March  2020, 10(1): 1-26. doi: 10.3934/mcrf.2019027

Quantitative approximation properties for the fractional heat equation

1. 

Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04105 Leipzig, Germany

2. 

Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

* Corresponding author: Angkana Rüland

Received  February 2018 Revised  February 2018 Published  April 2019

In this article we analyse quantitative approximation properties of a certain class of nonlocal equations: Viewing the fractional heat equation as a model problem, which involves both local and nonlocal pseudodifferential operators, we study quantitative approximation properties of solutions to it. First, relying on Runge type arguments, we give an alternative proof of certain qualitative approximation results from [9]. Using propagation of smallness arguments, we then provide bounds on the cost of approximate controllability and thus quantify the approximation properties of solutions to the fractional heat equation. Finally, we discuss generalizations of these results to a larger class of operators involving both local and nonlocal contributions.

Citation: Angkana Rüland, Mikko Salo. Quantitative approximation properties for the fractional heat equation. Mathematical Control & Related Fields, 2020, 10 (1) : 1-26. doi: 10.3934/mcrf.2019027
References:
[1]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004, 47pp. doi: 10.1088/0266-5611/25/12/123004.  Google Scholar

[2]

U. Biccari and V. Hernández-Santamarıa, Controllability of a one-dimensional fractional heat equation: Theoretical and numerical aspects, IMA Journal of Mathematical Control and Information. Google Scholar

[3]

U. Biccari, M. Warma and E. Zuazua, Local regularity for fractional heat equations, Recent Advances in PDEs: Analysis, Numerics and Control. Springer, Cham, 2018,233-249. Google Scholar

[4]

F. E. Browder, Approximation by solutions of partial differential equations, American Journal of Mathematics, 84 (1962), 134-160.  doi: 10.2307/2372809.  Google Scholar

[5]

F. E. Browder, Functional analysis and partial differential equations. Ⅱ, Mathematische Annalen, 145 (1961/1962), 81-226.  doi: 10.1007/BF01342796.  Google Scholar

[6]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 31, Elsevier, 2014, 23–53. doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[8]

L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 33, Elsevier, 2016, 767–807. doi: 10.1016/j.anihpc.2015.01.004.  Google Scholar

[9]

S. Dipierro, O. Savin and E. Valdinoci, Local approximation of arbitrary functions by solutions of nonlocal equations, The Journal of Geometric Analysis, 2018, 1-28. Google Scholar

[10]

S. DipierroO. Savin and E. Valdinoci, All functions are locally $s$-harmonic up to a small error, J. Eur. Math. Soc. (JEMS), 19 (2017), 957-966.  doi: 10.4171/JEMS/684.  Google Scholar

[11]

T. Duyckaerts and L. Miller, Resolvent conditions for the control of parabolic equations, Journal of Functional Analysis, 263 (2012), 3641-3673.  doi: 10.1016/j.jfa.2012.09.003.  Google Scholar

[12]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems, vol. 28 of Classics in Applied Mathematics, English edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.  Google Scholar

[13]

C. FabreJ.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 125 (1995), 31-61.  doi: 10.1017/S0308210500030742.  Google Scholar

[14]

M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Communications in Partial Differential Equations, 38 (2013), 1539-1573.  doi: 10.1080/03605302.2013.808211.  Google Scholar

[15]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Advances in Differential Equations, 5 (2000), 465-514.   Google Scholar

[16]

T. Ghosh, Y.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Comm. Partial Differential Equations, 42 (2017), 1923–1961, arXiv: 1708.00654. doi: 10.1080/03605302.2017.1390681.  Google Scholar

[17]

T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, arXiv preprint, arXiv: 1609.09248, to appear in Analysis and PDE. Google Scholar

[18]

G. Grubb, Regularity in $L^p$ Sobolev spaces of solutions to fractional heat equations, Journal of Functional Analysis, 274 (2018), 2634-2660.  doi: 10.1016/j.jfa.2017.12.011.  Google Scholar

[19]

V. Isakov, Inverse Source Problems, 34, American Mathematical Soc., 1990. doi: 10.1090/surv/034.  Google Scholar

[20]

M. Kassmann, A new formulation of Harnack's inequality for nonlocal operators, Comptes Rendus Mathematique, 349 (2011), 637-640.  doi: 10.1016/j.crma.2011.04.014.  Google Scholar

[21]

M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations, Riv. Math. Univ. Parma (N.S.), 5 (2014), 183-212.   Google Scholar

[22]

H. Koch, A. Rüland and W. Shi, Higher regularity for the fractional thin obstacle problem, Adv. Differential Equations, 22 (2017), 793–866, arXiv: 1605.06662.  Google Scholar

[23]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fractional Calculus and Applied Analysis, 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[24]

