@article{14042,
  abstract     = {Long-time and large-data existence of weak solutions for initial- and boundary-value problems concerning three-dimensional flows of incompressible fluids is nowadays available not only for Navier–Stokes fluids but also for various fluid models where the relation between the Cauchy stress tensor and the symmetric part of the velocity gradient is nonlinear. The majority of such studies however concerns models where such a dependence is explicit (the stress is a function of the velocity gradient), which makes the class of studied models unduly restrictive. The same concerns boundary conditions, or more precisely the slipping mechanisms on the boundary, where the no-slip is still the most preferred condition considered in the literature. Our main objective is to develop a robust mathematical theory for unsteady internal flows of implicitly constituted incompressible fluids with implicit relations between the tangential projections of the velocity and the normal traction on the boundary. The theory covers numerous rheological models used in chemistry, biorheology, polymer and food industry as well as in geomechanics. It also includes, as special cases, nonlinear slip as well as stick–slip boundary conditions. Unlike earlier studies, the conditions characterizing admissible classes of constitutive equations are expressed by means of tools of elementary calculus. In addition, a fully constructive proof (approximation scheme) is incorporated. Finally, we focus on the question of uniqueness of such weak solutions.},
  author       = {Bulíček, Miroslav and Málek, Josef and Maringová, Erika},
  issn         = {1422-6952},
  journal      = {Journal of Mathematical Fluid Mechanics},
  number       = {3},
  publisher    = {Springer Nature},
  title        = {{On unsteady internal flows of incompressible fluids characterized by implicit constitutive equations in the bulk and on the boundary}},
  doi          = {10.1007/s00021-023-00803-w},
  volume       = {25},
  year         = {2023},
}

@article{10005,
  abstract     = {We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first-order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone p-coercive graph. We then establish the global-in-time and large-data existence of a (weak) solution and its uniqueness. To this end, we adopt and significantly generalize Minty’s method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable from the point of view of numerical approximations.},
  author       = {Bulíček, Miroslav and Maringová, Erika and Málek, Josef},
  issn         = {1793-6314},
  journal      = {Mathematical Models and Methods in Applied Sciences},
  keywords     = {Nonlinear parabolic systems, implicit constitutive theory, weak solutions, existence, uniqueness},
  number       = {09},
  publisher    = {World Scientific},
  title        = {{On nonlinear problems of parabolic type with implicit constitutive equations involving flux}},
  doi          = {10.1142/S0218202521500457},
  volume       = {31},
  year         = {2021},
}

@article{10575,
  abstract     = {The choice of the boundary conditions in mechanical problems has to reflect the interaction of the considered material with the surface. Still the assumption of the no-slip condition is preferred in order to avoid boundary terms in the analysis and slipping effects are usually overlooked. Besides the “static slip models”, there are phenomena that are not accurately described by them, e.g. at the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier–Stokes-like problems with a dynamic slip boundary condition, which requires a proper generalization of the Gelfand triplet and the corresponding function space setting.},
  author       = {Abbatiello, Anna and Bulíček, Miroslav and Maringová, Erika},
  issn         = {1793-6314},
  journal      = {Mathematical Models and Methods in Applied Sciences},
  number       = {11},
  pages        = {2165--2212},
  publisher    = {World Scientific Publishing},
  title        = {{On the dynamic slip boundary condition for Navier-Stokes-like problems}},
  doi          = {10.1142/S0218202521500470},
  volume       = {31},
  year         = {2021},
}