Niclas Bernhoff

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• PhD
• Professor (Associate) at Karlstads Universitet

Consider the steady Boltzmann equation with slab symmetry for a monatomic, hard sphere gas in a half space. At the boundary of the half space, it is assumed that the gas is in contact with its condensed phase. The present paper discusses the existence and uniqueness of a uniformly decaying boundary layer type solution of the Boltzmann equation in t...

In this work, a general formulation, which is based on steady boundary layer problems for the Boltzmann equation, of a half-space problem is considered. The number of conditions on the indata at the interface needed to obtain well-posedness is investigated. The solutions will converge exponentially fast “far away” from the interface. For linearized...

The linearized Boltzmann collision operator has a central role in many important applications of the Boltzmann equation. Recently some important classical properties of the linearized collision operator for monatomic single species were extended to multicomponent monatomic gases and polyatomic single species. For multicomponent polyatomic gases, th...

At higher altitudes near space shuttles moving at hypersonic speed the air is excited to high temperatures. Then not only mechanical collisions are affecting the gas flow, but also chemical reactions have an impact on such hypersonic flows. In this work we insert chemical reactions, in form of dissociations and associations, in a model for a mixtur...

We consider some extensions of the classical discrete Boltzmann equation to the cases of multicomponent mixtures, polyatomic molecules (with a finite number of different internal energies), and chemical reactions, but also general discrete quantum kinetic Boltzmann-like equations; discrete versions of the Nordheim-Boltzmann (or Uehling-Uhlenbeck) e...

In this article, we recall various existing kinetic models of non-reactive polyatomic gases. We also review the results, all recently obtained, about the compactness of the associated linearized Boltzmann operator, and briefly investigate the mixture case. Eventually, with a specific collision kernel, we present a discussion about the physical rele...

A semi-classical approach to the study of the evolution of bosonic or fermionic excitations is through the Nordheim—Boltzmann- or, Uehling—Uhlenbeck—equation, also known as the quantum Boltzmann equation. In some low ranges of temperatures—e.g., in the presence of a Bose condensate—also other types of collision operators may render in essential con...

The linearized Boltzmann collision operator has a central role in many important applications of the Boltzmann equation. Recently some important classical properties of the linearized collision operator for monatomic single species were extended to multicomponent monatomic gases and polyatomic single species. For multicomponent polyatomic gases, th...

The linearized Boltzmann collision operator appears in many important applications of the Boltzmann equation. Therefore, knowing its main properties is of great interest. This work extends some classical results for the linearized Boltzmann collision operator for monatomic single species to the case of polyatomic single species, while also reviewin...

The linearized Boltzmann collision operator is fundamental in many studies of the Boltzmann equation and its main properties are of substantial importance. The decomposition into a sum of a positive multiplication operator, the collision frequency, and an integral operator is trivial. Compactness of the integral operator for monatomic single specie...

Half-space problems in the kinetic theory of gases are of great importance in the study of the asymptotic behavior of solutions of boundary value problems for the Boltzmann equation for small Knudsen numbers. They provide the boundary conditions for the fluid-dynamic-type equations and Knudsen-layer corrections to the solution of the fluid-dynamic-...

Half-space problems in the kinetic theory of gases are of great importance in the study of the asymptotic behavior of solutions of boundary value problems for the Boltzmann equation for small Knudsen numbers. In this work a generally formulated half-space problem, based on generalizations of stationary half-space problems in one spatial variable fo...

The linearized collision operator of the Boltzmann equation can in a natural way be written as a sum of a positive multiplication operator, the collision frequency, and an integral operator. Compactness of the integral operator for monatomic single species is a classical result, while corresponding result for mixtures is more recently obtained. In...

The linearized collision operator of the Boltzmann equation can in a natural way be written as a sum of a positive multiplication operator, the collision frequency, and an integral operator. Compactness of the integral operator for monatomic single species is a classical result, while corresponding result for mixtures is more recently obtained. In...

