# American Institute of Mathematical Sciences

February  2018, 38(2): 509-546. doi: 10.3934/dcds.2018023

## Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach

 1 Instituto de Matematica Interdisciplinar & Dpto. de Matematica Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias, 3,28040 Madrid, Spain 2 Université de Poitiers, Laboratoire de Mathématiques et Applications -UMR CNRS 7348 -SP2MI, France, Bd Marie et Pierre Curie, Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France 3 Institute for Scientific Computation and Applied Mathematics, Indiana University, Bloomington, Indiana 47405, USA

* Corresponding author: Jean-Michel.Rakotoson@math.univ-poitiers.fr

Received  April 2017 Published  February 2018

Fund Project: The research of D. G´omez-Castro was supported by a FPU fellowship from the Spanish government. The research of J.I. D´ıaz and D. G´omez-Castro was partially supported by the project ref. MTM 2014-57113-P of the DGISPI (Spain). Roger Temam was partially supported by NSF grant DMS 1510249 and by the Research Fund of Indiana University.

In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\text{IR}^N$. In most of the paper we consider homogeneous Dirichlet boundary conditions but we prove that when the potential function grows faster than the distance to the boundary to the power -2 then no boundary condition is required to get the uniqueness of very weak solutions. This result is new in the literature and must be distinguished from other previous results in which such uniqueness of solutions without any boundary condition was proved for degenerate diffusion operators (which is not our case). Our approach, based on the treatment on some distance to the boundary weighted spaces, uses a suitable regularity of the solution of the associated dual problem which is here established. We also consider the delicate question of the differentiability of the very weak solution and prove that some suitable additional hypothesis on the data is required since otherwise the gradient of the solution may not be integrable on the domain.

