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

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.
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. BaeD. 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. BakerX. 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. CarrilloL. 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. ChaeA. CordobaD. 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. CórdobaD. 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. BaeD. 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. BakerX. 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. CarrilloL. 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. ChaeA. CordobaD. 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. CórdobaD. 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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