December  2017, 10(6): 1233-1256. doi: 10.3934/dcdss.2017067

Stress-diffusive regularizations of non-dissipative rate-type materials

1. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland

2. 

OxPDE, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom

3. 

Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovská 83,186 75 Prague 8, Czech Republic

4. 

University of Warsaw, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw, Poland

5. 

Heidelberg University, Interdisciplinary Center for Scientific Computing, Im Neuenheimer Feld 205,69120 Heidelberg, Germany

* Corresponding author: Piotr Minakowski: minak@mimuw.edu.pl

To Tomáš Roubíček on the occasion of his 60th birthday.

Received  July 2016 Revised  February 2017 Published  June 2017

We consider non-dissipative (elastic) rate-type material models that are derived within the Gibbs-potential-based thermodynamic framework. Since the absence of any dissipative mechanism in the model prevents us from establishing even a local-in-time existence result in two spatial dimensions for a spatially periodic problem, we propose two regularisations. For such regularized problems we obtain well-posedness of the planar, spatially periodic problem. In contrast with existing results, we prove ours for a regularizing term present solely in the evolution equation for the stress.

Citation: Jan Burczak, Josef Málek, Piotr Minakowski. Stress-diffusive regularizations of non-dissipative rate-type materials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1233-1256. doi: 10.3934/dcdss.2017067
References:
[1]

R. Adams and J. Fournier, Sobolev Spaces, volume 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, second edition, 2003. Google Scholar

[2]

J. W. Barrett, Y. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Communications in Mathematical Sciences (Accepted, 22 January 2017), Available from: arXiv: 1608.04229, 2016.Google Scholar

[3]

J. W. Barrett and S. Boyaval, Existence and approximation of a (regularized) Oldroyd-B model, Mathematical Models and Methods in Applied Sciences, 21 (2011), 1783-1837. doi: 10.1142/S0218202511005581. Google Scholar

[4]

J. W. Barrett and E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers Ⅰ: Finitely extensible nonlinear bead-spring chains, Mathematical Models and Methods in Applied Sciences, 21 (2011), 1211-1289. doi: 10.1142/S0218202511005313. Google Scholar

[5]

O. Bejaoui and M. Majdoub, Global weak solutions for some Oldroyd models, Journal of Differential Equations, 254 (2013), 660-685. doi: 10.1016/j.jde.2012.09.010. Google Scholar

[6]

H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis: Theory, Methods & Applications, 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1. Google Scholar

[7]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Communications in Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154. Google Scholar

[8]

J. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM Journal on Mathematical Analysis, 33 (2001), 84-112. doi: 10.1137/S0036141099359317. Google Scholar

[9]

E. ChiodaroliE. Feireisl and O. Kreml, On the weak solutions to the equations of a compressible heat conducting gas, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 225-243. doi: 10.1016/j.anihpc.2013.11.005. Google Scholar

[10]

L. Chupin and S. Martin, Stationary Oldroyd model with diffusive stress: Mathematical analysis of the model and vanishing diffusion process, Journal of Non-Newtonian Fluid Mechanics, 218 (2015), 27-39. doi: 10.1016/j.jnnfm.2015.01.004. Google Scholar

[11]

P. Constantin and M. Kliegl, Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress, Archive for Rational Mechanics and Analysis, 206 (2012), 725-740. doi: 10.1007/s00205-012-0537-0. Google Scholar

[12]

De LellisSzékelyhidi and Jr., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010), 225-260. doi: 10.1007/s00205-008-0201-x. Google Scholar

[13]

De LellisSzékelyhidi and Jr., Dissipative continuous Euler flows, Invent. Math., 193 (2013), 377-407. doi: 10.1007/s00222-012-0429-9. Google Scholar

[14]

T. M. Elgindi and J. Liu, Global wellposedness to the generalized oldroyd type models in R3, Journal of Differential Equations, 259 (2015), 1958-1966. doi: 10.1016/j.jde.2015.03.026. Google Scholar

[15]

T. M. Elgindi and F. Rousset, Global regularity for some oldroyd-b type models, Communications on Pure and Applied Mathematics, 68 (2015), 2005-2021. doi: 10.1002/cpa.21563. Google Scholar

