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April  2019, 12(2): 215-229. doi: 10.3934/dcdss.2019015

Classical solutions for the system $\bf {\text{curl}\, v = g}$, with vanishing Dirichlet boundary conditions

Luigi C. Berselli , and  Placido Longo

Dipartimento di Matematica, Università di Pisa, Via F. Buonarroti 1/c, Pisa, I-56127, Italy

* Corresponding author: Luigi C. Berselli

Dedicated to Prof. Vicenţiu D. Rădulescu on the occasion of his 60 th birthday

Received  June 2017 Revised  November 2017 Published  August 2018

We consider the boundary value problem associated to the curl operator, with vanishing Dirichlet boundary conditions. We prove, under mild regularity of the data of the problem, existence of classical solutions.

Keywords: Curl and divergence equation, first order systems, classical solutions.
Mathematics Subject Classification: Primary: 26B12; Secondary: 35C05, 35F15.
Citation: Luigi C. Berselli, Placido Longo. Classical solutions for the system $\bf {\text{curl}\, v = g}$, with vanishing Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 215-229. doi: 10.3934/dcdss.2019015
References:
[1]

H. Beirão da Veiga, On the solutions in the large of the two-dimensional flow of a nonviscous incompressible fluid, J. Differential Equations, 54 (1984), 373–389. doi: 10.1016/0022-0396(84)90149-9.  Google Scholar

[2]

H. Beirão da Veiga, Concerning the existence of classical solutions to the Stokes system. On the minimal assumptions problem, J. Math. Fluid Mech., 16 (2014), 539–550. doi: 10.1007/s00021-014-0170-9.  Google Scholar

[3]

H. Beirão da Veiga, On some regularity results for the stationary Stokes system and the 2-D Euler equations, Port. Math., 72 (2015), 285–307. doi: 10.4171/PM/1969.  Google Scholar

[4]

H. Beirão da Veiga, Elliptic boundary value problems in spaces of continuous functions, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 43–52. doi: 10.3934/dcdss.2016.9.43.  Google Scholar

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L. C. Berselli and L. Bisconti, On the existence of almost-periodic solutions for the 2D dissipative Euler equations, Rev. Mat. Iberoam., 31 (2015), 267–290. doi: 10.4171/RMI/833.  Google Scholar

[6]

L. C. Berselli and P. Longo, Classical solutions of the divergence equation with Dini-continuous datum, Tech. Report, arXiv (2017) URL http://arXiv.org/abs/1712.07917. Google Scholar

[7]

M. E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators div and grad, in Theory of Cubature Formulas and the Application of Functional Analysis to Problems of Mathematical Physics, vol. 1980 of Trudy Sem. S. L. Soboleva, No. 1, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, (1980), 5–40,149.  Google Scholar

[8]

J. Bolik and W. von Wahl, Estimating $\nabla\textbf{u}$ in terms of ${\rm div}\, \textbf{u}$, ${\rm curl}\, \textbf{u}$ either $(ν, \textbf{u})$ or $ν×\textbf{u}$ and the topology, Math. Methods Appl. Sci., 20 (1997), 737–744. doi: 10.1002/(SICI)1099-1476(199706)20:9<737::AID-MMA863>3.3.CO;2-9.  Google Scholar

[9]

W. Borchers and H. Sohr, On the equations ${\rm rot}\, \textbf{v} = \textbf{g}$ and ${\rm div}\, \textbf{u} = f$ with zero boundary conditions, Hokkaido Math. J., 19 (1990), 67–87. doi: 10.14492/hokmj/1381517172.  Google Scholar

[10]

J. Bourgain and H. Brezis, New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc. (JEMS), 9 (2007), 277–315. doi: 10.4171/JEMS/80.  Google Scholar

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V. I. Burenkov, Sobolev Spaces on Domains, vol. 137 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998. doi: 10.1007/978-3-663-11374-4.  Google Scholar

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U. Dini, Fourier Series and Other Analytic Representations of Functions of one Real Variable. (Serie di Fourier ed Altre Rappresentazioni Analitiche Delle Funzioni Di Una Variabile Reale), Nistri, 1880. Google Scholar

[13]

