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

1.   Introduction

In this paper, we study transport equations with nonlocal velocity. One of the most well-known equation is the two dimensional Euler equation in vorticity form,

where the velocity $ u $ is recovered from the vorticity $ \omega $ through

Other nonlocal and quadratically nonlinear equations, such as the surface quasi-geostrophic equation, the incompressible porous medium equation, Stokes equations, magneto-geostrophic equation in multi-dimensions, have been studied intensively as one can see in [1,2,5,6,7,8,9,13,14,15,16,19,20,23,24,25] and references therein.

We here consider the 1D transport equations with nonlocal velocity field of the form

where $ \mathcal{N} $ is typically expressed by a Fourier multiplier. The differential operator $ \Lambda^{\gamma} = (\sqrt{-\Delta})^{\gamma} $ is defined by the action of the following kernels [10]:

where $ c_{\gamma}>0 $ is a normalized constant. Alternatively, we can define $ \Lambda^{\gamma} = (\sqrt{-\Delta})^{\gamma} $ as a Fourier multiplier: $ \widehat{\Lambda^{\gamma} f}(\xi) = |\xi|^{\gamma}\widehat{f}(\xi) $. The study of 1 is mainly motivated by [11] where Córdoba, Córdoba, and Fontelos proposed the following 1D model

for the 2D surface quasi-geostrophic equation and proved the finite time blow-up of smooth solutions. In this paper, we deal with 3a-3b and its variations with the following objectives.

(1) The existence of weak solution with rough initial data. The existence of global-in-time solutions is possible even if strong solutions blow up in finite time, as in the case of the Burgers' equation.

(2) The existence of strong solution when the velocity $ u $ is more singular than $ \theta $. 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: $ \mathcal{N} = -\mathcal{H} $ and $ \nu = 0 $. We first show the existence of local-in-time solution in a critical space under the scaling $ \theta_{0}(x)\mapsto \theta_{0}(\lambda x) $. We then introduce the notion of a weak super-solution and obtain a global-in-time weak super-solution with $ \theta_{0}\in L^{1}\cap L^{\infty} $ and $ \theta_{0}\ge 0 $.

● The model 2: $ \mathcal{N} = -\mathcal{H}(\partial_{xx} )^{-\alpha} $, $ \alpha>0 $, $ \nu = 1 $, and $ \gamma>0 $. This is a regularized version of 3a-3b 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 $ \alpha $ and $ \gamma $ compared to [3].

● The model 3: $ \mathcal{N} = -\mathcal{H}(\partial_{xx} )^{\beta} $, $ \beta>0 $, $ \nu = 1 $, and $ \gamma>0 $. Since $ \beta>0 $, the velocity field is more singular than the previous two models. In this case, we show the existence of strong solutions locally in time in two cases: (1) $ 0<\beta\leq \frac{\gamma}{4} $ when $ 0<\gamma<2 $ and (2) $ 0<\beta<1 $ when $ \gamma = 2 $. We also show the existence of strong solutions for $ 0<\beta<\frac{1}{2} $ and $ \gamma = 2 $ with rough initial data. We finally show the existence of strong solutions globally in time with $ 0<\beta<\frac{1}{4} $ and $ \gamma = 2 $.

We will give detailed statements and proofs of our results in Section 3–5.

2.   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 spatial derivatives are defined as

For $ l = 1, 2, 3 $, we also use the followings:

2.1.   Hilbert transform

We now give some properties of the Hilbert transform and related function spaces. The Hilbert transform is defined by

We will use the BMO space and its dual which is the Hardy space $ \mathcal{H}^1 $ which consists of those $ f $ such that $ f $ and $ \mathcal{H}f $ are integrable [17,Chapter 6]. By the following Cotlar formula [12]

we have $ f\mathcal{H}f \in \mathcal{H}^1 $ and for any $ f\in L^{2} $,

We have $ \Lambda f(x) = \mathcal{H}f_{x}(x) $ by using $ \widehat{Hf}(\xi) = -\text{sgn}(\xi) \widehat{f}(\xi) $, where $ \Lambda $ is defined in 2.

2.2.   Function spaces

Since we are dealing with equations on $ \mathbb{R} $, we state some definitions and function spaces on $ \mathbb{R} $.

Let $ f\in \mathcal{S}' $, a tempered distribution. Then, its Fourier transform is defined by

Let $ s\in \mathbb{R} $. The energy space $ H^{s} $ is defined by

We also define homogeneous spaces:

We note that for $ s>0 $ and $ \sigma>0 $,

because

In this paper, we also use two estimations in [18].