[16]

D. Fang and R. Zi, Strong solutions of 3d compressible Oldroyd-B fluids, Mathematical Methods in the Applied Sciences, 36 (2013), 1423-1439. doi: 10.1002/mma.2695. Google Scholar

[17]

E. Fernández-Cara, F. Guillén and R. R. Ortega, Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind, in Handbook of Numerical Analysis, Handbook of Numerical Analysis, 8, Elsevier, 2002,543-660. 2pt Google Scholar

[18]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. doi: 10.1017/CBO9780511546754. Google Scholar

[19]

C. Guillopé and J. C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., 15 (1990), 849-869. doi: 10.1016/0362-546X(90)90097-Z. Google Scholar

[20]

N. Gunther, On the motion of fluid in a moving container. izvestia akademia nauk ussr, Seriya Fizicheskaya-Mathematica, 20 (1927), 1323-1348,1503-1532. Google Scholar

[21]

T. Kato, On classical solutions of the two-dimensional non-stationary Euler equation, Archive for Rational Mechanics and Analysis, 25 (1967), 188-200. doi: 10.1007/BF00251588. Google Scholar

[22]

H. KozonoT. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Mathematische Zeitschrift, 242 (2002), 251-278. doi: 10.1007/s002090100332. Google Scholar

[23]

H. Kozono and Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Communications in Mathematical Physics, 214 (2000), 191-200. doi: 10.1007/s002200000267. Google Scholar

[24]

J. KratochvílJ. Málek and P. Minakowski, A Gibbs-potential-based framework for ideal plasticity of crystalline solids treated as a material flow through an adjustable crystal lattice space and its application to three-dimensional micropillar compression, International Journal of Plasticity, 87 (2016), 114-129. doi: 10.1016/j.ijplas.2016.09.006. Google Scholar

[25]

Z. Lei, Global existence of classical solutions for some Oldroyd-B model via the incompressible limit, Chinese Annals of Mathematics, Series B, 27 (2006), 565-580. doi: 10.1007/s11401-005-0041-z. Google Scholar

[26]

Z. LeiCh. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Archive for Rational Mechanics and Analysis, 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x. Google Scholar

[27]

L. Lichtenstein, Über einige Existenzprobleme der Hydrodynamik homogener, unzusammendrickbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsatze, Mathematische Zeitschrift, 23 (1925), 89-154. doi: 10.1007/BF01506223. Google Scholar

[28]

F.-H. LinCh. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Communications on Pure and Applied Mathematics, 58 (2005), 1437-1471. doi: 10.1002/cpa.20074. Google Scholar

[29]

P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Annals of Mathematics. Series B, 21 (2000), 131-146. doi: 10.1142/S0252959900000170. Google Scholar

[30]

M. Lukáčová-Medvid'ováH. Mizerová and Š. Nečasová, Global existence and uniqueness result for the diffusive Peterlin viscoelastic model, Nonlinear Analysis: Theory, Methods & Applications, 120 (2015), 154-170. doi: 10.1016/j.na.2015.03.001. Google Scholar

[31]

J. Málek, J. Nečas, M. Rokyta and M. Rȯžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, volume 13 of Applied Mathematics and Mathematical Computation, Chapman and Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1. Google Scholar

[32]

J. Málek and K. R. Rajagopal, Chapter 5 -mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in Handbook of Differential Equations Evolutionary Equations (eds. C. M. Dafermos and E. Feireisl), Handbook of Differential Equations: Evolutionary Equations, 2, North-Holland, 2005,371-459. 2ptGoogle Scholar

[33]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Non-Viscous Fluids, Applied Mathematical Sciences, 96, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0. Google Scholar

[34]

K. R. Rajagopal, The elasticity of elasticity, Z. Angew. Math. Phys., 58 (2007), 309-317. doi: 10.1007/s00033-006-6084-5. Google Scholar

[35]

K. R. Rajagopal, On a new class of models in elasticity, Math. Comput. Appl., 15 (2010), 506-528. Google Scholar

[36]