Citation: Jesus Ildefonso Díaz, David Gómez-Castro, Jean Michel Rakotoson, Roger Temam. Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 509-546. doi: 10.3934/dcds.2018023
##### References:
 [1] F. Abergel and J. M. Rakotoson, Gradient Blow-up in Zygmund spaces for the very weak solution of a linear equation, Discrete Continuous Dynamical Systems, Serie A, 33 (2013), 1809-1818.  Google Scholar [2] P. Ausher and M. Qafsaoui, Observations on $W^{1, p}$ estimates for divergence elliptic equations with VMO coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 5 (2002), 487-509.   Google Scholar [3] M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérant au bord, (French) C. R. Acad. Sci. Paris Sér. A-B, 262 (1966), A337-A340.  Google Scholar [4] M. S. Baouendi and C. Goulaouic, Régularité et théorie spectrale pour une classe d'opérateurs elliptiques dégénérés, (French) Arch. Rational Mech. Anal., 34 (1969), 361-379. doi: 10.1007/BF00281438.  Google Scholar [5] Ph. Bénilan, L. Boccardo, Th. Gallouët, R. Gariespy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuala Norm. Sup. Pisa, 22 (1995), 241-273.   Google Scholar [6] M. F. Bidaut-Véron and L. Vivier, An elliptic semi linear equation with source term involving boundary measures the subcritical case, Rev. Mat. Iberoamrica, 16 (2000), 477-513.  doi: 10.4171/RMI/281.  Google Scholar [7] H. Brézis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital., 1 (1998), 223-262.   Google Scholar [8] H. Brézis and T. Kato, Remarks on the Scrödinger operator with singular complex potentials, J. Pure Appl. Math., 58 (1979), 137-151.   Google Scholar [9] S. Byun, Elliptic equations with BMO coefficients in Lipschtiz domains, Trans. Amer. Math. Soc., 375 (2005), 1025-1046.  doi: 10.1090/S0002-9947-04-03624-4.  Google Scholar [10] S. Campanato, Equizioni ellitiche del secondo ordino e spazi ${\mathcal L}^{2, λ}(Ω)$, Annali di Matematica(4), 69 (1965), 321-381.  doi: 10.1007/BF02414377.  Google Scholar [11] D. C. Chang, The dual of Hardy spaces on a bounded domain in IRn, Forum Math., 6 (1994), 65-81.  doi: 10.1515/form.1994.6.65.  Google Scholar [12] D. R. Chang, G. Dafni and E. Stein, Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in I$\text{IR}^N$, Trans of AMS, 351 (1999), 1605-1661.  doi: 10.1090/S0002-9947-99-02111-X.  Google Scholar [13] P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, 1988.  Google Scholar [14] G. Dal Maso and U. Mosco, Wiener's criterion and $Γ$-convergence, Appl. Math. Optim., 15 (1987), 15-63.  doi: 10.1007/BF01442645.  Google Scholar [15] J. I. Díaz, On the ambiguous treatment of Schrödinger equations for the infinite potential well and an alternative via flat solutions : The one-dimensional case, Interfaces and Free Boundaries, 17 (2015), 333-351. doi: 10.4171/IFB/345.  Google Scholar [16] J. I. Díaz, On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via singular potentials: The multi-dimensional case, SeMA-Journal, 74 (2017), 255-278.  doi: 10.1007/s40324-017-0115-3.  Google Scholar [17] J. I. Díaz and D. Gómez-Castro, Shape differentiation : an application to the effectiveness of a steady state reaction-diffusion problem in chemical engineering, Electron. J. Diff. Eqns Conf., 22 (2015), 31-45.   Google Scholar [18] J. I. Díaz and J. M. Rakotoson, On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary, J. Functional Analysis, 257 (2009), 807-831.  doi: 10.1016/j.jfa.2009.03.002.  Google Scholar [19] J. I. Díaz and J. M. Rakotoson, On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary, DCDS Serie A, 27 (2010), 1037-1958.  doi: 10.3934/dcds.2010.27.1037.  Google Scholar [20] J. I. Díaz and J. M. Rakotoson, Elliptic problems on the space of weighted with the distance to the boundary integrable functions revisited, Variational and Topological methods: Theory, Applications, Numerical simulation and Open Problems, Electon. J. Diff. Equations, 21 (2014), 45-59.   Google Scholar [21] J. I. Díaz and L. Tello, On a nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica, 50 (1999), 19-51.   Google Scholar [22] D. Gilbarg and S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.  Google Scholar [23] D. Goldberg, A local version of real Hardy spaces, Duke Math Notes, 46 (1979), 27-42.  