(1) Fractional product rule. For $ s>0 $ and $ p $, $ p_{i} $, and $ q_{i} $ such that

we have the following estimations:

and

where $ (I-\Delta) $ is defined as the Fourier multiplier whose symbol is $ 1+|\xi|^{2} $.

(2) Commutator estimate.

2.3.   Littlewood-Paley theory

We here briefly introduce the Littlewood-Paley theory based on [4]. We first provide notation and definitions in the Littlewood-Paley theory. Let $ \mathcal{C} $ be the ring of center 0, of small radius $ \frac{3}{4} $ and great radius $ \frac{8}{3} $. We take smooth radial functions $ (\chi, \phi) $ with values in $ [0, 1] $ that are supported on the ball $ B_{\frac{4}{3}}(0) $ and $ \mathcal{C} $, respectively, and satisfy

From now on, we use the notation

We define dyadic blocks and lower frequency cut-off functions.

Then, the homogeneous Littlewood-Paley decomposition is given by

where $ \mathcal{S}^{'}_{h} $ is the space of tempered distributions $ u\in \mathcal{S}^{'} $ such that

We now define the homogeneous Besov spaces:

We recall Bernstein's inequality in 1D : for $ 1\leq p\leq q \leq \infty $ and $ k\in \mathbb{N} $,

Moreover, the Besov spaces enjoy nice scaling properties. Let $ f_{\lambda}(x) = f(\lambda x) $. Then, there exists a constant $ C $ such that

We also have the following commutator estimate.

Lemma 2.1 (Commutator estimate). For $ f, g \in \mathcal{S} $ (Schwarz class)

We finally introduce Simon's compactness.

Lemma 2.2. [26] 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\leq p<\infty $, the set $ \left\{v\in L^{p}_{T}X_{0}: \ \frac{\partial v}{\partial t}\in L^{1}_{T}X_{2}\right\} $ is compactly embedded in $ L^{p}_{T}X_{1} $.

3.   The model 1

We now study 1a-1b with $ \mathcal{N} = -\mathcal{H} $ and $ \nu = 0 $ which is nothing but 3a-3b:

3.1.   Local well-posedness

The local well-posedness of 14a-14b is established in $ H^{2} $ ([2]) and $ H^{\frac{3}{2}-\gamma} $ with the viscous term $ \Lambda^{\gamma}\theta $ ([14]). To improve these results, we notice that 14a-14b has the following scaling invariant property: if $ \theta(t, x) $ is a solution of 14a-14b, then so is $ \theta_{\lambda}(t, x) = \theta(\lambda t, \lambda x) $. So, we take initial data in a space whose norm is closely invariant under the scaling:

In this paper, we take the space $ \dot{B}^{\frac{3}{2}}_{2, 1} $ because there is a constant $ C $ such that

by taking $ s = \frac{3}{2} $, $ p = 2 $, and $ r = 1 $ in 13. The first result in this paper is the following theorem.

Theorem 3.1. For any $ \theta_{0}\in \dot{B}^{\frac{3}{2}}_{2, 1} $, there exists $ T = T(\|\theta_0\|) $ such that a unique solution of 14a-14b exists in $ C_{T}\dot{B}^{\frac{3}{2}}_{2, 1} $.

Proof. We only provide a priori estimates of $ \theta $ in the space stated in Theorem 3.1. The other parts, including the approximation procedure, are rather standard.

We apply $ \Delta_{j} $ to 14a, multiply by $ \Delta_{j}\theta $, and integrate the resulting equation over $ \mathbb{R} $ to get

By the Bernstein inequality, we have

We then apply Lemma 2.1 to the second term in the right-hand side of 15 to obtain

By 15, 16, and 17, we have

from which we deduce

This completes the proof.

3.2.   Global weak super-solution

We next consider 14a-14b with rough initial data. More precisely, we assume that $ \theta_{0} $ satisfies the following conditions

Since $ \theta $ satisfies the transport equation, we have

If we follow the usual weak formulation of 14a-14b, for all $ \psi\in C^{\infty}_{c}([0, T)\times \mathbb{R}) $

For $ \theta_{0}\ge 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 20 as

However, the $ \dot{H}^{\frac{1}{2}} $ regularity derived from 21 is not enough to pass to the limit in

from the $ \epsilon $-regularized equations described below. So, we introduce a new notion of solution. Let

Definition 3.2. We say $ \theta $ is a weak super-solution of 14a-14b on the time interval $ [0, T] $ if $ \theta(t, x)\ge 0 $ for all $ t\in [0, T] $, $ \theta\in \mathcal{A}_{T} $, and for each nonnegative $ \psi\in C^{\infty}_{c}([0, T)\times\mathbb{R}) $,

To deal with the third term in 22, we use the following Lemma.

