American Institute of Mathematical Sciences

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September  2019, 14(3): 471-487. doi: 10.3934/nhm.2019019

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

Hantaek Bae 1,, , Rafael Granero-Belinchón 2, and  Omar Lazar 3,

1. 

Department of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), Ulsan, Republic of Korea
2. 

Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Santander, Spain
3. 

Departamento de Análisis Matemático & IMUS, Universidad de Sevilla, Sevilla, Spain

* Corresponding author

Received  June 2018 Revised  February 2019 Published  May 2019

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 and
$ \Lambda^{\gamma} $
 is a Fourier multiplier defined by
$ \widehat{\Lambda^{\gamma} f}(\xi) = |\xi|^{\gamma}\widehat{f}(\xi) $
. In this paper, we show the existence of solutions of this model locally and globally in time for various types of nonlocal operators.
Keywords: Fluid equations, 1D models, global weak solution.
Mathematics Subject Classification: Primary: 35A01; Secondary: 35D30, 35Q35, 35Q86.
Citation: Hantaek Bae, Rafael Granero-Belinchón, Omar Lazar. On the local and global existence of solutions to 1d transport equations with nonlocal velocity. Networks and Heterogeneous Media, 2019, 14 (3) : 471-487. doi: 10.3934/nhm.2019019
References:
[1]

H. Bae, D. Chae and H. Okamoto, On the well-posedness of various one-dimensional model equations for fluid motion, Nonlinear Anal., 160 (2017), 25-43.  doi: 10.1016/j.na.2017.05.002.

[2]

H. Bae and R. Granero-Belinchón, Global existence for some transport equations with nonlocal velocity, Adv. Math., 269 (2015), 197-219.  doi: 10.1016/j.aim.2014.10.016.

[3]

H. Bae, R. Granero-Belinchón and O. Lazar, Global existence of weak solutions to dissipative transport equations with nonlocal velocity, Nonlinearity, 31 (2018) 1484–1515. doi: 10.1088/1361-6544/aaa2e0.

[4]

H. Bahouri, J-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, 2011. doi: 10.1007/978-3-642-16830-7.

[5]

G. R. Baker, X. Li and A. C. Morlet, Analytic structure of 1D transport equations with nonlocal fluxes, Physica D., 91 (1996), 349-375.  doi: 10.1016/0167-2789(95)00271-5.

[6]

J. A. Carrillo, L. C. F. Ferreira and J. C. Precioso, A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity, Adv. Math., 231 (2012), 306-327.  doi: 10.1016/j.aim.2012.03.036.

[7]

A. Castro and D. Córdoba, Global existence, singularities and ill-posedness for a nonlocal flux, Adv. Math., 219 (2008), 1916-1936.  doi: 10.1016/j.aim.2008.07.015.

[8]

A. Castro and D. Córdoba, Self-similar solutions for a transport equation with non-local flux, Chinese Annals of Mathematics, Series B, 30 (2009), 505-512.  doi: 10.1007/s11401-009-0180-8.

[9]

D. Chae, A. Cordoba, D. Cordoba and M. Fontelos, Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. Math., 194 (2005), 203-223.  doi: 10.1016/j.aim.2004.06.004.

[10]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528.  doi: 10.1007/s00220-004-1055-1.

[11]

A. Córdoba, D. Córdoba and M. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math., 162 (2005), 1-13.  doi: 10.4007/annals.2005.162.1377.

[12]

M. Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana, 1 (1955), 105-167. 

[13]

S. De Gregorio, On a one-dimensional model for the 3D vorticity equation, J. Statist. Phys., 59 (1990), 1251-1263.  doi: 10.1007/BF01334750.

[14]

H. Dong, Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal., 255, (2008), 3070–3097. doi: 10.1016/j.jfa.2008.08.005.

[15]

H. Dong, On a multi-dimensional transport equation with nonlocal velocity, Adv. Math., 264 (2014), 747–761. doi: 10.1016/j.aim.2014.07.028.

[16]

H. Dong and D. Li, On a one-dimensional $\alpha$-patch model with nonlocal drift and fractional dissipation, Trans. Amer. Math. Soc., 366 (2014), 2041–2061. doi: 10.1090/S0002-9947-2013-06075-8.

[17]

J. Duoandikoetxea, Fourier Analysis, Translated and revised from the 1995 Spanish original by David Cruz-Uribe, Graduate Studies in Mathematics, 29, American Mathematical Society, 2000.

[18]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[19]

A. Kiselev, Regularity and blow up for active scalars, Math. Model. Math. Phenom., 5 (2010), 225-255.  doi: 10.1051/mmnp/20105410.

[20]

O. Lazar, On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion, Journal of Diff. Eq., 261 (2016), 4974-4996.  doi: 10.1016/j.jde.2016.07.009.

[21]

O. Lazar and P.-G. Lemarié-Rieusset, Infinite energy solutions for a 1D transport equation with nonlocal velocity, Dynamics of PDEs, 13 (2016), 107-131. doi: 10.4310/DPDE.2016.v13.n2.a2.