K. R. Rajagopal, Conspectus of concepts of elasticity, Math. Mech. Solids, 16 (2011), 536-562. doi: 10.1177/1081286510387856. Google Scholar

[37]

K. R. Rajagopal and A. R. Srinivasa, On a class of non-dissipative materials that are not hyperelastic, Royal Society of London Proceedings Series A, 465 (2009), 493-500. doi: 10.1098/rspa.2008.0319. Google Scholar

[38]

K. R. Rajagopal and A. R. Srinivasa, A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials, Proc. R. Soc. A, 467 (2011), 39-58. doi: 10.1098/rspa.2010.0136. Google Scholar

[39]

J. Simon, Compact sets in the space Lp (0, t; B), Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[40]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series 30, Princeton University Press, 1970. Google Scholar

[41]

R. Sureshkumar and A. N. Beris, Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows, Journal of Non-Newtonian Fluid Mechanics, 60 (1995), 53-80. doi: 10.1016/0377-0257(95)01377-8. Google Scholar

[42]

C. A. Truesdell, Hypo-elasticity, J. Ration. Mech. Anal., 4 (1955), 83-133. Google Scholar

[43]

E. Wiedemann, Existence of weak solutions for the incompressible Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 727-730. doi: 10.1016/j.anihpc.2011.05.002. Google Scholar

[44]

W. Wolibner, Un théoréme sur l'existence du mouvement plan d'un fluide parfait, homogene, incompressible, pendant un temps infiniment long, Mathematische Zeitschrift, 37 (1933), 698-726. doi: 10.1007/BF01474610. Google Scholar

show all references

References:
[1]

R. Adams and J. Fournier, Sobolev Spaces, volume 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, second edition, 2003. Google Scholar

[2]

J. W. Barrett, Y. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Communications in Mathematical Sciences (Accepted, 22 January 2017), Available from: arXiv: 1608.04229, 2016.Google Scholar

[3]

J. W. Barrett and S. Boyaval, Existence and approximation of a (regularized) Oldroyd-B model, Mathematical Models and Methods in Applied Sciences, 21 (2011), 1783-1837. doi: 10.1142/S0218202511005581. Google Scholar

[4]

J. W. Barrett and E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers Ⅰ: Finitely extensible nonlinear bead-spring chains, Mathematical Models and Methods in Applied Sciences, 21 (2011), 1211-1289. doi: 10.1142/S0218202511005313. Google Scholar

[5]

O. Bejaoui and M. Majdoub, Global weak solutions for some Oldroyd models, Journal of Differential Equations, 254 (2013), 660-685. doi: 10.1016/j.jde.2012.09.010. Google Scholar

[6]

H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis: Theory, Methods & Applications, 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1. Google Scholar

[7]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Communications in Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154. Google Scholar

[8]

J. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM Journal on Mathematical Analysis, 33 (2001), 84-112. doi: 10.1137/S0036141099359317. Google Scholar

[9]

E. ChiodaroliE. Feireisl and O. Kreml, On the weak solutions to the equations of a compressible heat conducting gas, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 225-243. doi: 10.1016/j.anihpc.2013.11.005. Google Scholar

[10]

L. Chupin and S. Martin, Stationary Oldroyd model with diffusive stress: Mathematical analysis of the model and vanishing diffusion process, Journal of Non-Newtonian Fluid Mechanics, 218 (2015), 27-39. doi: 10.1016/j.jnnfm.2015.01.004. Google Scholar

[11]

P. Constantin and M. Kliegl, Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress, Archive for Rational Mechanics and Analysis, 206 (2012), 725-740. doi: 10.1007/s00205-012-0537-0. Google Scholar

[12]

De LellisSzékelyhidi and Jr., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010), 225-260. doi: 10.1007/s00205-008-0201-x. Google Scholar

[13]

De LellisSzékelyhidi and Jr., Dissipative continuous Euler flows, Invent. Math., 193 (2013), 377-407. doi: 10.1007/s00222-012-0429-9. Google Scholar

[14]

T. M. Elgindi and J. Liu, Global wellposedness to the generalized oldroyd type models in R3, Journal of Differential Equations, 259 (2015), 1958-1966. doi: 10.1016/j.jde.2015.03.026. Google Scholar