Lemma 3.3. [3] For $ f\in L^{\frac{3}{2}} $, $ g\in L^{\frac{3}{2}} $ and $ \psi \in W^{1, \infty} $, we have

The second result in our paper is the following theorem.

Theorem 3.4. For any $ \theta_{0} $ satisfying 18, there exists a weak super-solution of 14a-14b in $ \mathcal{A}_{T} $.

Proof. We first regularize initial data as $ \theta^{\epsilon}_{0} = \rho_{\epsilon}\ast \theta_{0} $ where $ \rho_{\epsilon} $ 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 $ \epsilon>0 $, namely

For the proof of the existence of a global-in-time smooth solution we refer to [21]. Moreover, $ \theta^{\epsilon} $ satisfies that $ \theta^{\epsilon}\ge 0 $ and

Therefore, $ (\theta^{\epsilon}) $ is bounded in $ \mathcal{A}_{T} $ uniformly in $ \epsilon>0 $.

The first two terms on the left-hand side of 24 imply

for any $ p, q\in[1, \infty] $. In particular,

These two bounds imply

by the embedding 6. From $ \theta^{\epsilon}\in L^{2}_{T}\dot{H}^{\frac{1}{2}} $, we also have

by the embedding 6. Moreover, for any $ \phi \in H^{2} $,

which implies that

Combining all together, we obtain

To pass to the limit into the weak super-solution formulation, we extract a subsequence of $ \left(\theta^{\epsilon}\right) $, using the same index $ \epsilon $ for simplicity, and a function $ \theta \in \mathcal{A}_T $ such that

where we use Lemma 2.2 for the strong convergence with

We now multiply 23 by a nonnegative test function $ \psi\in \mathcal{C}^{\infty}_{c}\left([0, T)\times\mathbb{R}\right) $ and integrate over $ \mathbb{R} $. Then,

We note that we are able to rearrange terms in the usual weak formulation into 26 since $ \theta^{\epsilon} $ is smooth. By the strong convergence in 25, we can pass to the limit to $ \text{I} $. Moreover, since

strongly in $ L^{2}_{T}L^{6} $ by Lemma 3.3 and the weak convergence in 25, we can pass to the limit to $ \text{II} $. Lastly, define

We then have that $ g^\epsilon\rightharpoonup g $ in $ L^2([0, T]\times\mathbb{R}) $. Since the $ L^2 $ norm is weakly lower semicontinuous, we find that

or, equivalently,

Combining all the limits together, we obtain that

This completes the proof.

4.   The model 2

We now consider the following equation:

where $ \alpha, \gamma>0 $. In this case, we focus on the existence of weak solutions under some conditions of $ (\alpha, \gamma) $. As before, we assume that $ \theta_{0} $ satisfies the following conditions

Let

Definition 4.1. We say $ \theta $ is a weak solution of 28a-28b on the time interval $ [0, T] $ if $ \theta(t, x)\ge 0 $ for all $ t\in [0, T] $, $ \theta\in \mathcal{B}_{T} $, and for each $ \psi\in C^{\infty}_{c}([0, T)\times\mathbb{R}) $,

The third result in the paper is the following.

Theorem 4.2. Suppose that two positive numbers $ \alpha $ and $ \gamma $ satisfy

Then, for any $ \theta_{0} $ satisfying 29, there exists a weak solution of 28a-28b in $ \mathcal{B}_{T} $ for all $ T>0 $.

Proof. As in the proof of Theorem 3.4, we regularize $ \theta_{0} $ and the equation as

Then, the corresponding $ \theta^{\epsilon} $ satisfies

and

We next multiply 31 by $ \theta^{\epsilon} $ and integrate over $ \mathbb{R} $. Then,

where we use the fractional product rule 8 to obtain

By this bound and 32, we have

We now consider $ \left\|(1-\Delta)^{-\frac{\gamma}{4}}\Lambda (\partial_{xx})^{-\alpha}\theta^{\epsilon}(t)\right\|_{L^{2}} $. For $ |\xi|\leq 1 $,

For $ |\xi|\ge 1 $,

So,

By 34 and 35, we obtain

and so

By Gronwall's inequality,

where we use 33 to bound the time integral of $ \left\|\Lambda^{\frac{1}{2}}(\partial_{xx})^{-\frac{\alpha}{2}}\theta^{\epsilon}(t)\right\|^{2}_{L^{2}} $. Hence we finally derive the following

Therefore, $ (\theta^{\epsilon}) $ is bounded in $ \mathcal{B}_{T} $ uniformly in $ \epsilon>0 $.