A Correction to this paper has been published: https://doi.org/10.1007/s00205-021-01608-9

Consider the steady Boltzmann equation with slab symmetry for a monatomic, hard sphere gas in a half space. At the boundary of the half space, it is assumed that the gas is in contact with its condensed phase. The present paper discusses the existence and uniqueness of a uniformly decaying boundary layer type solution of the Boltzmann equation in t...

In this work, we consider a Boltzmann equation for anyons. In particular, we study a general discrete velocity model of the equation, where the velocity variable is assumed to only take values from a given finite - such that the (finite) number of velocities is arbitrary - set of velocities. Included, as two limiting cases, is the discrete quantum...

An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. Unlike for the Boltzmann equation, for DVMs there can appear extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invar...

An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that, in difference to in the continuous case, DVMs can have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only...

An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that DVMs can also have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and so...

We consider the existence of nonlinear boundary layers and the typically nonlinear problem of existence of shock profiles for the Broadwell model, which is a simplified discrete velocity model for the Boltzmann equation. We find explicit expressions for the nonlinear boundary layers and the shock profiles. In spite of the few velocities used for th...

We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for a general discrete model of a quantum kinetic equation for excitations in a Bose gas. In the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations. These systems are studied clos...

Half-space problems for the Boltzmann equation are of great importance in the study of the asymptotic behavior of the solutions of boundary value problems of the Boltzmann equation for small Knudsen numbers. Half-space problems provide the boundary conditions for the fluid-dynamic-type equations and Knudsen-layer corrections to the solution of the...

We consider a non-linear half-space problem related to the condensation problem for the discrete Boltzmann equation and extend some known results for a single-component gas to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian at infinity, as the non-condensable gas tends to zero at infinity. We a...

We consider the discrete Boltzmann equation for binary gas mixtures. Some known results for half-space problems and shock profile solutions of the discrete Boltzmann for single-component gases are extended to the case of two-component gases. These results include well-posedness results for half-space problems for the linearized discrete Boltzmann e...

Existence of solutions of weakly non-linear half-space problems for the general discrete velocity (with arbitrarily finite number of velocities) model of the Boltzmann equation are studied. The solutions are assumed to tend to an assigned Maxwellian at infinity, and the data for the outgoing particles at the boundary are assigned, possibly linearly...

| We study typical half-space problems of rareed gas dynamics, including the problems of Milne and Kramer, for the discrete Boltzmann equation (a general discrete velocity model, DVM, with an arbitrary nite number of velocities). Then the discrete Boltzmann equation reduces to a system of ODEs. The data for the outgoing particles at the boundary ar...

In this paper we study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for the discrete Boltzmann equation. The discrete Boltzmann equation reduces to a system of ODEs for plane stationary problems. These systems are studied, and for general boundary conditions at the "wall" a classification of well...

The analytically difficult problem of existence of shock wave solutions is studied for the general discrete velocity model (DVM) with an arbitrary finite number of velocities (the discrete Boltzmann equation in terminology of H. Cabannes). For the shock wave problem the discrete Boltzmann equation becomes a system of ordinary differential equations...

We study the shock wave problem for the general discrete velocity model (DVM), with an arbitrary finite number of velocities. In this case the discrete Boltzmann equation becomes a system of ordinary differential equations (dynamical system). Then the shock waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians). In th...

We study some questions related to general discrete velocity (with arbitrarily number of velocities) models (DVMs) of the Boltzmann equation. In the case of plane stationary problems the typical DVM reduces to a dynamical system (system of ODEs). Properties of such systems are studied in the most general case. In particular, a topological classific...

We study some questions related to discrete velocity models (DVMs) of the Boltzmann equation. In the case of plane stationary problems the typical DVM reduces to a dynamical system (system of ODEs). Properties of such systems are studied in this paper in the most general case. In particular, a topological classification of their singular points is...

We study some questions related to general discrete velocity (with arbitrarily number of velocities) models (DVMs) of the Boltzmann equation. In the case of plane stationary problems the typical DVM reduces to a dynamical system (system of ODEs). Properties of such systems are studied in the most general case. In particular, a topological classific...