[15]

T. M. Elgindi and F. Rousset, Global regularity for some oldroyd-b type models, Communications on Pure and Applied Mathematics, 68 (2015), 2005-2021. doi: 10.1002/cpa.21563. Google Scholar

[16]

D. Fang and R. Zi, Strong solutions of 3d compressible Oldroyd-B fluids, Mathematical Methods in the Applied Sciences, 36 (2013), 1423-1439. doi: 10.1002/mma.2695. Google Scholar

[17]

E. Fernández-Cara, F. Guillén and R. R. Ortega, Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind, in Handbook of Numerical Analysis, Handbook of Numerical Analysis, 8, Elsevier, 2002,543-660. 2pt Google Scholar

[18]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. doi: 10.1017/CBO9780511546754. Google Scholar

[19]

C. Guillopé and J. C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., 15 (1990), 849-869. doi: 10.1016/0362-546X(90)90097-Z. Google Scholar

[20]

N. Gunther, On the motion of fluid in a moving container. izvestia akademia nauk ussr, Seriya Fizicheskaya-Mathematica, 20 (1927), 1323-1348,1503-1532. Google Scholar

[21]

T. Kato, On classical solutions of the two-dimensional non-stationary Euler equation, Archive for Rational Mechanics and Analysis, 25 (1967), 188-200. doi: 10.1007/BF00251588. Google Scholar

[22]

H. KozonoT. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Mathematische Zeitschrift, 242 (2002), 251-278. doi: 10.1007/s002090100332. Google Scholar

[23]

H. Kozono and Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Communications in Mathematical Physics, 214 (2000), 191-200. doi: 10.1007/s002200000267. Google Scholar

[24]

J. KratochvílJ. Málek and P. Minakowski, A Gibbs-potential-based framework for ideal plasticity of crystalline solids treated as a material flow through an adjustable crystal lattice space and its application to three-dimensional micropillar compression, International Journal of Plasticity, 87 (2016), 114-129. doi: 10.1016/j.ijplas.2016.09.006. Google Scholar

[25]

Z. Lei, Global existence of classical solutions for some Oldroyd-B model via the incompressible limit, Chinese Annals of Mathematics, Series B, 27 (2006), 565-580. doi: 10.1007/s11401-005-0041-z. Google Scholar

[26]

Z. LeiCh. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Archive for Rational Mechanics and Analysis, 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x. Google Scholar

[27]

L. Lichtenstein, Über einige Existenzprobleme der Hydrodynamik homogener, unzusammendrickbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsatze, Mathematische Zeitschrift, 23 (1925), 89-154. doi: 10.1007/BF01506223. Google Scholar

[28]

F.-H. LinCh. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Communications on Pure and Applied Mathematics, 58 (2005), 1437-1471. doi: 10.1002/cpa.20074. Google Scholar

[29]

P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Annals of Mathematics. Series B, 21 (2000), 131-146. doi: 10.1142/S0252959900000170. Google Scholar

[30]

M. Lukáčová-Medvid'ováH. Mizerová and Š. Nečasová, Global existence and uniqueness result for the diffusive Peterlin viscoelastic model, Nonlinear Analysis: Theory, Methods & Applications, 120 (2015), 154-170. doi: 10.1016/j.na.2015.03.001. Google Scholar

[31]

J. Málek, J. Nečas, M. Rokyta and M. Rȯžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, volume 13 of Applied Mathematics and Mathematical Computation, Chapman and Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1. Google Scholar

[32]

J. Málek and K. R. Rajagopal, Chapter 5 -mathematical issues concerning the Navier-Stokes equations and some of its generalizations, in Handbook of Differential Equations Evolutionary Equations (eds. C. M. Dafermos and E. Feireisl), Handbook of Differential Equations: Evolutionary Equations, 2, North-Holland, 2005,371-459. 2ptGoogle Scholar

[33]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Non-Viscous Fluids, Applied Mathematical Sciences, 96, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4284-0. Google Scholar

[34]

K. R. Rajagopal, The elasticity of elasticity, Z. Angew. Math. Phys., 58 (2007), 309-317. doi: 10.1007/s00033-006-6084-5. Google Scholar