By 32 and 33,

We next consider $ \mathcal{H}(\partial_{xx})^{-\alpha}\theta^{\epsilon} $. We first choose $ \beta \in \left[0, \frac{1}{2}\right) $ also satisfying

Then,

and so

Moreover, by Sobolev embedding,

where $ \beta $ is defined in 39. By 37, we also have

Combining all together, we derive that

Finally, 7 and 38 imply that

To pass the limit to this formulation, we extract a subsequence of $ \left(\theta^{\epsilon}\right) $, using the same index $ \epsilon $ for simplicity, and a function $ \theta \in \mathcal{B}_T $ such that

Here, we use Lemma 2.2 with

to obtain 42b. Similarly, we use Lemma 2.2 with the condition 30 and

to obtain 42c.

We now multiply 31 by a test function $ \psi\in \mathcal{C}^{\infty}_{c}\left([0, T)\times\mathbb{R}\right) $ and integrate over $ \mathbb{R} $. Then,

By 40 and the strong convergence in 42b, we can pass to the limit to $ \text{I} $. By 41 and the strong convergence in 42c, we can also pass to the limit to $ \text{II} $. Therefore, we obtain

This completes the proof of Theorem 4.2.

Remark 1. Theorem 4.2 improves Theorem 1.4 in [3], where $ (\alpha, \gamma) $ is assumed to satisfy $ \alpha\ge \frac{1}{2}-\frac{\gamma}{4} $. The main idea of taking weaker regularization in 28a-28b is that the Hilbert transform in front of $ (1-\partial_{xx})^{-\alpha} $ gives 33 which makes to obtain 37. We choose $ \alpha> \frac{1}{2}-\frac{\gamma}{2} $ instead of $ \alpha\ge \frac{1}{2}-\frac{\gamma}{2} $ to apply compactness argument when we pass to the limit to $ \epsilon $-regularized equations.

5.   The model 3

In this section, we consider the following equation

where $ \beta, \gamma>0 $. Depending on the range of $ \beta $ and $ \gamma $, we will have four different results.

5.1.   Local well-posedness

We begin with the local well-posedness result.

Theorem 5.1. Let $ 0<\gamma<2 $ and $ 0<\beta\leq \frac{\gamma}{4} $. For $ \theta_0 \in H^2 (\Bbb R) $ there exists $ T = T(\|\theta_0\|_{H^2}) $ such that a unique solution of 44a-44b exists in $ C_{T}H^2 $. Moreover, we have the following blow-up criterion:

where $u = -\mathcal{H}(\partial_{xx})^{\beta}\theta$.

Proof. Operating $\partial^{l}$ on 44a, taking its $L^2$ inner product with $\partial^l \theta$, and summing over $l = 0, 1, 2$,

Using the commutator estimate 9, we have

And by integration by parts,

Since $ \beta\leq \frac{\gamma}{4} $, for a sufficiently small $ \kappa>0 $

By 47 and 48, we obtain

from which we deduce that there is $ T = T(\|\theta_{0}\|_{H^{2}}) $ such that

49 also implies 45.

To show the uniqueness, let $ \theta_{1} $ and $ \theta_{2} $ be two solutions of 44a-44b, and let $ \theta = \theta_{1}-\theta_{2} $ and $ u = u_{1}-u_{2} = -\mathcal{H}(\partial_{xx})^{\beta}\theta_{1} + -\mathcal{H}(\partial_{xx})^{\beta}\theta_{2} $. Then, $ (\theta, u) $ satisfies the following equations

By taking the $ L^{2} $ product of the equation with $ \theta $,

So, $ \theta = 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 $ \beta \nearrow \frac{1}{2} $ as $ \gamma\nearrow 2 $. But, we can increase the range of $ \beta $ when we deal with 44a-44b directly with $ \gamma = 2 $ because we can do the integration by parts.

Theorem 5.2. Let $ \gamma = 2 $ and $ 0<\beta<1 $. For $ \theta_0 \in H^2 (\Bbb R) $ there exists $ T = T(\|\theta_0\|_{H^2}) $ such that a unique solution of 44a-44b exists in $ C_{T}H^2 $.