P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Communications on Pure and Applied Mathematics, 9 (1956), 747-766.  doi: 10.1002/cpa.3160090407.  Google Scholar

[25]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Communications in Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[26]

T. LeonoriI. PeralA. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete & Continuous Dynamical Systems-A, 35 (2015), 6031-6068.  doi: 10.3934/dcds.2015.35.6031.  Google Scholar

[27]

J.-L. Lions, Remarks on approximate controllability, Journal dAnalyse Mathématique, 59 (1992), 103–116. doi: 10.1007/BF02790220.  Google Scholar

[28]

S. Łojasiewicz, Sur le problème de la division, Studia Math., 18 (1959), 87-136.  doi: 10.4064/sm-18-1-87-136.  Google Scholar

[29]

S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation, SIAM journal on Control and Optimization, 44 (2006), 1950-1972.  doi: 10.1137/S036301290444263X.  Google Scholar

[30]

L. Miller, On the controllability of anomalous diffusions generated by the fractional Laplacian, Mathematics of Control, Signals and Systems, 18 (2006), 260-271.  doi: 10.1007/s00498-006-0003-3.  Google Scholar

[31]

L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete and Continuous Dynamical Systems-Series B, 14 (2010), 1465-1485.  doi: 10.3934/dcdsb.2010.14.1465.  Google Scholar

[32]

K.-D. Phung, Note on the cost of the approximate controllability for the heat equation with potential, Journal of Mathematical Analysis and Applications, 295 (2004), 527-538.  doi: 10.1016/j.jmaa.2004.03.059.  Google Scholar

[33]

L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques, Asymptotic Analysis, 10 (1995), 95-115.   Google Scholar

[34]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat, 60 (2016), 3-26.   Google Scholar

[35]

A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Communications in Partial Differential Equations, 40 (2015), 77-114.  doi: 10.1080/03605302.2014.905594.  Google Scholar

[36]

A. Rüland, Quantitative invertibility and approximation for the truncated Hilbert and Riesz transforms, arXiv preprint, arXiv: 1708.04285, to appear in Revista Matemática Iberoamericana. Google Scholar

[37]

A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, arXiv Preprint, August. Google Scholar

[38]

A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003, 21pp. doi: 10.1088/1361-6420/aaac5a.  Google Scholar

[39]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Communications in Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[40]

H. Yu, Unique continuation for fractional orders of elliptic equations, Ann. PDE, 3 (2017), Art. 16, 21 pp, arXiv: 1609.01376. doi: 10.1007/s40818-017-0033-9.  Google Scholar

[41]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 527-621.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

show all references

References:
[1]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004, 47pp. doi: 10.1088/0266-5611/25/12/123004.  Google Scholar

[2]

U. Biccari and V. Hernández-Santamarıa, Controllability of a one-dimensional fractional heat equation: Theoretical and numerical aspects, IMA Journal of Mathematical Control and Information. Google Scholar

[3]

U. Biccari, M. Warma and E. Zuazua, Local regularity for fractional heat equations, Recent Advances in PDEs: Analysis, Numerics and Control. Springer, Cham, 2018,233-249. Google Scholar

[4]

F. E. Browder, Approximation by solutions of partial differential equations, American Journal of Mathematics, 84 (1962), 134-160.  doi: 10.2307/2372809.  Google Scholar

[5]

F. E. Browder, Functional analysis and partial differential equations. Ⅱ, Mathematische Annalen, 145 (1961/1962), 81-226.  doi: 10.1007/BF01342796.  Google Scholar

[6]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 31, Elsevier, 2014, 23–53. doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[8]

L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 33, Elsevier, 2016, 767–807. doi: 10.1016/j.anihpc.2015.01.004.  Google Scholar

[9]

S. Dipierro, O. Savin and E. Valdinoci, Local approximation of arbitrary functions by solutions of nonlocal equations, The Journal of Geometric Analysis, 2018, 1-28. Google Scholar

[10]

S. DipierroO. Savin and E. Valdinoci, All functions are locally $s$-harmonic up to a small error, J. Eur. Math. Soc. (JEMS), 19 (2017), 957-966.  doi: 10.4171/JEMS/684.  Google Scholar

[11]

T. Duyckaerts and L. Miller, Resolvent conditions for the control of parabolic equations, Journal of Functional Analysis, 263 (2012), 3641-3673.  doi: 10.1016/j.jfa.2012.09.003.  Google Scholar

[12]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems, vol. 28 of Classics in Applied Mathematics, English edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.  Google Scholar

[13]

C. FabreJ.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 125 (1995), 31-61.  doi: 10.1017/S0308210500030742.  Google Scholar

[14]

M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Communications in Partial Differential Equations, 38 (2013), 1539-1573.  doi: 10.1080/03605302.2013.808211.  Google Scholar

