41 research outputs found
The matricial relaxation of a linear matrix inequality
Given linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask:
(Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X)
PsD? (Q2) when do they have the same solution set? Such questions can be
NP-hard. This paper describes a natural relaxation of an LMI, based on
substituting matrices for the variables x_j. With this relaxation, the
domination questions (Q1) and (Q2) have elegant answers, indeed reduce to
constructible semidefinite programs. Assume there is an X such that L_1(X) and
L_2(X) are both PD, and suppose the positivity domain of L_1 is bounded. For
our "matrix variable" relaxation a positive answer to (Q1) is equivalent to the
existence of matrices V_j such that L_2(x)=V_1^* L_1(x) V_1 + ... + V_k^*
L_1(x) V_k. As for (Q2) we show that, up to redundancy, L_1 and L_2 are
unitarily equivalent.
Such algebraic certificates are typically called Positivstellensaetze and the
above are examples of such for linear polynomials. The paper goes on to derive
a cleaner and more powerful Putinar-type Positivstellensatz for polynomials
positive on a bounded set of the form {X | L(X) PsD}.
An observation at the core of the paper is that the relaxed LMI domination
problem is equivalent to a classical problem. Namely, the problem of
determining if a linear map from a subspace of matrices to a matrix algebra is
"completely positive".Comment: v1: 34 pages, v2: 41 pages; supplementary material is available in
the source file, or see http://srag.fmf.uni-lj.si
Phase Transition in a Vlasov-Boltzmann Binary Mixture
There are not many kinetic models where it is possible to prove bifurcation
phenomena for any value of the Knudsen number. Here we consider a binary
mixture over a line with collisions and long range repulsive interaction
between different species. It undergoes a segregation phase transition at
sufficiently low temperature. The spatially homogeneous Maxwellian equilibrium
corresponding to the mixed phase, minimizing the free energy at high
temperature, changes into a maximizer when the temperature goes below a
critical value, while non homogeneous minimizers, corresponding to coexisting
segregated phases, arise. We prove that they are dynamically stable with
respect to the Vlasov-Boltzmann evolution, while the homogeneous equilibrium
becomes dynamically unstable
Decay and Continuity of Boltzmann Equation in Bounded Domains
Boundaries occur naturally in kinetic equations and boundary effects are
crucial for dynamics of dilute gases governed by the Boltzmann equation. We
develop a mathematical theory to study the time decay and continuity of
Boltzmann solutions for four basic types of boundary conditions: inflow,
bounce-back reflection, specular reflection, and diffuse reflection. We
establish exponential decay in norm for hard potentials for
general classes of smooth domains near an absolute Maxwellian. Moreover, in
convex domains, we also establish continuity for these Boltzmann solutions away
from the grazing set of the velocity at the boundary. Our contribution is based
on a new decay theory and its interplay with delicate
decay analysis for the linearized Boltzmann equation, in the presence of many
repeated interactions with the boundary.Comment: 89 pages
Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials
In this paper it is shown that unique solutions to the relativistic Boltzmann
equation exist for all time and decay with any polynomial rate towards their
steady state relativistic Maxwellian provided that the initial data starts out
sufficiently close in . If the initial data are continuous then
so is the corresponding solution. We work in the case of a spatially periodic
box. Conditions on the collision kernel are generic in the sense of
(Dudy{\'n}ski and Ekiel-Je{\.z}ewska, Comm. Math. Phys., 1988); this resolves
the open question of global existence for the soft potentials.Comment: 64 page