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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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