doi: 10.1215/S0012-7094-79-04603-9.  Google Scholar [24] D. Gómez-Castro, Shape differentiation of a steady-state reaction-diffusion problem arising in chemical engineering: the case of non-smooth kinetic with dead core, To appear in Electronic Journal of Dierential Equations, 2017. Google Scholar [25] J. Hadamard, Sur le probléme d'analyse relatif á l'équilibre des plaques élastiques encastrées, Mémoire couronné en 1907 par l'Académie des Sciences, 33 (), 515-629.   Google Scholar [26] J. Hernandez, F. Mancebo and J. Vega, On the linearization of some singular, nonlinear elliptic problems and applications, Annales de l'I.H.P. Analyse non linéaire, 19 (2002), 777-813.  doi: 10.1016/S0294-1449(02)00102-6.  Google Scholar [27] P. W. Jones, Extension theorems for BMO, Indiana Univ. Notes, 29 (1980), 41-66.  doi: 10.1512/iumj.1980.29.29005.  Google Scholar [28] J. L. Lions, Quelques Méthodes de Résolution de Problémes Aux Limites Non Linéaires, (French) Dunod; Gauthier-Villars, Paris 1969.  Google Scholar [29] J. Merker and J. M. Rakotoson, Very weak solutions of Poisson's equation with singular data under Neumann boundary conditions, Cal. of Var. and P.D.E., 52 (2015), 705-726.  doi: 10.1007/s00526-014-0730-0.  Google Scholar [30] M. Marcus and L. Véron, Nonlinear Second Order Elliptic Equations Involving Measures, de Gruyter, Berlin, 2013.  Google Scholar [31] J. Mossino and R. Temam, Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics, Duke Math. J., 48 (1981), 475-495.  doi: 10.1215/S0012-7094-81-04827-4.  Google Scholar [32] F. Murat and J. Simon, Sur le contrôle par un domaine géométrique N$^o$76015 Prépublications du laboratoire d'Analyse Numérique, Univ. Paris VI, 1976. Google Scholar [33] L. Orsina and A. Ponce, Hopf potentials for the Schrösinger operator v1, in arXiv: 1702.04572 11 Mar 2017. Google Scholar [34] L. Orsina and A. Ponce, Hopf potentials for the Schrösinger operator v2, personal communication 12 April 2017. Google Scholar [35] J. M. Rakotoson, Réarrangement Relatif, Un Instrument D'estimation Dans Les Problémes Aux Limites, Springer Berlin, 2008. doi: 10.1007/978-3-540-69118-1.  Google Scholar [36] J. M. Rakotoson, Few natural extensions of the regularity of a very weak solution, Differential Integral Equations, 24 (2011), 1125-1146.   Google Scholar [37] J. M. Rakotoson, New Hardy inequalities and behaviour of linear elliptic equations, Journal of Functional Analysis, 263 (2012), 2893-2920.  doi: 10.1016/j.jfa.2012.08.001.  Google Scholar [38] J. M. Rakotoson, Linear equation with data in non standard spaces, Atti Accad. Naz. Lincei, Math., Appl., 26 (2015), 241-262.  doi: 10.4171/RLM/705.  Google Scholar [39] J. M. Rakotoson, Sufficient condition for a blow-up in the space of absolutely continuous functions for the very weak solution, Applied Math. and Optimization, 73 (2016), 153-163.  doi: 10.1007/s00245-015-9297-1.  Google Scholar [40] J. M. Rakotoson, Linear equations with variable coefficient and Banach functions spaces, Book to appear. Google Scholar [41] J. M. Rakotoson and R. Temam, A co-area formula with applications to monotone rearrangement and to regularity, Arch. Ration. Mach. Anal., 109 (1990), 213-238.  doi: 10.1007/BF00375089.  Google Scholar [42] C. Simader, On Dirichlet's Boundary Value Problem, Springer, Berlin, 1972.  Google Scholar [43] J. Simon, Differentiation with respect to the domains in boundary value problems, Numerical Funct. Anl. and Optimiz., 2 (1980), 649-687.  doi: 10.1080/01630563.1980.10120631.  Google Scholar [44] D.A. Stegenga, Bounded Toeplitz operators on ${\mathcal H}^1$ and applications of duality between ${\mathcal H}^1$ and the functions of Bounded Mean Oscillations, Amer. J. Math., 98 (1976), 573-589.  doi: 10.2307/2373807.  Google Scholar [45] G. Talenti, Elliptic equations and Rearrangements, Ann. Scuola. Norm. Sup. Pisa Cl Sci., 3 (1976), 697-718.   Google Scholar [46] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second edition. CBMS-NSF Regional Conference Series in Applied Mathematics, 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.  Google Scholar [47] A. Torchinsky, Real-variable Methods in Harmonic Analysis, Academic Press, 1986.  Google Scholar