Proof. We begin the $ L^{2} $ bound:

We next estimate $ \theta_{xx} $. Indeed, after several integration parts, we have

When $ 0<\beta<1 $,

And

Therefore, we obtain

This implies that there exists $ T = T(\|\theta_0\|_{H^2}) $ such that there exists a unique solution of 44a-44b in $ C_{T}H^2 $.

We may lower the regularity of the initial data to prove a local existence result of a weak solution by considering initial data in $ \dot H^{\frac{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 51a-51b via $ \theta \mapsto -\theta $. For this equation, we do $ \dot H^{\frac{1}{2}} $ estimates and prove that there exists a local existence of a unique solution in that special case.

Theorem 5.3. Let $ \gamma = 2 $ and $ 0<\beta<\frac{1}{2} $. For any $ \theta_0 \in \dot H^{\frac{1}{2}} (\Bbb R) $, there exists $ T = T(\Vert \theta_{0} \Vert_{\dot H^{\frac{1}{2}}}) $ such that there exists a unique local-in-time solution in $ C_{T}\dot H^{\frac{1}{2}} \cap L^2_{T}\dot{H}^{\frac{3}{2}} $.

Proof. By recalling that $ \Lambda^{2\beta} = (-\partial_{xx})^{\beta} $ we get

We now use the $ \mathcal{H}^1 $-BMO duality to estimate the right hand side of the last equality. By using the estimate 5 and $ \dot H^{\frac{1}{2}} \hookrightarrow BMO $, we obtain

and thus we have

By fixing $ \gamma = 2 $ and by using the interpolation inequalities

where we use $ \frac{1}{2}\leq 2\beta+\frac{1}{2} \leq \frac{3}{2} $ for $ \beta \in \left(0, \frac{1}{2}\right) $ to get the second inequality. Hence, we obtain

where we use the condition $ \beta \in \left(0, \frac{1}{2}\right) $ again to derive the inequality. This implies local existence of a unique solution up to some time $ T = T(\Vert \theta_{0} \Vert_{\dot H^{\frac{1}{2}}}) $.

Remark 2. In the case $ \beta = 1/2 $, the equation reduces to the following Hamilton-Jacobi equation (or primitive Burgers equation)

For this equation, it seems that a naive approach based in energy methods cannot work. Indeed, if we multiply by $ \Lambda\theta $ and integrate by parts the nonlinearity takes a commutator structure

However, it seems that, at this level of regularity, this commutator is comparable to an energy estimate:

which is also equivalent to the use of Hardy-BMO duality:

Also, the best estimate that one has for the commutator $ [\mathcal{H}, \theta_{x}] \theta_{x} $ in $ L^{2} $ is that it is controlled by $ \Vert\theta_{x}\Vert_{BMO}\Vert\theta\Vert_{\dot H^{1}} $ (see e.g. [22]) which is, once again, similar to the use of the Hardy-BMO duality. So the commutator structure is not that useful in this special case.

Remark 3. It is also unclear whether the local solution starting from an arbitrary initial data becomes smooth. However, for smooth initial data satisfying size restriction in appropriate spaces, one can prove the desired smoothing effect.

5.2.   Global well-posedness

We finally deal with 51a-51b with $ \gamma = 2 $.

Theorem 5.4. Let $ \gamma = 2 $ and $ \beta<\frac{1}{4} $. For any $ \theta_0 \in H^2 (\Bbb R) $, there exists a unique global-in-time solution in $ C_{T}H^2 $.

Proof. By Theorem 5.1, we only need to control the quantities in 45. Let $ u = -\mathcal{H}(\partial_{xx})^{\beta}\theta $. We first note that 51a-51b satisfies the maximum principle and so

We take the $ L^2 $ inner product of 51a with $ \theta $. Then,

Since

we have

We next take $ \partial_x $ to 51a, take its $ L^2 $ inner product with $ \theta_x $, and integrate by parts to obtain

Since

we obtain

We also obtain

By 53, 54 and 55, we finally obtain

and so we complete the proof of Theorem 5.4.

Acknowledgments

The authors acknowledge the referees for their valuable comments and suggestions that highly improved the manuscript.

H.B. was supported by NRF-2018R1D1A1B07049015.

R.G.B. was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR), and by the Universidad de Cantabria.

O.L. was partially supported by the Marie-Curie Grant, acronym: TRANSIC, from the FP7-IEF program and by the ERC through the Starting Grant project H2020-EU.1.1.-63922.

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).