57 research outputs found
Weak-strong uniqueness for measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank free energy
We analyze the Ericksen-Leslie system equipped with the Oseen-Frank energy in
three space dimensions. Recently, the author introduced the concept of
measure-valued solutions to this system and showed the global existence of
these generalized solutions. In this paper, we show that suitable
measure-valued solutions, which fulfill an associated energy inequality, enjoy
the weak-strong uniqueness property, i. e. the measure-valued solution agrees
with a strong solution if the latter exists. The weak-strong uniqueness is
shown by a relative energy inequality for the associated nonconvex energy
functional
Measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank energy
In this article, we prove the existence of measure-valued solutions to the
Ericksen-Leslie system equipped with the Oseen-Frank energy. We introduce the
concept of generalized gradient Young measures. Via a Galerkin approximation,
we show the existence of weak solutions to a regularized system and attain
measure-valued solutions for vanishing regularization. Additionally, it is
shown that the measure-valued solution fulfills an energy inequality.Comment: arXiv admin note: text overlap with arXiv:1711.0337
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On the existence of weak solutions in the context of multidimensional incompressible fluid dynamics
We define the concept of energy-variational solutions for the Navier--Stokes and Euler equations. This concept is shown to be equivalent to weak solutions with energy conservation. Via a standard Galerkin discretization, we prove the existence of energy-variational solutions and thus weak solutions in any space dimension for the Navier--Stokes equations. In the limit of vanishing viscosity the same assertions are deduced for the incompressible Euler system. Via the selection criterion of maximal dissipation we deduce well-posedness for these equations
Maximal dissipative solutions for incompressible fluid dynamics
We introduce the new concept of maximal dissipative solutions for the Navier--Stokes and Euler equations and show that these solutions exist and the solution set is closed and convex. The concept of maximal dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique. A maximal dissipative solution is defined as the minimizer of a convex functional and we argue that this definition bears several advantages
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Maximal dissipative solutions for incompressible fluid dynamics
We introduce the new concept of maximal dissipative solutions for the Navier--Stokes and Euler equations and show that these solutions exist and the solution set is closed and convex. The concept of maximal dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique. A maximal dissipative solution is defined as the minimizer of a convex functional and we argue that this definition bears several advantages
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Dissipative solution to the Ericksen-Leslie system equipped with the Oseen-Frank energy
We analyze the EricksenLeslie system equipped with the OseenFrank
energy in three space dimensions. The new concept of dissipative solutions is
introduced. Recently, the author introduced the concept of measure-valued
solutions to the considered system and showed global existence as well as
weak-strong uniqueness of these generalized solutions. In this paper, we show
that the expectation of the measure valued solution is a dissipative
solution. The concept of a dissipative solution itself relies on an
inequality instead of an equality, but is described by functions instead of
parametrized measures. These solutions exist globally and fulfill the
weak-strong uniqueness property. Additionally, we generalize the relative
energy inequality to solutions fulfilling different nonhomogeneous Dirichlet
boundary conditions and incorporate the influence of a temporarily constant
electromagnetic field. Relying on this generalized energy inequality, we
investigate the long-time behavior and show that all solutions converge for
the large time limit to a certain steady state
Dissipative solution to the Ericksen--Leslie system equipped with the Oseen--Frank energy
We analyze the Ericksen-Leslie system equipped with the Oseen?Frank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measure-valued solutions to the considered system and showed global existence as well as weak-strong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized measures. These solutions exist globally and fulfill the weak-strong uniqueness property. Additionally, we generalize the relative energy inequality to solutions fulfilling different nonhomogeneous Dirichlet boundary conditions and incorporate the influence of a temporarily constant electromagnetic field. Relying on this generalized energy inequality, we investigate the long-time behavior and show that all solutions converge for the large time limit to a certain steady state
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Measure-valued solutions to the Ericksen-Leslie model equipped with the Oseen-Frank energy
In this article, we prove the existence of measure-valued solutions to
the EricksenLeslie system equipped with the OseenFrank energy. We introduce
the concept of generalized gradient Young measures. Via a Galerkin
approximation, we show the existence of weak solutions to a regularized
system and attain measure-valued solutions for vanishing regularization.
Additionally, it is shown that the measure-valued solution fulfills an energy
inequality
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