Weak Solutions for a Class of Nonlinear Systems of Viscoelasticity

An Eulerian Description of Fluids Containing Visco-Elastic Particles

“…We pose the basic equations in a purely Eulerian description; numerical simulation of such a system will only require a single mesh for the Eulerian domain. The system of equations we propose contains the classical visco-hyperelasticity equations for which there is no satisfactory theory of existence and uniqueness [7,16]. However, we consider an approximation for which it is possible to develop a reasonable existence theory.…”

An Approach to Nonlinear Viscoelasticity via Metric Gradient Flows

“…Condition (1.2) is quite difficult to handle analytically due to the fact that it is not compatible with some common hypotheses (e.g., monotonicity with respect to the strain rate) on the stress (see [12], [29,Section 2.3]). The existence of a dissipation potential R means that there exists a real-valued function R(x, F,Ḟ ) with R(x, F,Ḟ ) ≥ R(x, F, 0) = 0 such that ∂Ḟ R(x, F,Ḟ ) = S(x, F,Ḟ ).…”

“…The fully dynamical equation of nonlinear viscoelasticity of rate type (which corresponds to (1.1) together with the inertia term) has been well-studied by various authors such as [12,14,31,27,28,25,26] for existence and long-time behavior of solutions. The only theory for the existence of solutions for this problem with frame indifferent S(x, F,Ḟ ) is that of Potier-Ferry [25,26], who established global existence and uniqueness of solutions for initial data close to a smooth equilibrium for pure displacement boundary conditions.…”

“…The only theory for the existence of solutions for this problem with frame indifferent S(x, F,Ḟ ) is that of Potier-Ferry [25,26], who established global existence and uniqueness of solutions for initial data close to a smooth equilibrium for pure displacement boundary conditions. Demoulini [12], on the other hand, obtained Young measure solutions and also imposed the so-called uniform strict monotonicity condition on the strain rate to prove existence of classical weak solutions. As shown in [29] this condition is not compatible with frame indifference.…”

Global existence and uniqueness of solutions for a viscoelastic two-phase model with nonlocal capillarity

“…For non-negative φ , the existence of weak solutions to system (1.1) for given initial values (on the spatial domain R) was proven for arbitrary Lipschitz continuous σ with a variational technique (see [13], Theorem 5.4) relying on the ideas in [4]. This result is extended in this contribution in various directions: initial boundary value problems are addressed, partly negative kernels are treated, and optimal regularity is proven under quite rough initial conditions.…”