On partial convolution and mean comparison
We prove mean comparison from a different perspective, where we introduce the concept of partial convolution.
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arXiv subjects
Joerg Kampen
Publications and source records attributed to Joerg Kampen.
The dilemma of turbulence modelling
A new construction technique of multiple solutions of the Euler equa- tion in strong spaces is introduced which reveals the relationship to multi- ple Navier Stokes equation solutions with special force terms while avoid- ing viscosity limit constructions. This shows that severe restrictions have to be imposed on time dependent external force terms in order to ob- tain uniqueness for the Navier Stokes equation Cauchy problem. Such restrictions are imposed in the statement of the so-called millenium prob- lem. Minimal turbulence models should arguably incorporate weaker force terms in order to account for boundary conditions and forces. However, we show that models of this type which have been proposed recently, do not have a unique solution. This lack of determinism of mimimal turbu- lence models indicates a dilemma: either models are too simple to capture turbulence but may have unique smooth solutions, or there is a modelling gap as the model does not determine a unique solution.
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Some new consequences of the CKN-theory
It is a simple consequence of the Cafarelli-Kohn-Nirenberg theory that every possible singularity in a thin Haussdorff-measurable set of a Leray- Hopf solution of the incompressible Navier Stokes equation is on the tip of a small open cone, where the solution is smooth. Using global regularity results for weak Hopf-Leray solutions this potential singularity can be analyzed by investigation of the asymptotic behavior at infinite time of a solution of a related initial-boundary value problem posed in transformed coordinates on a cylinder. Next to some new consequences such as global regularity of the Leray Hopf solution after finite time, many known results can be recovered with this method succinctly, especially the result that H1-regularity implies global existence and smoothness.
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Weak singularities of 3-D Euler equations and restricted regularity of Navier Stokes equation solutions with time dependent force terms
Classical vorticity solution branches of the three dimensional incompressible Euler equation are constructed where a velocity component can blow up at some point after finite time for regular data in H2. Furthermore, vorticity can blow up after finite time for data in H2, and there are classical solution branches with regular data which develop weak singularities at some point of space time after finite time. The construction of these time-local solution branches is by viscosity limits of viscosity extensions of time-reversed Euler-type equations. The short time (weak) singularities are then initial values of local solution branches of the time-reversed Euler type equations, which are constructed via (not time-reversible) viscosity extensions. These solutions of the 3-D Euler equation have a straightforward interpretation as solution of incompressible Navier Stokes equation with time dependent force terms with restricted regularity or blow up of a velocity component at some point, as the time-dependent force term can be chosen such that it cancels the viscosity term of the incompressible Navier Stokes equation.
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Singular vorticity solutions of the incompressible Euler equation via inviscid limits
Singular vorticty solutions of the incompressible 3D-Euler equation are constructed which satisfy the BKM criterion (cf. [2]). The construction is done by inviscid limits of vorticity solutions of transformed incompressible Navier Stokes type equations with a damping potential term, where the latter equations admit a global regular solution for positive viscosity. The inviscid limit vorticity solution of the incompressible Euler vorticity equation becomes singular at a point of the boundary of a finite domain.
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Construction of unshielded singular solutions of the harmonic field equations
Singular solutions of the harmonic Einstein evolution equation are constructed which are related to spatially global and time-local solutions for a certain class of quasilinear hyperbolic systems of second order. The constructed singularities of curvature invariants occur generically and are accessible by g.a.p. curves. The singularities are not strongly censored, and for strongly asymptotically predictable space-times, they are located in the causal past of the future null infinity, and are, hence, not shielded by a black hole. This is an alternative construction of singularities, which may be applied to other hyperbolic equations such as the Euler equation (cf. [3] for a different construction method- both of our constructions are fundamentally different from supercritical blow-up constructions in the Katz-Pavlovic model or singular solution constructions for heat-flow maps in specific dimensions).
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Singular solutions of Navier Stokes equations with time-dependent external force terms in L2
It is shown that Navier Stokes equation models with time dependent external forces in L2 can have singular solutions.
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A 3D Euler equation solution with 2D sets of singularities and data with Hoelder continuous first order derivatives
An example of a solution branch of the three dimensional Euler equation Cauchy problem is constructed which develops a singular velocity component and a singular vorticity component after finite time for some data which have Hoelder continuous first order spatial derivatives. Such a solution branch can be extended beyond a time section at some positive finite time where a two dimensional set of singularities is located.
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Trotter product formulas and global regular upper bounds of the Navier Stokes equation solution
Global upper bounds with respect to regular norms of the incompressible Navier Stokes equation solution with regular data are constructed by an infinite scheme, where we work in bounded ZFC with bounded quantifiers and explicit infinitesimals. Trotter product formulas with infinitesimal error are obtained, which simplify for calculi with explicit infinitesimal and make the spatial effects needed in order to obtain global schemes more transparent.
