185 research outputs found
Blow-up of the hyperbolic Burgers equation
The memory effects on microscopic kinetic systems have been sometimes
modelled by means of the introduction of second order time derivatives in the
macroscopic hydrodynamic equations. One prototypical example is the hyperbolic
modification of the Burgers equation, that has been introduced to clarify the
interplay of hyperbolicity and nonlinear hydrodynamic evolution. Previous
studies suggested the finite time blow-up of this equation, and here we present
a rigorous proof of this fact
Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non BV perturbations
We develop a theory based on relative entropy to show the uniqueness and L^2
stability (up to a translation) of extremal entropic Rankine-Hugoniot
discontinuities for systems of conservation laws (typically 1-shocks, n-shocks,
1-contact discontinuities and n-contact discontinuities of large amplitude)
among bounded entropic weak solutions having an additional trace property. The
existence of a convex entropy is needed. No BV estimate is needed on the weak
solutions considered. The theory holds without smallness condition. The
assumptions are quite general. For instance, strict hyperbolicity is not needed
globally. For fluid mechanics, the theory handles solutions with vacuum.Comment: 29 page
Global generalized solutions for Maxwell-alpha and Euler-alpha equations
We study initial-boundary value problems for the Lagrangian averaged alpha
models for the equations of motion for the corotational Maxwell and inviscid
fluids in 2D and 3D. We show existence of (global in time) dissipative
solutions to these problems. We also discuss the idea of dissipative solution
in an abstract Hilbert space framework.Comment: 27 pages, to appear in Nonlinearit
Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations
The Navier--Stokes equations are commonly used to model and to simulate flow
phenomena. We introduce the basic equations and discuss the standard methods
for the spatial and temporal discretization. We analyse the semi-discrete
equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index
and quantify the numerical difficulties in the fully discrete schemes, that are
induced by the strangeness of the system. By analyzing the Kronecker index of
the difference-algebraic equations, that represent commonly and successfully
used time stepping schemes for the Navier--Stokes equations, we show that those
time-integration schemes factually remove the strangeness. The theoretical
considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909,
https://doi.org/10.5281/zenodo.99890
Weak-strong uniqueness property for the full Navier-Stokes-Fourier system
The Navier-Stokes-Fourier system describing the motion of a compressible,
viscous, and heat conducting fluid is known to possess global-in-time weak
solutions for any initial data of finite energy. We show that a weak solution
coincides with the strong solution, emanating from the same initial data, as
long as the latter exists. In particular, strong solutions are unique within
the class of weak solutions
Uniform regularity for the Navier-Stokes equation with Navier boundary condition
We prove that there exists an interval of time which is uniform in the
vanishing viscosity limit and for which the Navier-Stokes equation with Navier
boundary condition has a strong solution. This solution is uniformly bounded in
a conormal Sobolev space and has only one normal derivative bounded in
. This allows to get the vanishing viscosity limit to the
incompressible Euler system from a strong compactness argument
Relative entropies, suitable weak solutions, and weak strong uniqueness for the compressible Navier-Stokes system
We introduce the notion of relative entropy for the weak solutions of the
compressible Navier-Stokes system. We show that any finite energy weak solution
satisfies a relative entropy inequality for any pair of sufficiently smooth
test functions. As a corollary we establish weak-strong uniqueness principle
for the compressible Navier-Stokes system
On Landau damping
Going beyond the linearized study has been a longstanding problem in the
theory of Landau damping. In this paper we establish exponential Landau damping
in analytic regularity. The damping phenomenon is reinterpreted in terms of
transfer of regularity between kinetic and spatial variables, rather than
exchanges of energy; phase mixing is the driving mechanism. The analysis
involves new families of analytic norms, measuring regularity by comparison
with solutions of the free transport equation; new functional inequalities; a
control of nonlinear echoes; sharp scattering estimates; and a Newton
approximation scheme. Our results hold for any potential no more singular than
Coulomb or Newton interaction; the limit cases are included with specific
technical effort. As a side result, the stability of homogeneous equilibria of
the nonlinear Vlasov equation is established under sharp assumptions. We point
out the strong analogy with the KAM theory, and discuss physical implications.Comment: News: (1) the main result now covers Coulomb and Newton potentials,
and (2) some classes of Gevrey data; (3) as a corollary this implies new
results of stability of homogeneous nonmonotone equilibria for the
gravitational Vlasov-Poisson equatio
The role of population PK-PD modelling in paediatric clinical research
Children differ from adults in their response to drugs. While this may be the result of changes in dose exposure (pharmacokinetics [PK]) and/or exposure response (pharmacodynamics [PD]) relationships, the magnitude of these changes may not be solely reflected by differences in body weight. As a consequence, dosing recommendations empirically derived from adults dosing regimens using linear extrapolations based on body weight, can result in therapeutic failure, occurrence of adverse effect or even fatalities. In order to define rational, patient-tailored dosing schemes, population PK-PD studies in children are needed. For the analysis of the data, population modelling using non-linear mixed effect modelling is the preferred tool since this approach allows for the analysis of sparse and unbalanced datasets. Additionally, it permits the exploration of the influence of different covariates such as body weight and age to explain the variability in drug response. Finally, using this approach, these PK-PD studies can be designed in the most efficient manner in order to obtain the maximum information on the PK-PD parameters with the highest precision. Once a population PK-PD model is developed, internal and external validations should be performed. If the model performs well in these validation procedures, model simulations can be used to define a dosing regimen, which in turn needs to be tested and challenged in a prospective clinical trial. This methodology will improve the efficacy/safety balance of dosing guidelines, which will be of benefit to the individual child
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