show all references

##### References:
 [1] F. Abergel and J. M. Rakotoson, Gradient Blow-up in Zygmund spaces for the very weak solution of a linear equation, Discrete Continuous Dynamical Systems, Serie A, 33 (2013), 1809-1818.  Google Scholar [2] P. Ausher and M. Qafsaoui, Observations on $W^{1, p}$ estimates for divergence elliptic equations with VMO coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 5 (2002), 487-509.   Google Scholar [3] M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérant au bord, (French) C. R. Acad. Sci. Paris Sér. A-B, 262 (1966), A337-A340.  Google Scholar [4] M. S. Baouendi and C. Goulaouic, Régularité et théorie spectrale pour une classe d'opérateurs elliptiques dégénérés, (French) Arch. Rational Mech. Anal., 34 (1969), 361-379. doi: 10.1007/BF00281438.  Google Scholar [5] Ph. Bénilan, L. Boccardo, Th. Gallouët, R. Gariespy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuala Norm. Sup. Pisa, 22 (1995), 241-273.   Google Scholar [6] M. F. Bidaut-Véron and L. Vivier, An elliptic semi linear equation with source term involving boundary measures the subcritical case, Rev. Mat. Iberoamrica, 16 (2000), 477-513.  doi: 10.4171/RMI/281.  Google Scholar [7] H. Brézis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital., 1 (1998), 223-262.   Google Scholar [8] H. Brézis and T. Kato, Remarks on the Scrödinger operator with singular complex potentials, J. Pure Appl. Math., 58 (1979), 137-151.   Google Scholar [9] S. Byun, Elliptic equations with BMO coefficients in Lipschtiz domains, Trans. Amer. Math. Soc., 375 (2005), 1025-1046.  doi: 10.1090/S0002-9947-04-03624-4.  Google Scholar [10] S. Campanato, Equizioni ellitiche del secondo ordino e spazi ${\mathcal L}^{2, λ}(Ω)$, Annali di Matematica(4), 69 (1965), 321-381.  doi: 10.1007/BF02414377.  Google Scholar [11] D. C. Chang, The dual of Hardy spaces on a bounded domain in IRn, Forum Math., 6 (1994), 65-81.  doi: 10.1515/form.1994.6.65.  Google Scholar [12] D. R. Chang, G. Dafni and E. Stein, Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in I$\text{IR}^N$, Trans of AMS, 351 (1999), 1605-1661.  doi: 10.1090/S0002-9947-99-02111-X.  Google Scholar [13] P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, 1988.  Google Scholar [14] G. Dal Maso and U. Mosco, Wiener's criterion and $Γ$-convergence, Appl. Math. Optim., 15 (1987), 15-63.  doi: 10.1007/BF01442645.  Google Scholar [15] J. I. Díaz, On the ambiguous treatment of Schrödinger equations for the infinite potential well and an alternative via flat solutions : The one-dimensional case, Interfaces and Free Boundaries, 17 (2015), 333-351. doi: 10.4171/IFB/345.  Google Scholar [16] J. I. Díaz, On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via singular potentials: The multi-dimensional case, SeMA-Journal, 74 (2017), 255-278.  doi: 10.1007/s40324-017-0115-3.  Google Scholar [17] J. I. Díaz and D. Gómez-Castro, Shape differentiation : an application to the effectiveness of a steady state reaction-diffusion problem in chemical engineering, Electron. J. Diff. Eqns Conf., 22 (2015), 31-45.   Google Scholar [18] J. I. Díaz and J. M. Rakotoson, On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary, J. Functional Analysis, 257 (2009), 807-831.  doi: 10.1016/j.jfa.2009.03.002.  Google Scholar [19] J. I. Díaz and J. M. Rakotoson, On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary, DCDS Serie A, 27 (2010), 1037-1958.  doi: 10.3934/dcds.2010.27.1037.  Google Scholar [20] J. I. Díaz and J. M. Rakotoson, Elliptic problems on the space of weighted with the distance to the boundary integrable functions revisited, Variational and Topological methods: Theory, Applications, Numerical simulation and Open Problems, Electon. J. Diff. Equations, 21 (2014), 45-59.   Google Scholar [21] J. I. Díaz and L. Tello, On a nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica, 50 (1999), 19-51.   Google Scholar [22] D. Gilbarg and S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.  Google Scholar [23] D. Goldberg, A local version of real Hardy spaces, Duke Math Notes, 46 (1979), 27-42.  doi: 10.1215/S0012-7094-79-04603-9.  