# zbMATH — the first resource for mathematics

On a transport equation with nonlocal drift. (English) Zbl 1334.35254
Summary: In [Ann. Math. (2) 162, No. 3, 1377–1389 (2005; Zbl 1101.35052)], A. Córdoba et al. proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions $\partial _t \theta +u \; \partial _x \theta = 0, \quad u = H \theta ,$ where $$H$$ is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion $\partial _t \theta + u \; \partial _x \theta + \Lambda ^\gamma \theta = 0, \quad u = H \theta,$ where $$\Lambda = (-\Delta )^{1/2}$$, and $$1/2 \leq \gamma <1$$. Our results also apply to the model with velocity field $$u = \Lambda ^s H \theta$$, where $$s \in (-1,1)$$. We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the Hölder class in $$C^{(s+1)/2}$$, for all positive time.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35L60 First-order nonlinear hyperbolic equations
##### Keywords:
Burgers equation; Hilbert transform
Full Text:
##### References:
 [1] Alibaud, Natha\"el; Droniou, J\'er\^ome; Vovelle, Julien, Occurrence and non-appearance of shocks in fractal Burgers equations, J. Hyperbolic Differ. Equ., 4, 3, 479-499, (2007) · Zbl 1144.35038 [2] Baker, Gregory R.; Li, Xiao; Morlet, Anne C., Analytic structure of two $$1$$D-transport equations with nonlocal fluxes, Phys. D, 91, 4, 349-375, (1996) · Zbl 0899.76104 [3] Caffarelli, Luis A.; Vasseur, Alexis, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171, 3, 1903-1930, (2010) · Zbl 1204.35063 [4] Cardaliaguet, Pierre; Silvestre, Luis, H\"older continuity to Hamilton-Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side, Comm. Partial Differential Equations, 37, 9, 1668-1688, (2012) · Zbl 1255.35066 [5] Castro, A.; C\'ordoba, D., Global existence, singularities and ill-posedness for a nonlocal flux, Adv. Math., 219, 6, 1916-1936, (2008) · Zbl 1186.35002 [6] Castro, Angel; C\'ordoba, Diego, Self-similar solutions for a transport equation with non-local flux, Chin. Ann. Math. Ser. B, 30, 5, 505-512, (2009) · Zbl 1186.35154 [7] Chae, Dongho; C\'ordoba, Antonio; C\'ordoba, Diego; Fontelos, Marco A., Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. Math., 194, 1, 203-223, (2005) · Zbl 1128.76372 [8] Chan, Chi Hin; Czubak, Magdalena; Silvestre, Luis, Eventual regularization of the slightly supercritical fractional Burgers equation, Discrete Contin. Dyn. Syst., 27, 2, 847-861, (2010) · Zbl 1194.35320 [9] [CheskidovZaya13] A. Cheskidov and K. Zaya, Regularizing effect of the forward energy cascade in the inviscid dyadic model, arXiv: 1310.7612, 10 2013. · Zbl 1328.35159 [10] [ChoiHouKiselevLuoSverakYao14] K. Choi, T. Y. Hou, A. Kiselev, G. Luo, V. Sverak, and Y. Yao, On the finite-time blowup of a 1D model for the 3D axisymmetric Euler equations, arXiv:1407.4776, 2014. [11] Choi, Kyudong; Kiselev, Alexander; Yao, Yao, Finite time blow up for a 1D model of 2D Boussinesq system, Comm. Math. Phys., 334, 3, 1667-1679, (2015) · Zbl 1309.35072 [12] Constantin, P.; Lax, P. D.; Majda, A., A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math., 38, 6, 715-724, (1985) · Zbl 0615.76029 [13] Constantin, Peter; Majda, Andrew J.; Tabak, Esteban, Formation of strong fronts in the $$2$$-D quasigeostrophic thermal active scalar, Nonlinearity, 7, 6, 1495-1533, (1994) · Zbl 0809.35057 [14] Constantin, Peter; Tarfulea, Andrei; Vicol, Vlad, Long Time Dynamics of Forced Critical SQG, Comm. Math. Phys., 335, 1, 93-141, (2015) · Zbl 1316.35238 [15] Constantin, Peter; Vicol, Vlad, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22, 5, 1289-1321, (2012) · Zbl 1256.35078 [16] C\'ordoba, Antonio; C\'ordoba, Diego; Fontelos, Marco A., Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2), 162, 3, 1377-1389, (2005) · Zbl 1101.35052 [17] C\'ordoba, Antonio; C\'ordoba, Diego; Fontelos, Marco A., Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. Math. Pures Appl. (9), 86, 6, 529-540, (2006) · Zbl 1106.35059 [18] Dabkowski, Michael, Eventual regularity of the solutions to the