U. Dini, Sur la méthode des approximations successives pour les équations aux derivées partielles du deuxième ordre, Acta Math., 25 (1902), 185–230. doi: 10.1007/BF02419026.  Google Scholar

[14]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems, Springer Monographs in Mathematics, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.  Google Scholar

[16]

R. Griesinger, Decompositions of $L^q$ and $H^{1, q}_0$ with respect to the operator ${\rm rot}$, Math. Ann., 288 (1990), 245–262. doi: 10.1007/BF01444533.  Google Scholar

[17]

R. Griesinger, The boundary value problem ${\rm rot}\, u = f, \; u$ vanishing at the boundary and the related decompositions of $L^q$ and $H^{1, q}_0$: Existence, Ann. Univ. Ferrara Sez. VII (N. S.), 36 (1990), 15–43 (1991), URL http://dx.doi.org/10.1007/BF02837204.  Google Scholar

[18]

H. Helmholtz, Ueber die Theorie der Elektrodynamik. Zweite Abhandlung. Kritisches, J. Reine Angew. Math., 75 (1873), 35–66. doi: 10.1515/crll.1873.75.35.  Google Scholar

[19]

H. Koch, Transport and instability for perfect fluids, Math. Ann., 323 (2002), 491–523. doi: 10.1007/s002080200312.  Google Scholar

[20]

A. Korn, Über Minimalflächen, Deren Randkurven Wenig Von Ebenen Kurven Abweichen, Preuss Akad Wiss., 1909. Google Scholar

[21]

H. Kozono and T. Yanagisawa, $L^r$-variational inequality for vector fields and the HelmholtzWeyl decomposition in bounded domains, Indiana Univ. Math. J., 58 (2009), 1853–1920. doi: 10.1512/iumj.2009.58.3605.  Google Scholar

[22]

S. L. Sobolev, On a theorem of functional analysis, Mat. Sbornik, English Transl. : Amer. Math. Soc. Transl., 34 (1963), 39–68. Google Scholar

[23]

S. L. Sobolev, Introduction to the Theory of Cubature Formulae, Nauka, 1974, (Russian).  Google Scholar

[24]

W. von Wahl, On necessary and sufficient conditions for the solvability of the equations ${\rm rot}\, u = γ$ and ${\rm div}\, u = \epsilon$ with u vanishing on the boundary, in The Navier-Stokes equations (Oberwolfach, 1988), vol. 1431 of Lecture Notes in Math., Springer, Berlin, 1990, 152–157. doi: 10.1007/BFb0086065.  Google Scholar

[25]

H. Weyl, The method of orthogonal projection in potential theory, Duke Math. J., 7 (1940), 411–444. doi: 10.1215/S0012-7094-40-00725-6.  Google Scholar

show all references

References:
[1]

H. Beirão da Veiga, On the solutions in the large of the two-dimensional flow of a nonviscous incompressible fluid, J. Differential Equations, 54 (1984), 373–389. doi: 10.1016/0022-0396(84)90149-9.  Google Scholar

[2]

H. Beirão da Veiga, Concerning the existence of classical solutions to the Stokes system. On the minimal assumptions problem, J. Math. Fluid Mech., 16 (2014), 539–550. doi: 10.1007/s00021-014-0170-9.  Google Scholar

[3]

H. Beirão da Veiga, On some regularity results for the stationary Stokes system and the 2-D Euler equations, Port. Math., 72 (2015), 285–307. doi: 10.4171/PM/1969.  Google Scholar

[4]

H. Beirão da Veiga, Elliptic boundary value problems in spaces of continuous functions, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 43–52. doi: 10.3934/dcdss.2016.9.43.  Google Scholar

[5]

L. C. Berselli and L. Bisconti, On the existence of almost-periodic solutions for the 2D dissipative Euler equations, Rev. Mat. Iberoam., 31 (2015), 267–290. doi: 10.4171/RMI/833.  Google Scholar

[6]

L. C. Berselli and P. Longo, Classical solutions of the divergence equation with Dini-continuous datum, Tech. Report, arXiv (2017) URL http://arXiv.org/abs/1712.07917. Google Scholar

[7]

M. E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators div and grad, in Theory of Cubature Formulas and the Application of Functional Analysis to Problems of Mathematical Physics, vol. 1980 of Trudy Sem. S. L. Soboleva, No. 1, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, (1980), 5–40,149.  Google Scholar