Google Scholar [24] D. Gómez-Castro, Shape differentiation of a steady-state reaction-diffusion problem arising in chemical engineering: the case of non-smooth kinetic with dead core, To appear in Electronic Journal of Dierential Equations, 2017. Google Scholar [25] J. Hadamard, Sur le probléme d'analyse relatif á l'équilibre des plaques élastiques encastrées, Mémoire couronné en 1907 par l'Académie des Sciences, 33 (), 515-629.   Google Scholar [26] J. Hernandez, F. Mancebo and J. Vega, On the linearization of some singular, nonlinear elliptic problems and applications, Annales de l'I.H.P. Analyse non linéaire, 19 (2002), 777-813.  doi: 10.1016/S0294-1449(02)00102-6.  Google Scholar [27] P. W. Jones, Extension theorems for BMO, Indiana Univ. Notes, 29 (1980), 41-66.  doi: 10.1512/iumj.1980.29.29005.  Google Scholar [28] J. L. Lions, Quelques Méthodes de Résolution de Problémes Aux Limites Non Linéaires, (French) Dunod; Gauthier-Villars, Paris 1969.  Google Scholar [29] J. Merker and J. M. Rakotoson, Very weak solutions of Poisson's equation with singular data under Neumann boundary conditions, Cal. of Var. and P.D.E., 52 (2015), 705-726.  doi: 10.1007/s00526-014-0730-0.  Google Scholar [30] M. Marcus and L. Véron, Nonlinear Second Order Elliptic Equations Involving Measures, de Gruyter, Berlin, 2013.  Google Scholar [31] J. Mossino and R. Temam, Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics, Duke Math. J., 48 (1981), 475-495.  doi: 10.1215/S0012-7094-81-04827-4.  Google Scholar [32] F. Murat and J. Simon, Sur le contrôle par un domaine géométrique N$^o$76015 Prépublications du laboratoire d'Analyse Numérique, Univ. Paris VI, 1976. Google Scholar [33] L. Orsina and A. Ponce, Hopf potentials for the Schrösinger operator v1, in arXiv: 1702.04572 11 Mar 2017. Google Scholar [34] L. Orsina and A. Ponce, Hopf potentials for the Schrösinger operator v2, personal communication 12 April 2017. Google Scholar [35] J. M. Rakotoson, Réarrangement Relatif, Un Instrument D'estimation Dans Les Problémes Aux Limites, Springer Berlin, 2008. doi: 10.1007/978-3-540-69118-1.  Google Scholar [36] J. M. Rakotoson, Few natural extensions of the regularity of a very weak solution, Differential Integral Equations, 24 (2011), 1125-1146.   Google Scholar [37] J. M. Rakotoson, New Hardy inequalities and behaviour of linear elliptic equations, Journal of Functional Analysis, 263 (2012), 2893-2920.  doi: 10.1016/j.jfa.2012.08.001.  Google Scholar [38] J. M. Rakotoson, Linear equation with data in non standard spaces, Atti Accad. Naz. Lincei, Math., Appl., 26 (2015), 241-262.  doi: 10.4171/RLM/705.  Google Scholar [39] J. M. Rakotoson, Sufficient condition for a blow-up in the space of absolutely continuous functions for the very weak solution, Applied Math. and Optimization, 73 (2016), 153-163.  doi: 10.1007/s00245-015-9297-1.  Google Scholar [40] J. M. Rakotoson, Linear equations with variable coefficient and Banach functions spaces, Book to appear. Google Scholar [41] J. M. Rakotoson and R. Temam, A co-area formula with applications to monotone rearrangement and to regularity, Arch. Ration. Mach. Anal., 109 (1990), 213-238.  doi: 10.1007/BF00375089.  Google Scholar [42] C. Simader, On Dirichlet's Boundary Value Problem, Springer, Berlin, 1972.  Google Scholar [43] J. Simon, Differentiation with respect to the domains in boundary value problems, Numerical Funct. Anl. and Optimiz., 2 (1980), 649-687.  doi: 10.1080/01630563.1980.10120631.  Google Scholar [44] D.A. Stegenga, Bounded Toeplitz operators on ${\mathcal H}^1$ and applications of duality between ${\mathcal H}^1$ and the functions of Bounded Mean Oscillations, Amer. J. Math., 98 (1976), 573-589.  doi: 10.2307/2373807.  Google Scholar [45] G. Talenti, Elliptic equations and Rearrangements, Ann. Scuola. Norm. Sup. Pisa Cl Sci., 3 (1976), 697-718.   Google Scholar [46] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second edition. CBMS-NSF Regional Conference Series in Applied Mathematics, 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.  Google Scholar [47] A. Torchinsky, Real-variable Methods in Harmonic Analysis, Academic Press, 1986.  Google Scholar
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