[22]

D. Li, On Kato-Ponce and fractional Leibniz, arXiv: 1609.01780. doi: 10.4171/rmi/1049.

[23]

D. Li and J. Rodrigo, Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation, Adv. Math., 217 (2008), 2563-2568.  doi: 10.1016/j.aim.2007.11.002.

[24]

D. Li and J. Rodrigo, On a one-dimensional nonlocal flux with fractional dissipation, SIAM J. Math. Anal., 43 (2011), 507-526.  doi: 10.1137/100794924.

[25]

A. Morlet, Further properties of a continuum of model equations with globally defined flux, J. Math. Anal. Appl., 221 (1998), 132-160.  doi: 10.1006/jmaa.1997.5801.

[26]

J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

show all references

References:
[1]

H. Bae, D. Chae and H. Okamoto, On the well-posedness of various one-dimensional model equations for fluid motion, Nonlinear Anal., 160 (2017), 25-43.  doi: 10.1016/j.na.2017.05.002.

[2]

H. Bae and R. Granero-Belinchón, Global existence for some transport equations with nonlocal velocity, Adv. Math., 269 (2015), 197-219.  doi: 10.1016/j.aim.2014.10.016.

[3]

H. Bae, R. Granero-Belinchón and O. Lazar, Global existence of weak solutions to dissipative transport equations with nonlocal velocity, Nonlinearity, 31 (2018) 1484–1515. doi: 10.1088/1361-6544/aaa2e0.

[4]

H. Bahouri, J-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, 2011. doi: 10.1007/978-3-642-16830-7.

[5]

G. R. Baker, X. Li and A. C. Morlet, Analytic structure of 1D transport equations with nonlocal fluxes, Physica D., 91 (1996), 349-375.  doi: 10.1016/0167-2789(95)00271-5.

[6]

J. A. Carrillo, L. C. F. Ferreira and J. C. Precioso, A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity, Adv. Math., 231 (2012), 306-327.  doi: 10.1016/j.aim.2012.03.036.

[7]

A. Castro and D. Córdoba, Global existence, singularities and ill-posedness for a nonlocal flux, Adv. Math., 219 (2008), 1916-1936.  doi: 10.1016/j.aim.2008.07.015.

[8]

A. Castro and D. Córdoba, Self-similar solutions for a transport equation with non-local flux, Chinese Annals of Mathematics, Series B, 30 (2009), 505-512.  doi: 10.1007/s11401-009-0180-8.

[9]

D. Chae, A. Cordoba, D. Cordoba and M. Fontelos, Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. Math., 194 (2005), 203-223.  doi: 10.1016/j.aim.2004.06.004.

[10]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528.  doi: 10.1007/s00220-004-1055-1.

[11]

A. Córdoba, D. Córdoba and M. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math., 162 (2005), 1-13.  doi: 10.4007/annals.2005.162.1377.

[12]

M. Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana, 1 (1955), 105-167. 

[13]

S. De Gregorio, On a one-dimensional model for the 3D vorticity equation, J. Statist. Phys., 59 (1990), 1251-1263.  doi: 10.1007/BF01334750.

[14]

H. Dong, Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal., 255, (2008), 3070–3097. doi: 10.1016/j.jfa.2008.08.005.

[15]

H. Dong, On a multi-dimensional transport equation with nonlocal velocity, Adv. Math., 264 (2014), 747–761. doi: 10.1016/j.aim.2014.07.028.

[16]

H. Dong and D. Li, On a one-dimensional $\alpha$-patch model with nonlocal drift and fractional dissipation, Trans. Amer. Math. Soc., 366 (2014), 2041–2061. doi: 10.1090/S0002-9947-2013-06075-8.

[17]

J. Duoandikoetxea, Fourier Analysis, Translated and revised from the 1995 Spanish original by David Cruz-Uribe, Graduate Studies in Mathematics, 29, American Mathematical Society, 2000.

[18]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[19]

A. Kiselev, Regularity and blow up for active scalars, Math. Model. Math. Phenom., 5 (2010), 225-255.  doi: 10.1051/mmnp/20105410.

[20]

O. Lazar, On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion, Journal of Diff. Eq., 261 (2016), 4974-4996.  doi: 10.1016/j.jde.2016.07.009.

[21]

O. Lazar and P.-G. Lemarié-Rieusset, Infinite energy solutions for a 1D transport equation with nonlocal velocity, Dynamics of PDEs, 13 (2016), 107-131. doi: 10.4310/DPDE.2016.v13.n2.a2.

[22]

D. Li, On Kato-Ponce and fractional Leibniz, arXiv: 1609.01780. doi: 10.4171/rmi/1049.

[23]

D. Li and J. Rodrigo, Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation, Adv. Math., 217 (2008), 2563-2568.  doi: 10.1016/j.aim.2007.11.002.

[24]

D. Li and J. Rodrigo, On a one-dimensional nonlocal flux with fractional dissipation, SIAM J. Math. Anal., 43 (2011), 507-526.  doi: 10.1137/100794924.

[25]

A. Morlet, Further properties of a continuum of model equations with globally defined flux, J. Math. Anal. Appl., 221 (1998), 132-160.  doi: 10.1006/jmaa.1997.5801.

[26]

J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

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