[15]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Advances in Differential Equations, 5 (2000), 465-514.   Google Scholar

[16]

T. Ghosh, Y.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Comm. Partial Differential Equations, 42 (2017), 1923–1961, arXiv: 1708.00654. doi: 10.1080/03605302.2017.1390681.  Google Scholar

[17]

T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, arXiv preprint, arXiv: 1609.09248, to appear in Analysis and PDE. Google Scholar

[18]

G. Grubb, Regularity in $L^p$ Sobolev spaces of solutions to fractional heat equations, Journal of Functional Analysis, 274 (2018), 2634-2660.  doi: 10.1016/j.jfa.2017.12.011.  Google Scholar

[19]

V. Isakov, Inverse Source Problems, 34, American Mathematical Soc., 1990. doi: 10.1090/surv/034.  Google Scholar

[20]

M. Kassmann, A new formulation of Harnack's inequality for nonlocal operators, Comptes Rendus Mathematique, 349 (2011), 637-640.  doi: 10.1016/j.crma.2011.04.014.  Google Scholar

[21]

M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations, Riv. Math. Univ. Parma (N.S.), 5 (2014), 183-212.   Google Scholar

[22]

H. Koch, A. Rüland and W. Shi, Higher regularity for the fractional thin obstacle problem, Adv. Differential Equations, 22 (2017), 793–866, arXiv: 1605.06662.  Google Scholar

[23]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fractional Calculus and Applied Analysis, 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[24]

P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Communications on Pure and Applied Mathematics, 9 (1956), 747-766.  doi: 10.1002/cpa.3160090407.  Google Scholar

[25]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Communications in Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[26]

T. LeonoriI. PeralA. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete & Continuous Dynamical Systems-A, 35 (2015), 6031-6068.  doi: 10.3934/dcds.2015.35.6031.  Google Scholar

[27]

J.-L. Lions, Remarks on approximate controllability, Journal dAnalyse Mathématique, 59 (1992), 103–116. doi: 10.1007/BF02790220.  Google Scholar

[28]

S. Łojasiewicz, Sur le problème de la division, Studia Math., 18 (1959), 87-136.  doi: 10.4064/sm-18-1-87-136.  Google Scholar

[29]

S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation, SIAM journal on Control and Optimization, 44 (2006), 1950-1972.  doi: 10.1137/S036301290444263X.  Google Scholar

[30]

L. Miller, On the controllability of anomalous diffusions generated by the fractional Laplacian, Mathematics of Control, Signals and Systems, 18 (2006), 260-271.  doi: 10.1007/s00498-006-0003-3.  Google Scholar

[31]

L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete and Continuous Dynamical Systems-Series B, 14 (2010), 1465-1485.  doi: 10.3934/dcdsb.2010.14.1465.  Google Scholar

[32]

K.-D. Phung, Note on the cost of the approximate controllability for the heat equation with potential, Journal of Mathematical Analysis and Applications, 295 (2004), 527-538.  doi: 10.1016/j.jmaa.2004.03.059.  Google Scholar

[33]

L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques, Asymptotic Analysis, 10 (1995), 95-115.   Google Scholar

[34]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat, 60 (2016), 3-26.   Google Scholar

[35]

A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Communications in Partial Differential Equations, 40 (2015), 77-114.  doi: 10.1080/03605302.2014.905594.  Google Scholar

[36]

A. Rüland, Quantitative invertibility and approximation for the truncated Hilbert and Riesz transforms, arXiv preprint, arXiv: 1708.04285, to appear in Revista Matemática Iberoamericana. Google Scholar

[37]

A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, arXiv Preprint, August. Google Scholar

[38]

A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003, 21pp. doi: 10.1088/1361-6420/aaac5a.  Google Scholar

[39]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Communications in Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[40]

H. Yu, Unique continuation for fractional orders of elliptic equations, Ann. PDE, 3 (2017), Art. 16, 21 pp, arXiv: 1609.01376. doi: 10.1007/s40818-017-0033-9.  Google Scholar

[41]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 527-621.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

Figure 1.  The annulus from the proof of Corollary 1. The shaded area corresponds to the annulus $ A_{R_1(\tilde{k}_1),R_2(\tilde{k}_1)}(\tilde{k}_1) $ with $ \tilde{k}_1 $ chosen to be $ i|k_0| $. The bold red line (see the online version for a colour figures) indicates the discontinuity line of the function $ e^{s_1 \log(k_1^2 + |k_0|^2)} $
[1]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[2]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[3]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[4]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[5]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

[6]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[7]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385

[8]

Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020118

[9]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158

[10]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[11]

Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $ (n, m) $-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117

[12]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[13]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[14]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270

[15]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[16]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[17]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116

[18]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[19]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[20]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (281)
  • HTML views (689)
  • Cited by (4)

Other articles
by authors

[Back to Top]