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On global regularity and singularities of Navier-Stokes- and Euler equation solutions
Euler-Leray data functions of first and second order are defined by first and second order derivatives of the nonlinear spatial part of the incompressible Euler equation operator in Leray projection form applied to Cauchy data. The Lipschitz continuity of these functions for a certain class of strongly regular Cauchy data is sufficient for the existence of global regular upper bounds of incompressible Navier Stokes equation solutions. Global regular upper bounds of global solution branches of the incompressible Euler equation can be obtained for a class of strongly regular Cauchy data, if the Cauchy data satisfy an additional condition of strong polynomial decay at spatial infinity. Furthermore, if a Lipschitz condition for the Euler- Leray data function of second order is satisfied, then there are long time vorticity blow ups of the incompressible Euler equation, and, correspondingly, short time and long time vorticity blow ups or singular solutions of incompressible Navier Stokes equations with time-dependent forces of lower regularity. A further consequence is that multiple global solutions of the incompressible Euler equations exists.
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Some interesting class of integrable partial differential equation systems
We determine a considerable class of nonlinear partial differential equation systems which have global regular solutions. Uniqueness is not a direct general consequence of this method. The scheme can be applied to the incompressible Navier Stokes equation.
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Some infinite matrix analysis, a Trotter product formula for dissipative operators, and an algorithm for the incompressible Navier-Stokes equation
We introduce a global scheme on the n-torus of a controlled incompressible Navier-Stokes equation in terms of a coupled controlled infinite ODE-system of Fourier-modes with smooth data. We construct a scheme of global approximations related to linear partial integrodifferential equations in dual space which are uniformly bounded in dual Sobolev spaces with polynomially decaying modes. The scheme is based on some infinite matrix algebra related to weakly singular integrals, and a Trotter-product formula for dissipative operators which leads to rigorous existence results and a uniform bound for the solutions of the successive approximating linearized equations in dual space which may be otherwise represented as formal solutions in the sense of an iterated Dyson formalism.
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Generalisation of Hajek s stochastic comparison results to stochastic sums
Hajek's stochastic comparison result is generalised to multivariate stochastic sum processes with univariate convex data functions and for univariate monoton nondecreasing convex data functions for processes with and without drift respectively. The univariate result is recovered.
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On an auto-controlled global existence scheme of the incompressible Navier Stokes equation
We propose a global scheme for the incompressible Navier Stokes equation, where at each time step a damping potential term is introduced via a time dilation transformation of the equation itself. This leads a global upper bounds of the value function and its spatial derivatives. The regularity is limited only by the regularity of the viscosity coefficient function and by the regularity and polynomial decay of the data. On an analytical level the scheme proposed is an alternative to schemes with external control functions.
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On the multivariate Burgers equation and the incompressible Navier-Stokes equation (part III)
In this paper we first obtain local contraction results in a Hm-norm with respect to time and space for a local scheme. We show that a global controlled scheme preserves higher order regularity with respect to the spatial variables together with polynomial decay of order m of the data at each time step. Here, we simplify the controlled scheme considered in [1] and [2]. Especially, for the simplified controlled scheme we get an upper bound for the Leray projection term. Furthermore, the schemes discussed in [1] and [2] are simplified in the sense that the estimates are achieved without the use of some properties concerning the adjoint of a local fundamental solutions with variable drift terms. We note that the pointwise and absolute convergence of the local functional series and their first order time derivatives and their spatial derivatives leads to a constructive approach of local and global classical solutions.
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A global scheme for the incompressible Navier-Stokes equation on compact Riemannian manifolds
We propose a global scheme for a controlled Navier-Stokes equation system on compact smooth Riemannian manifolds.
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On global schemes for highly degenerate Navier Stokes equation systems
First order semi-linear coupling of scalar hypoelliptic equations of second order leads to a natural class of incompressible Navier Stokes equation systems, which encompasses systems with variable viscosity and essentially Navier Stokes equation systems on manifolds. We introduce a controlled global solution scheme which is based on a) local contraction results in function spaces with polynomial decay of some order at spatial infinity related to the polynomial growth factors of standard a priori estimates of densities and their derivatives for hypoelliptic diffusions of Hoermander type (cf. [15]), and b) on a controlled equation system where we discuss variations of the scheme we considered in [10]. We supplement our notes on global bounds of the Leray projection term in that paper and related controlled Navier Stokes equation schemes in [6, 7, 9, 10, 12]. Some arguments concerning linear upper bounds of the control function are added.
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Density estimates for differential equations of second order satisfying a weak Hoermander condition
We prove an extension of Hoermander's classical result on hypoelliptic second order equations, where the coefficients of the related vector fields are globally Lipschitz and satisfy the classical Hoermander condition on a dense set while the density still exists in a classical sense. Furthermore, Hoermander's classical result and related density estimates based on Malliavin calculus are recovered from an analytical point of view.
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