1, Λf (x) = Hf x (x).
We finally introduce Simon's compactness.
Lemma 2.1. [22] Let X 0 , X 1 , and X 2 be Banach spaces such that X 0 is compactly embedded in X 1 and X 1 is a subset of X 2 . Then, for 1 ≤ p < ∞, the set v ∈ L p T X 0 : ∂v ∂t ∈ L 1 T X 2 is compactly embedded in L p T X 1 .

The model 1
We now study (1.1) with N = −H and ν = 0 which is nothing but (1.2): By the Bernstein inequality, we have We then apply Lemma 3.1 to the second term in the right-hand side of (3.2) to obtain 3.2. Global weak super-solution. We next consider (3.1) with rough initial data. More precisely, we assume that θ 0 satisfies the following conditions Since θ satisfies the transport equation, we have If we follow the usual weak formulation of (3.1), For θ 0 ≥ 0, there is gain of a half derivative from the structure of the nonlinearity, that is So, we can rewrite the left-hand side of (3.7) as However, theḢ 1 2 regularity derived from (3.8) is not enough to pass to the limit in from the ǫ-regularized equations described below. So, we introduce a new notion of solution. Let To prove Theorem 3.3, we need to estimate a commutator term involving Λ 1 2 : which is proved in [3].
. The second result in our paper is the following theorem. Proof. We first regularize initial data as θ ǫ 0 = ρ ǫ * θ 0 where ρ ǫ is a standard mollifier that preserve the positivity of the regularized initial data. We then regularize the equation by introducing the Laplacian term with a coefficient ǫ > 0, namely For the proof of the existence of a global-in-time smooth solution we refer to [17]. Moreover, θ ǫ satisfies that θ ǫ ≥ 0 and From this, we have uniform bounds Moreover, for any φ ∈ H 2 , Combining all together, we obtain To pass to the limit into the weak super-solution formulation, we extract a subsequence of (θ ǫ ), using the same index ǫ for simplicity, and a function θ ∈ A T such that where we use Lemma 2.1 for the strong convergence with We now multiply (3.10) by a test function ψ ∈ C ∞ c ([0, T ) × R) and integrate over R. Then, (3.12) We note that we are able to rearrange terms in the usual weak formulation into (3.12) since θ ǫ is smooth. By the strong convergence in (3.11), we can pass to the limit to I. Moreover, since T L 6 by Lemma 3.2 and the strong convergence in (3.11), we can pass to the limit to II. Lastly, by Fatou's lemma, Combining all the limits together, we obtain that This completes the proof.

The model 2
We now consider the following equation: where α, γ > 0. In this case, we focus on the existence of weak solutions under some conditions of (α, γ). As before, we assume that θ 0 satisfies the following conditions Definition 4.1. We say θ is a weak solution of (4.1) on the time interval The third result in the paper is the following.
Theorem 4.2. Suppose that two positive numbers α and γ satisfy Then, for any θ 0 satisfying (4.2), there exists a weak solution of (4.1) in B T for all T > 0.
Proof. As in the proof of Theorem 3.3, we regularize θ 0 and the equation as Then, the corresponding θ ǫ satisfies and We next multiply (4.4) by θ ǫ and integrate over R. Then, By (4.3), (4.5) and (4.6), we obtain Therefore, (θ ǫ ) is bounded in B T uniformly in ǫ > 0. From this, we have uniform bounds Moreover, the condition (4.3) implies that Combining all together, we also derive that θ ǫ t ∈ L 1 T H −2 . We now multiply (4.4) by a test function ψ ∈ C ∞ c ([0, T ) × R) and integrate over R. Then, (4.8) To pass the limit to this formulation, we extract a subsequence of (θ ǫ ), using the same index ǫ for simplicity, and a function θ ∈ B T such that where we use Lemma 2.1 for the strong convergence with the condition (4.3) and By the strong convergence in (4.9), we can pass to the limit to I and II in (4.8). Therefore, we obtain This completes the proof of Theorem 4.2.

The model 3
In this section, we consider the following equation where β, γ > 0. Depending on the range of β and γ, we will have four different results.
To show the uniqueness, let θ 1 and θ 2 be two solutions of (5.8), and let θ = θ 1 −θ 2 and u = u 1 −u 2 . Then, (θ, u) satisfies the following equations By taking the L 2 product of the equation with θ, So, θ = 0 in L 2 and thus a solution is unique. This completes the proof of Theorem 5.1.
Theorem 5.1 provides a local existence result for β ր 1 2 as γ ր 2. But, we can increase the range of β when we deal with (5.8) directly with γ = 2 because we can do the integration by parts. Proof. We begin the L 2 bound: We next estimate θ xx . Indeed, after several integration parts, we have When 0 < β < 1, And Therefore, we obtain This implies that there exists T = T ( θ 0 H 2 ) such that there exists a unique solution of (5.8) in We may lower the regularity of the initial data to prove a local existence result of a weak solution by considering initial data inḢ 1 2 . The main tools to achieve this will be the use of the Hardy-BMO duality together with interpolation arguments. However, in order to simplify the computation, we consider an equivalent equation by changing the sign of the nonlinearity: This can be obtained from (5.8) via θ → −θ. For this equation, we doḢ 1 2 estimates and prove that there exists a local existence of a unique solution in that special case. Proof. By recalling that Λ 2β = (−∂ xx ) β we get We now use the H 1 -BMO duality to estimate the right hand side of the last equality. By using the estimate (2.1) andḢ where we use the condition β ∈ 0, 1 2 again to derive the inequality. This implies local existence of a unique solution up to some time T = T ( θ 0 Ḣ 1 2 ).

5.2.
Global well-posedness. We finally deal with (5.8) with γ = 2. Proof. By Theorem 5.1, we only need to control the quantities in (5.2). Let u = −H(∂ xx ) β θ. We first note that (5.8) satisfies the maximum principle and so We take the L 2 inner product of (5.8) with θ. Then, we have We next take ∂ x to (5.8), take its L 2 inner product with θ x , and integrate by parts to obtain Since ( θ x (s) L 2 + θ xx (s) L 2 ) ds ≤ C (t, θ 0 L 1 , θ 0 H 2 ) and so we complete the proof of Theorem 5.4.

Appendix
This appendix is briefly written based on [4]. We first provide notation and definitions in the Littlewood-Paley theory. Let C be the ring of center 0, of small radius 3 4 and great radius 8 3 . We take smooth radial functions (χ, φ) with values in [0, 1] that are supported on the ball B 4 3 (0) and C, respectively, and satisfy (6.1) From now on, we use the notation φ j (ξ) = φ 2 −j ξ . We define dyadic blocks and lower frequency cut-off functions. We now define the homogeneous Besov spaces: We also recall Bernstein's inequality in 1D : for 1 ≤ p ≤ q ≤ ∞ and k ∈ N, Both O. L. and R.G.B. were partially supported by the Grant MTM2014-59488-P from the former Ministerio de Economía y Competitividad (MINECO, Spain).