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• 1. Acedo, L.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow2002Inngår i: J. Statist. Phys, Vol. 109:5-6Artikkel i tidsskrift (Fagfellevurdert)
• 2.
Univ Autonoma Madrid, ICMAT, E-28049 Madrid, Spain.;Univ Autonoma Madrid, Dept Matemat, E-28049 Madrid, Spain..
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik. Osaka Kyoiku Univ, Dept Math, Kashiwara, Osaka 5828582, Japan.. Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
Advances in Operator Cauchy-Schwarz Inequalities and their Reverses2015Inngår i: Annals of Functional Analysis, ISSN 2008-8752, E-ISSN 2008-8752, Vol. 6, nr 3, s. 275-295Artikkel i tidsskrift (Fagfellevurdert)

The Cauchy-Schwarz (C-S) inequality is one of the most famous inequalities in mathematics. In this survey article, we first give a brief history of the inequality. Afterward, we present the C-S inequality for inner product spaces. Focusing on operator inequalities, we then review some significant recent developments of the C-S inequality and its reverses for Hilbert space operators and elements of Hilbert C*-modules. In particular, we pay special attention to an operator Wielandt inequality.

• 3.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Embedding Theorems for Mixed Norm Spaces and Applications2010Doktoravhandling, monografi (Annet vitenskapelig)

This thesis is devoted to the study of mixed norm spaces that arise in connection with embeddings of Sobolev and Besov type spaces. We study different structural, integrability, and smoothness properties of functions satisfying certain mixed norm conditions. Conditions of this type are determined by the behaviour of linear sections of functions. The work in this direction originates in a paper due to Gagliardo (1958), and was further developed by Fournier (1988), by Blei and Fournier (1989), and by Kolyada (2005).

Here we continue these studies. We obtain some refinements of known embeddings for certain mixed norm spaces introduced by Gagliardo, and we study general properties of these spaces. In connection with these results, we consider a scale of intermediate mixed norm spaces, and prove intrinsic embeddings in this scale.

We also consider more general, fully anisotropic, mixed norm spaces. Our main theorem states an embedding of these spaces to Lorentz spaces. Applying this result, we obtain sharp embedding theorems for anisotropic Sobolev-Besov spaces, and anisotropic fractional Sobolev spaces. The methods used are based on non-increasing rearrangements, and on estimates of sections of functions and sections of sets. We also study limiting relations between embeddings of spaces of different type. More exactly, mixed norm estimates enable us to get embedding constants with sharp asymptotic behaviour. This gives an extension of the results obtained for isotropic Besov spaces by Bourgain, Brezis, and Mironescu, and for anisotropic Besov spaces by Kolyada.

We study also some basic properties (in particular the approximation properties) of special weak type spaces that play an important role in the construction of mixed norm spaces, and in the description of Sobolev type embeddings.

In the last chapter, we study mixed norm spaces consisting of functions that have smooth sections. We prove embeddings of these spaces to Lorentz spaces. From this result, known properties of Sobolev-Liouville spaces follow.

• 4.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Embedding Theorems for Mixed Norm Spaces and Applications2008Licentiatavhandling, monografi (Annet vitenskapelig)

This thesis is devoted to the study of mixed norm spaces that arise in connection with embeddings of Sobolev and Besov type spaces. The work in this direction originates in a paper due to Gagliardo (1958), and was continued by Fournier (1988) and by Kolyada (2005).

We consider fully anisotropic mixed norm spaces. Our main theorem states an embedding of these spaces into Lorentz spaces. Applying this result, we obtain sharp embedding theorems for anisotropic fractional Sobolev spaces and anisotropic Sobolev-Besov spaces. The methods used are based on non-increasing rearrangements and on estimates of sections of functions and sections of sets. We also study limiting relations between embeddings of spaces of different type. More exactly, mixed norm estimates enable us to get embedding constants with sharp asymptotic behaviour. This gives an extension of the results obtained for isotropic Besov spaces $B_p^\alpha$ by Bourgain, Brezis, and Mironescu, and for Besov spaces $B^{\alpha_1,\dots,\alpha_n}_p$ by Kolyada.

We study also some basic properties (in particular the approximation properties) of special weak type spaces that play an important role in the construction of mixed norm spaces and in the description of Sobolev type embeddings.

• 5.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
On Fournier-Gagliardo mixed norm spaces2011Inngår i: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, Vol. 36, s. 493-508Artikkel i tidsskrift (Fagfellevurdert)

We study mixed norm spaces

V (Rn)

that arise in connection with embeddings of

Sobolev spaces

W

1

1

(Rn). We prove embeddings of V (Rn)

into Lorentz type spaces defined in terms

of iterative rearrangements. Basing on these results, we introduce the scale of mixed norm spaces

V

p

(Rn). We prove that V ½ V p

and we discuss some questions related to this embedding.