supercritical dissipative quasi-geostrophic equation, Geom. Funct. Anal., 21, 1, 1-13, (2011) · Zbl 1210.35185 [19] Dabkowski, Michael; Kiselev, Alexander; Silvestre, Luis; Vicol, Vlad, Global well-posedness of slightly supercritical active scalar equations, Anal. PDE, 7, 1, 43-72, (2014) · Zbl 1294.35092 [20] De Gregorio, Salvatore, On a one-dimensional model for the three-dimensional vorticity equation, J. Statist. Phys., 59, 5-6, 1251-1263, (1990) · Zbl 0712.76027 [21] De Gregorio, Salvatore, A partial differential equation arising in a $$1$$D model for the $$3$$D vorticity equation, Math. Methods Appl. Sci., 19, 15, 1233-1255, (1996) · Zbl 0860.35101 [22] Do, Tam, On a 1D transport equation with nonlocal velocity and supercritical dissipation, J. Differential Equations, 256, 9, 3166-3178, (2014) · Zbl 1286.35058 [23] Dong, Hongjie, Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal., 255, 11, 3070-3097, (2008) · Zbl 1170.35004 [24] Dong, Hongjie; Du, Dapeng; Li, Dong, Finite time singularities and global well-posedness for fractal Burgers equations, Indiana Univ. Math. J., 58, 2, 807-821, (2009) · Zbl 1166.35030 [25] Dong, Hongjie; Li, Dong, On a one-dimensional $$α$$-patch model with nonlocal drift and fractional dissipation, Trans. Amer. Math. Soc., 366, 4, 2041-2061, (2014) · Zbl 1302.35334 [26] Evans, Lawrence C., Partial differential equations, Graduate Studies in Mathematics 19, xviii+662 pp., (1998), American Mathematical Society, Providence, RI · Zbl 0902.35002 [27] Gancedo, Francisco, Existence for the $$α$$-patch model and the QG sharp front in Sobolev spaces, Adv. Math., 217, 6, 2569-2598, (2008) · Zbl 1148.35099 [28] Kiselev, A., Regularity and blow up for active scalars, Math. Model. Nat. Phenom., 5, 4, 225-255, (2010) · Zbl 1194.35490 [29] Kiselev, Alexander, Nonlocal maximum principles for active scalars, Adv. Math., 227, 5, 1806-1826, (2011) · Zbl 1244.35022 [30] Kiselev, Alexander; Nazarov, Fedor; Shterenberg, Roman, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ., 5, 3, 211-240, (2008) · Zbl 1186.35020 [31] Kiselev, A.; Nazarov, F.; Volberg, A., Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167, 3, 445-453, (2007) · Zbl 1121.35115 [32] Li, Dong; Rodrigo, Jose, Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation, Adv. Math., 217, 6, 2563-2568, (2008) · Zbl 1138.35381 [33] Li, Dong; Rodrigo, Jos\'e L., On a one-dimensional nonlocal flux with fractional dissipation, SIAM J. Math. Anal., 43, 1, 507-526, (2011) · Zbl 1231.35172 [34] [LuoHou13] G. Luo and T. Y. Hou, Potentially singular solutions of the 3D incompressible Euler equations, arXiv:1310.0497, 2013. [35] Morlet, Anne C., Further properties of a continuum of model equations with globally defined flux, J. Math. Anal. Appl., 221, 1, 132-160, (1998) · Zbl 0916.35049 [36] Oberman, Adam M., Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems, SIAM J. Numer. Anal., 44, 2, 879-895 (electronic), (2006) · Zbl 1124.65103 [37] Okamoto, Hisashi; Sakajo, Takashi; Wunsch, Marcus, On a generalization of the Constantin-Lax-Majda equation, Nonlinearity, 21, 10, 2447-2461, (2008) · Zbl 1221.35300 [38] Sakajo, Takashi, On global solutions for the Constantin-Lax-Majda equation with a generalized viscosity term, Nonlinearity, 16, 4, 1319-1328, (2003) · Zbl 1140.76332 [39] Seregin, Gregory; Silvestre, Luis; Sver\'ak, Vladim\'\i r; Zlato\vs, Andrej, On divergence-free drifts, J. Differential Equations, 252, 1, 505-540, (2012) · Zbl 1232.35027 [40] Silvestre, Luis, Eventual regularization for the slightly supercritical quasi-geostrophic equation, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 27, 2, 693-704, (2010) · Zbl 1187.35186 [41] Silvestre, Luis, On the differentiability of the solution to an equation with drift and fractional diffusion, Indiana Univ. Math. J., 61, 2, 557-584, (2012) · Zbl 1308.35042 [42] [VasseurChan] A. Vasseur and C. H. Chan, \newblock Personal communication, \newblock2014.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.