[35]

K. R. Rajagopal, On a new class of models in elasticity, Math. Comput. Appl., 15 (2010), 506-528. Google Scholar

[36]

K. R. Rajagopal, Conspectus of concepts of elasticity, Math. Mech. Solids, 16 (2011), 536-562. doi: 10.1177/1081286510387856. Google Scholar

[37]

K. R. Rajagopal and A. R. Srinivasa, On a class of non-dissipative materials that are not hyperelastic, Royal Society of London Proceedings Series A, 465 (2009), 493-500. doi: 10.1098/rspa.2008.0319. Google Scholar

[38]

K. R. Rajagopal and A. R. Srinivasa, A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials, Proc. R. Soc. A, 467 (2011), 39-58. doi: 10.1098/rspa.2010.0136. Google Scholar

[39]

J. Simon, Compact sets in the space Lp (0, t; B), Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[40]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series 30, Princeton University Press, 1970. Google Scholar

[41]

R. Sureshkumar and A. N. Beris, Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows, Journal of Non-Newtonian Fluid Mechanics, 60 (1995), 53-80. doi: 10.1016/0377-0257(95)01377-8. Google Scholar

[42]

C. A. Truesdell, Hypo-elasticity, J. Ration. Mech. Anal., 4 (1955), 83-133. Google Scholar

[43]

E. Wiedemann, Existence of weak solutions for the incompressible Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 727-730. doi: 10.1016/j.anihpc.2011.05.002. Google Scholar

[44]

W. Wolibner, Un théoréme sur l'existence du mouvement plan d'un fluide parfait, homogene, incompressible, pendant un temps infiniment long, Mathematische Zeitschrift, 37 (1933), 698-726. doi: 10.1007/BF01474610. Google Scholar

[1]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10

[2]

Hongwei Wang, Amin Esfahani. Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation. Evolution Equations & Control Theory, 2019, 8 (4) : 709-735. doi: 10.3934/eect.2019035

[3]

K. R. Rajagopal. The thermo-mechanics of rate-type fluids. Discrete & Continuous Dynamical Systems - S, 2012, 5 (6) : 1133-1145. doi: 10.3934/dcdss.2012.5.1133

[4]

Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203

[5]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181

[6]

Yuri Trakhinin. On well-posedness of the plasma-vacuum interface problem: the case of non-elliptic interface symbol. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1371-1399. doi: 10.3934/cpaa.2016.15.1371

[7]

Jaime Angulo, Carlos Matheus, Didier Pilod. Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system. Communications on Pure & Applied Analysis, 2009, 8 (3) : 815-844. doi: 10.3934/cpaa.2009.8.815

[8]

Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic & Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006

[9]

Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure & Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605

[10]

Amina Amassad, Mircea Sofonea. Analysis of some nonlinear evolution systems arising in rate-type viscoplasticity. Conference Publications, 1998, 1998 (Special) : 58-71. doi: 10.3934/proc.1998.1998.58

[11]

Markus Grasmair. Well-posedness and convergence rates for sparse regularization with sublinear $l^q$ penalty term. Inverse Problems & Imaging, 2009, 3 (3) : 383-387. doi: 10.3934/ipi.2009.3.383

[12]

Francesco Maddalena, Danilo Percivale, Franco Tomarelli. Adhesive flexible material structures. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 553-574. doi: 10.3934/dcdsb.2012.17.553

[13]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387

[14]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032

[15]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095

[16]

Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197

[17]

Gabriela Marinoschi. Well posedness of a time-difference scheme for a degenerate fast diffusion problem. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 435-454. doi: 10.3934/dcdsb.2010.13.435

[18]

Jihong Zhao, Ting Zhang, Qiao Liu. Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 555-582. doi: 10.3934/dcds.2015.35.555

[19]

Toyohiko Aiki, Joost Hulshof, Nobuyuki Kenmochi, Adrian Muntean. Analysis of non-equilibrium evolution problems: Selected topics in material and life sciences. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : i-iii. doi: 10.3934/dcdss.2014.7.1i

[20]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (31)
  • HTML views (92)
  • Cited by (0)

Other articles
by authors

[Back to Top]