[8]

J. Bolik and W. von Wahl, Estimating $\nabla\textbf{u}$ in terms of ${\rm div}\, \textbf{u}$, ${\rm curl}\, \textbf{u}$ either $(ν, \textbf{u})$ or $ν×\textbf{u}$ and the topology, Math. Methods Appl. Sci., 20 (1997), 737–744. doi: 10.1002/(SICI)1099-1476(199706)20:9<737::AID-MMA863>3.3.CO;2-9.  Google Scholar

[9]

W. Borchers and H. Sohr, On the equations ${\rm rot}\, \textbf{v} = \textbf{g}$ and ${\rm div}\, \textbf{u} = f$ with zero boundary conditions, Hokkaido Math. J., 19 (1990), 67–87. doi: 10.14492/hokmj/1381517172.  Google Scholar

[10]

J. Bourgain and H. Brezis, New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc. (JEMS), 9 (2007), 277–315. doi: 10.4171/JEMS/80.  Google Scholar

[11]

V. I. Burenkov, Sobolev Spaces on Domains, vol. 137 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998. doi: 10.1007/978-3-663-11374-4.  Google Scholar

[12]

U. Dini, Fourier Series and Other Analytic Representations of Functions of one Real Variable. (Serie di Fourier ed Altre Rappresentazioni Analitiche Delle Funzioni Di Una Variabile Reale), Nistri, 1880. Google Scholar

[13]

U. Dini, Sur la méthode des approximations successives pour les équations aux derivées partielles du deuxième ordre, Acta Math., 25 (1902), 185–230. doi: 10.1007/BF02419026.  Google Scholar

[14]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems, Springer Monographs in Mathematics, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.  Google Scholar

[16]

R. Griesinger, Decompositions of $L^q$ and $H^{1, q}_0$ with respect to the operator ${\rm rot}$, Math. Ann., 288 (1990), 245–262. doi: 10.1007/BF01444533.  Google Scholar

[17]

R. Griesinger, The boundary value problem ${\rm rot}\, u = f, \; u$ vanishing at the boundary and the related decompositions of $L^q$ and $H^{1, q}_0$: Existence, Ann. Univ. Ferrara Sez. VII (N. S.), 36 (1990), 15–43 (1991), URL http://dx.doi.org/10.1007/BF02837204.  Google Scholar

[18]

H. Helmholtz, Ueber die Theorie der Elektrodynamik. Zweite Abhandlung. Kritisches, J. Reine Angew. Math., 75 (1873), 35–66. doi: 10.1515/crll.1873.75.35.  Google Scholar

[19]

H. Koch, Transport and instability for perfect fluids, Math. Ann., 323 (2002), 491–523. doi: 10.1007/s002080200312.  Google Scholar

[20]

A. Korn, Über Minimalflächen, Deren Randkurven Wenig Von Ebenen Kurven Abweichen, Preuss Akad Wiss., 1909. Google Scholar

[21]

H. Kozono and T. Yanagisawa, $L^r$-variational inequality for vector fields and the HelmholtzWeyl decomposition in bounded domains, Indiana Univ. Math. J., 58 (2009), 1853–1920. doi: 10.1512/iumj.2009.58.3605.  Google Scholar

[22]

S. L. Sobolev, On a theorem of functional analysis, Mat. Sbornik, English Transl. : Amer. Math. Soc. Transl., 34 (1963), 39–68. Google Scholar

[23]

S. L. Sobolev, Introduction to the Theory of Cubature Formulae, Nauka, 1974, (Russian).  Google Scholar

[24]

W. von Wahl, On necessary and sufficient conditions for the solvability of the equations ${\rm rot}\, u = γ$ and ${\rm div}\, u = \epsilon$ with u vanishing on the boundary, in The Navier-Stokes equations (Oberwolfach, 1988), vol. 1431 of Lecture Notes in Math., Springer, Berlin, 1990, 152–157. doi: 10.1007/BFb0086065.  Google Scholar

[25]

H. Weyl, The method of orthogonal projection in potential theory, Duke Math. J., 7 (1940), 411–444. doi: 10.1215/S0012-7094-40-00725-6.  Google Scholar

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