• 6. Baba, S.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Genus 2 Curves with Quaternionic Multiplication2008Inngår i: Canadian Journal of Mathematics, Vol. 60 (4)Artikkel i tidsskrift (Fagfellevurdert)

We explicitly construct the canonical rational models of Shimuracurves, both analytically in terms of modular forms andalgebraically in terms of coefficients of genus 2 curves, in thecases of quaternion algebras of discriminant 6 and 10. Thisemulates the classical construction in the elliptic curvecase. We also give families of genus 2 QM curves, whose Jacobiansare the corresponding abelian surfaces on the Shimura curve, andwith coefficients that are modular forms of weight 12. We applythese results to show that our j-functions are supported exactlyat those primes where the genus 2 curve does not admitpotentially good reduction, and construct fields where thispotentially good reduction is attained. Finally, using j, weconstruct the fields of moduli and definition for some moduliproblems associated to the Atkin-Lehner group actions

• 7. Baba, Srinath
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Primes of superspecial reduction for QM abelian surfaces2008Inngår i: Bulletin of the London Mathematical Society, Vol. 40Artikkel i tidsskrift (Fagfellevurdert)
• 8.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
On the structure of some irrotational vector fields2005Inngår i: Gen. Math. 13Artikkel i tidsskrift (Fagfellevurdert)
• 9.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Blaschke produkt generated covering surfaces2009Inngår i: Math.Bohem., ISSN 0862-7959, Vol. 134, nr 2, s. 173-182Artikkel i tidsskrift (Fagfellevurdert)
• 10.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Blaschke self-mappings of the real projective plane2007Konferansepaper (Fagfellevurdert)
• 11.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Measure and integration on nonorientable Klein surfaces2002Inngår i: Int. J.Pure Appl.Math 1Artikkel i tidsskrift (Fagfellevurdert)
• 12.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Nonorientable Compactifications of Riemann SurfacesI2003Inngår i: n the volume "Progress in Analysis", Worlds ScientificArtikkel i tidsskrift (Fagfellevurdert)
• 13.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
The geometry of Blaschke products mappings2009Konferansepaper (Fagfellevurdert)
• 14.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Vector Fields on Nonorientable Surfaces2003Inngår i: International Journal of Mathematics and Mathematical SciencesArtikkel i tidsskrift (Fagfellevurdert)
• 15.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Dynamics of Dianalytic Transformations of Klein Surfaces2004Inngår i: Mathematica BohemicaArtikkel i tidsskrift (Fagfellevurdert)
• 16.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
The Normal Derivative to the Border of the Möbius Strip2002Inngår i: Revue Roumaine de Mathématiques Pures et AppliqueesArtikkel i tidsskrift (Fagfellevurdert)
• 17.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Cousin-I spaces and domains of holomorphy2009Inngår i: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 96, nr 1, s. 51-60Artikkel i tidsskrift (Fagfellevurdert)
• 18.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Multidimensional integral inequalities with homogeneous weights2005Inngår i: Gen. Math. Vol. 13, No. 1 (2005)Artikkel i tidsskrift (Fagfellevurdert)
• 19.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Rumänien. Kroatien.
New characterizations of asymptotic stability for evolution families on Banach spaces2004Inngår i: Electronic Journal of Differential Equations, ISSN 1550-6150, E-ISSN 1072-6691, s. 1-9, artikkel-id 38Artikkel i tidsskrift (Fagfellevurdert)
• 20.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Duality theorems over the cone of monotone functions and sequences in higher dimensions2002Inngår i: J. of Inequal. & and Appl., 2002, Vol.7 (1), 79-108Artikkel i tidsskrift (Fagfellevurdert)
• 21.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
A Sawyer duality principle for radialy monotone functions in R^n2005Inngår i: JIPAM 6, (2005), electronic, 1-13Artikkel i tidsskrift (Fagfellevurdert)
• 22.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Mixed norm and multidimensional Lorentz spaces2006Inngår i: POsitivity, 10 (2006), no. 3Artikkel i tidsskrift (Fagfellevurdert)
• 23.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik. Department of Applied Mathematics and Analysis, University of Barcelona, Spain.
Sharp constants related to the triangle inequality in Lorentz spaces2009Inngår i: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 361, nr 10, s. 5555-5574Artikkel i tidsskrift (Fagfellevurdert)
• 24.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Matriceal Lebesgue spaces and Hölder inequality2005Inngår i: JFSA, 3(3)(2005)Artikkel i tidsskrift (Fagfellevurdert)
• 25.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Approximation of infinite marices by matricial Haar polynomials2005Inngår i: Ark.Mat.,43(2005), 251-269Artikkel i tidsskrift (Fagfellevurdert)
• 26.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
A new variational characterization of Sobolev spaces2015Inngår i: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 25, nr 4, s. 2185-2195Artikkel i tidsskrift (Fagfellevurdert)

We obtain a new variational characterization of the Sobolev space $W_p^1(\Omega)$ (where $\Omega\subseteq\R^n$ and $p>n$). This is a generalization of a classical result of F. Riesz. We also consider some related results.

• 27.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, Romania. Department of Mathematics, Luleå University of Technology.
Best constants between equivalent norms in Lorentz sequence spaces2012Inngår i: Journal of Function Spaces and Applications, ISSN 0972-6802, E-ISSN 1758-4965Artikkel i tidsskrift (Fagfellevurdert)
• 28.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik. Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
Technical University of Civil Engineering Bucharest, