June  2020, 9(2): 301-339. doi: 10.3934/eect.2020007

On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations

Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, INDIA

*Corresponding author: Manil T. Mohan

Received  December 2018 Revised  February 2019 Published  August 2019

In this work, we consider the three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt fluids in bounded and unbounded domains. We investigate the global solvability results, asymptotic behavior and also address some control problems of such viscoelastic fluid flow equations with "fading memory" and "memory of length $ \tau $". A local monotonicity property of the linear and nonlinear operators and a localized version of the Minty-Browder technique are used to obtain global solvability results. Since we are not using compactness arguments in the proofs, the global solvability results are also valid in unbounded domains like Poincaré domains. We also remark that using an $ m $-accretive quantization of the linear and nonlinear operators, one can establish the existence and uniqueness of strong solutions for the Navier-Stokes-Voigt equations and avoid the tedious Galerkin approximation scheme. We examine the asymptotic behavior of the stationary solutions and also establish the exponential stability results. Finally, under suitable assumptions on the Galerkin basis, we consider the controlled Galerkin approximated 3D Kelvin-Voigt fluid flow equations. Using the Hilbert uniqueness method combined with Schauder's fixed point theorem, the exact controllability of the finite dimensional Galerkin approximated system is established.

Citation: Manil T. Mohan. On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations. Evolution Equations & Control Theory, 2020, 9 (2) : 301-339. doi: 10.3934/eect.2020007
References:
[1]

F. D. ArarunaF. W. Chaves-Silva and M. A. Rojas-Medar, Exact controllability of Galerkin's approximations of micropolar fluids, Proceedings of the American Mathematical Society, 138 (2010), 1361-1370.  doi: 10.1090/S0002-9939-09-10154-5.  Google Scholar

[2]

C. T. Anh and P. T. Trang, Pull-back attractors for three-dimensional Navier-Stokes-Voigt equations in some unbounded domains, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 143 (2013), 223-251.  doi: 10.1017/S0308210511001491.  Google Scholar

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.  Google Scholar

[4]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[5]

V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity, Zeitschrift für angewandte Mathematik und Physik, 54 (2003), 449-461.  doi: 10.1007/s00033-003-1087-y.  Google Scholar

[6]

V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, Journal of Mathematical Analysis and Applications, 255 (2001), 281-307.  doi: 10.1006/jmaa.2000.7256.  Google Scholar

[7] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993.   Google Scholar
[8]

M. Böhm, On Navier-Stokes and Kelvin-Voigt equations in three dimensions in interpolation spaces, Mathematische Nachrichten, 155 (1992), 151-165.  doi: 10.1002/mana.19921550112.  Google Scholar

[9]

Y. P. CaoE. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.  Google Scholar

[10]

T. Caraballo and J. Real, Asymptotic behavior of Navier-Stokes equations with delays, R. Soc. London A, 459 (2003), 3181-3194.  doi: 10.1098/rspa.2003.1166.  Google Scholar

[11]

F. Di PlinioA. GiorginiV. Pata and R. Temam, Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity, Journal of Nonlinear Science, 28 (2018), 653-686.  doi: 10.1007/s00332-017-9422-1.  Google Scholar

[12]

L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, American Mathematical Society, 2nd Ed, 2010. doi: 10.1090/gsm/019.  Google Scholar

[13]

B. P. W. FernandoS. S. Sritharan and M. Xu, A simple proof of global solvability for 2-D Navier-Stokes equations in unbounded domains, Differential and Integral Equations, 23 (2010), 223-235.   Google Scholar

[14]

C. G. Gal and T. T. Medjo, A Navier-Stokes-Voight model with memory, Math. Meth. Appl. Sci., 36 (2013), 2507-2523.  doi: 10.1002/mma.2771.  Google Scholar

[15]

V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math., 30 (2009), 697-714.  doi: 10.1007/s11401-009-0205-3.  Google Scholar

[16]

S. Kundu and A. K. Pani, Stabilization of Kelvin-Voigt viscoelastic fluid flow model, Applicable Analysis, 2018, https://doi.org/10.1080/00036811.2018.1460810. doi: 10.1080/00036811.2018.1460810.  Google Scholar

[17]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.  Google Scholar

[18]

J.-L. Lions and E. Zuazua, Contrôlabilité exacte des approximations de Galerkin des èquations de Navier-Stokes, C. R. Acad. Sci. Paris, Ser. I, 324 (1997), 1015-1021.  doi: 10.1016/S0764-4442(97)87878-0.  Google Scholar

[19]

J.-L. Lions and E. Zuazua, Exact boundary controllability of Galerkin approximations of Navier-Stokes equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 605-621.   Google Scholar

[20]

J.-L. Lions and E. Zuazua, On the cost of controlling unstable systems: The case of boundary controls, J. Anal. Math., 73 (1997), 225-249.  doi: 10.1007/BF02788145.  Google Scholar

[21]

J. Lukkarinen and M. S. Pakkanen, On the positivity of Riemann-Stieltjes integrals, Bull. Aust. Math. Soc., 87 (2013), 400-405.  doi: 10.1017/S0004972712000639.  Google Scholar

[22]

A. O. MarinhoA. T. Lourêdo and M. Milla Miranda, Exact controllability of Galerkin's approximations for the Oldroyd fluid system, International Journal of Modeling and Optimization, 4 (2014), 126-132.  doi: 10.7763/IJMO.2014.V4.359.  Google Scholar

[23]

M. T. Mohan, Global strong solutions of the stochastic three dimensional inviscid simplified Bardina turbulence model, Communications on Stochastic Analysis, 12 (2018), 249-270.  doi: 10.31390/cosa.12.3.03.  Google Scholar

[24]

M. T. Mohan, An extension of the Beale-Kato-Majda criterion for the 3D Navier-Stokes equation with hereditary viscosity, Accepted in Pure and Applied Functional Analysis, (2019). Google Scholar

[25]

M. T. Mohan, On the two dimensional tidal dynamics system: stationary solution and stability, Applicable Analysis, (2018), https://doi.org/10.1080/00036811.2018.1546002. doi: 10.1080/00036811.2018.1546002.  Google Scholar

[26]

M. T. Mohan, Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: Existence, uniqueness, exponential stability and invariant measures, Accepted in Stochastic Analysis and Applications, (2019). Google Scholar

[27]

M. T. Mohan, Global attractors and determining modes for the three dimensional Kelvin-Voigt fluids, Submitted. Google Scholar

[28]

M. T. Mohan and S. S. Sritharan, New methods for local solvability of quasilinear symmetric hyperbolic systems, Evolution Equations and Control Theory, 5 (2016), 273-302.  doi: 10.3934/eect.2016005.  Google Scholar

[29]

M. T. Mohan and S. S. Sritharan, Stochastic quasilinear symmetric hyperbolic system perturbed by Lévy noise, Pure and Applied Functional Analysis, 3 (2018), 137-178.   Google Scholar

[30]

M. T. Mohan and S. S. Sritharan, Stochastic Navier-Stokes equation perturbed by Lévy noise with hereditary viscosity, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 22 (2019), 1950006, 32 pp. doi: 10.1142/S0219025719500061.  Google Scholar

[31]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar

[32]

A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem. LOMI, 38 (1973), 98-136.   Google Scholar

[33]

A. P. Oskolkov, Theory of nonstationary flows of Kelvin-Voigt fluids, Zap. Nauchn. Sem. LOMI, 115 (1982), 191–202, 310.  Google Scholar

[34]

A. P. Oskolkov, Initial boundary value problems for the equations of motion of Kelvin-Voigt fluids and Oldroyd fluids, Proceedings of Steklov Institute of Mathematics, 2 (1989), 137-182.   Google Scholar

[35]

A. P. Oskolkov, Nonlocal problems for the equations of Kelvin- Voight fluids and their $\varepsilon$approximations in classes of smooth functions, Zap. Nauchn. Sem. POMI, 230 (1995), 214-242.  doi: 10.1007/BF02433999.  Google Scholar

[36]

A. P. Oskolkov, Nonlocal problems for the equations of motion of Kelvin-Voight fluids, Journal of Mathematical Sciences, 75 (1995), 2058-2078.  doi: 10.1007/BF02362946.  Google Scholar

[37]

A. P. Oskolkov and R. D. Shadiev, Nonlocal problems of the theory of the equations of motion for Kelvin-Voight fluids, Journal of Soviet Mathematics, 62 (1992), 2699-2723.  doi: 10.1007/BF01102639.  Google Scholar

[38]

A. P. Oskolkov and R. D. Shadiev, Nonlocal problems in the theory of equations of motion of Kelvin-Voight fluids. 2, Journal of Mathematical Sciences, 59 (1992), 1206-1214.  doi: 10.1007/BF01374082.  Google Scholar

[39]

A. P. Oskolkov and R. D. Shadiev, Towards a theory of global solvability on $[0, \infty)$ of initial-boundary value problems for the equations of motion of oldroyd and Kelvin-Voight fluids, Journal of Mathematical Sciences, 68 (1994), 240-253.  doi: 10.1007/BF01249338.  Google Scholar

[40]

R. T. Rockafellar, Convex Analysis, Reprint of the 1970 original. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ, 1997.  Google Scholar

[41]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050.  Google Scholar

[42]

V. G. Zvyagin and M. V. Turbin, The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, Journal of Mathematical Sciences, 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2.  Google Scholar

show all references

References:
[1]

F. D. ArarunaF. W. Chaves-Silva and M. A. Rojas-Medar, Exact controllability of Galerkin's approximations of micropolar fluids, Proceedings of the American Mathematical Society, 138 (2010), 1361-1370.  doi: 10.1090/S0002-9939-09-10154-5.  Google Scholar

[2]

C. T. Anh and P. T. Trang, Pull-back attractors for three-dimensional Navier-Stokes-Voigt equations in some unbounded domains, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 143 (2013), 223-251.  doi: 10.1017/S0308210511001491.  Google Scholar

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.  Google Scholar

[4]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[5]

V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity, Zeitschrift für angewandte Mathematik und Physik, 54 (2003), 449-461.  doi: 10.1007/s00033-003-1087-y.  Google Scholar

[6]

V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, Journal of Mathematical Analysis and Applications, 255 (2001), 281-307.  doi: 10.1006/jmaa.2000.7256.  Google Scholar

[7] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993.   Google Scholar
[8]

M. Böhm, On Navier-Stokes and Kelvin-Voigt equations in three dimensions in interpolation spaces, Mathematische Nachrichten, 155 (1992), 151-165.  doi: 10.1002/mana.19921550112.  Google Scholar

[9]

Y. P. CaoE. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.  Google Scholar

[10]

T. Caraballo and J. Real, Asymptotic behavior of Navier-Stokes equations with delays, R. Soc. London A, 459 (2003), 3181-3194.  doi: 10.1098/rspa.2003.1166.  Google Scholar

[11]

F. Di PlinioA. GiorginiV. Pata and R. Temam, Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity, Journal of Nonlinear Science, 28 (2018), 653-686.  doi: 10.1007/s00332-017-9422-1.  Google Scholar

[12]

L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, American Mathematical Society, 2nd Ed, 2010. doi: 10.1090/gsm/019.  Google Scholar

[13]

B. P. W. FernandoS. S. Sritharan and M. Xu, A simple proof of global solvability for 2-D Navier-Stokes equations in unbounded domains, Differential and Integral Equations, 23 (2010), 223-235.   Google Scholar

[14]

C. G. Gal and T. T. Medjo, A Navier-Stokes-Voight model with memory, Math. Meth. Appl. Sci., 36 (2013), 2507-2523.  doi: 10.1002/mma.2771.  Google Scholar

[15]

V. K. Kalantarov and E. S. Titi, Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math., 30 (2009), 697-714.  doi: 10.1007/s11401-009-0205-3.  Google Scholar

[16]

S. Kundu and A. K. Pani, Stabilization of Kelvin-Voigt viscoelastic fluid flow model, Applicable Analysis, 2018, https://doi.org/10.1080/00036811.2018.1460810. doi: 10.1080/00036811.2018.1460810.  Google Scholar

[17]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.  Google Scholar

[18]

J.-L. Lions and E. Zuazua, Contrôlabilité exacte des approximations de Galerkin des èquations de Navier-Stokes, C. R. Acad. Sci. Paris, Ser. I, 324 (1997), 1015-1021.  doi: 10.1016/S0764-4442(97)87878-0.  Google Scholar

[19]

J.-L. Lions and E. Zuazua, Exact boundary controllability of Galerkin approximations of Navier-Stokes equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 605-621.   Google Scholar

[20]

J.-L. Lions and E. Zuazua, On the cost of controlling unstable systems: The case of boundary controls, J. Anal. Math., 73 (1997), 225-249.  doi: 10.1007/BF02788145.  Google Scholar

[21]

J. Lukkarinen and M. S. Pakkanen, On the positivity of Riemann-Stieltjes integrals, Bull. Aust. Math. Soc., 87 (2013), 400-405.  doi: 10.1017/S0004972712000639.  Google Scholar

[22]

A. O. MarinhoA. T. Lourêdo and M. Milla Miranda, Exact controllability of Galerkin's approximations for the Oldroyd fluid system, International Journal of Modeling and Optimization, 4 (2014), 126-132.  doi: 10.7763/IJMO.2014.V4.359.  Google Scholar

[23]

M. T. Mohan, Global strong solutions of the stochastic three dimensional inviscid simplified Bardina turbulence model, Communications on Stochastic Analysis, 12 (2018), 249-270.  doi: 10.31390/cosa.12.3.03.  Google Scholar

[24]

M. T. Mohan, An extension of the Beale-Kato-Majda criterion for the 3D Navier-Stokes equation with hereditary viscosity, Accepted in Pure and Applied Functional Analysis, (2019). Google Scholar

[25]

M. T. Mohan, On the two dimensional tidal dynamics system: stationary solution and stability, Applicable Analysis, (2018), https://doi.org/10.1080/00036811.2018.1546002. doi: 10.1080/00036811.2018.1546002.  Google Scholar

[26]

M. T. Mohan, Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: Existence, uniqueness, exponential stability and invariant measures, Accepted in Stochastic Analysis and Applications, (2019). Google Scholar

[27]

M. T. Mohan, Global attractors and determining modes for the three dimensional Kelvin-Voigt fluids, Submitted. Google Scholar

[28]

M. T. Mohan and S. S. Sritharan, New methods for local solvability of quasilinear symmetric hyperbolic systems, Evolution Equations and Control Theory, 5 (2016), 273-302.  doi: 10.3934/eect.2016005.  Google Scholar

[29]

M. T. Mohan and S. S. Sritharan, Stochastic quasilinear symmetric hyperbolic system perturbed by Lévy noise, Pure and Applied Functional Analysis, 3 (2018), 137-178.   Google Scholar

[30]

M. T. Mohan and S. S. Sritharan, Stochastic Navier-Stokes equation perturbed by Lévy noise with hereditary viscosity, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 22 (2019), 1950006, 32 pp. doi: 10.1142/S0219025719500061.  Google Scholar

[31]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar

[32]

A. P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem. LOMI, 38 (1973), 98-136.   Google Scholar

[33]

A. P. Oskolkov, Theory of nonstationary flows of Kelvin-Voigt fluids, Zap. Nauchn. Sem. LOMI, 115 (1982), 191–202, 310.  Google Scholar

[34]

A. P. Oskolkov, Initial boundary value problems for the equations of motion of Kelvin-Voigt fluids and Oldroyd fluids, Proceedings of Steklov Institute of Mathematics, 2 (1989), 137-182.   Google Scholar

[35]

A. P. Oskolkov, Nonlocal problems for the equations of Kelvin- Voight fluids and their $\varepsilon$approximations in classes of smooth functions, Zap. Nauchn. Sem. POMI, 230 (1995), 214-242.  doi: 10.1007/BF02433999.  Google Scholar

[36]

A. P. Oskolkov, Nonlocal problems for the equations of motion of Kelvin-Voight fluids, Journal of Mathematical Sciences, 75 (1995), 2058-2078.  doi: 10.1007/BF02362946.  Google Scholar

[37]

A. P. Oskolkov and R. D. Shadiev, Nonlocal problems of the theory of the equations of motion for Kelvin-Voight fluids, Journal of Soviet Mathematics, 62 (1992), 2699-2723.  doi: 10.1007/BF01102639.  Google Scholar

[38]

A. P. Oskolkov and R. D. Shadiev, Nonlocal problems in the theory of equations of motion of Kelvin-Voight fluids. 2, Journal of Mathematical Sciences, 59 (1992), 1206-1214.  doi: 10.1007/BF01374082.  Google Scholar

[39]

A. P. Oskolkov and R. D. Shadiev, Towards a theory of global solvability on $[0, \infty)$ of initial-boundary value problems for the equations of motion of oldroyd and Kelvin-Voight fluids, Journal of Mathematical Sciences, 68 (1994), 240-253.  doi: 10.1007/BF01249338.  Google Scholar

[40]

R. T. Rockafellar, Convex Analysis, Reprint of the 1970 original. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ, 1997.  Google Scholar

[41]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050.  Google Scholar

[42]

V. G. Zvyagin and M. V. Turbin, The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids, Journal of Mathematical Sciences, 168 (2010), 157-308.  doi: 10.1007/s10958-010-9981-2.  Google Scholar

[1]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[2]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[3]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[4]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[5]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[6]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[7]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078

[8]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120

[9]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[10]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[11]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[12]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[13]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[14]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345

[15]

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

[16]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[17]

Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020269

[18]

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

[19]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[20]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (323)
  • HTML views (624)
  • Cited by (2)

Other articles
by authors

[Back to Top]