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We present and review several models of fractional viscous stresses from the literature, which generalise classical viscosity theories to fractional orders by replacing total strain derivatives in time with fractional time derivatives. We…
We consider inverse problems for non-linear hyperbolic and elliptic equations and give an introduction to the method based on the multiple linearization, or on the construction of artificial sources, to solve these problems. The method is…
We consider the dominant equations for the motion of the non-Newtonian fluid in a domain from an energetic point of view. We apply our energetic variational approaches and the first law of thermodynamics to derive the generalized…
The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a cost function.…
An account of the transversality conditions of variational problems gives rise to essential results in the analysis of different physical phenomena. This powerful and elegant approach has proven to be fruitful in a diversity of variational…
In this paper, we propose and analyze an abstract stabilized mixed finite element framework that can be applied to nonlinear incompressible elasticity problems. In the abstract stabilized framework, we prove that any mixed finite element…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
We consider control-constrained linear-quadratic optimal control problems on evolving surfaces. In order to formulate well-posed problems, we prove existence and uniqueness of weak solutions for the state equation, in the sense of…
In this paper we consider the motion of a rigid body in a viscous incompressible fluid when some Navier slip conditions are prescribed on the body's boundary. The whole `viscous incompressible fluid + rigid body' system is assumed to occupy…
This is a continuation of our study [Uhlmann-Zhai, JMPA, 2021] on an inverse boundary value problem for a nonlinear elastic wave equation. We prove that all the linear and nonlinear coefficients can be recovered from the…
We describe a variational calculation for the problem of screening of a point charge in a layered correlated metal for dopings close to the Mott transition where the screening is non-linear due to the proximity to the incompressible…
We compute the response matrix for a tracer particle in a compressible fluid with odd viscosity living on a two-dimensional surface. Unlike the incompressible case, we find that an odd compressible fluid can produce an odd lift force on a…
In this paper, we prove the existence of separable solutions to the equations of motion for self-gravitating hyperelastic matter, under an appropriate class of constitutive assumptions on the strain-energy function. Our framework includes…
Partial complementary energy densities are introduced through partial Legendre transforms from the strain energy density of linear elasticity. They have mixed components of the strain and stress tensors. Mixed variational principles based…
Modelling the large deformation of hyperelastic solids under plane stress conditions for arbitrary compressible and nearly incompressible material models is challenging. This is in contrast to the case of full incompressibility where the…
In this paper we establish that the time-harmonic elasticity problem in a half-strip with non-homogeneous Dirichlet conditions on its boundary section and traction-free conditions on its upper and lower boundaries, has a unique weak…
A number of important results of studying large deformations of hyper-elastic shells are obtained using discrete methods of mathematical physics. In the present paper, using the variational method for solving nonlinear boundary problems of…
Large scale features of a randomly isotropically forced incompressible and unbounded rotating fluid are examined in perturbation theory. At first order in both the random force amplitude and the angular velocity we find two types of…
We develop a mesh-based semi-Lagrangian discretization of the time-dependent incompressible Navier-Stokes equations with free boundary conditions recast as a non-linear transport problem for a momentum 1-form. A linearly implicit fully…
A number of boundary problems in multidimensional elasticity theory are solved. The solutions can be treated as the simplest cosmological models. Some specific properties of the solutions and experimental consequences of the theory are…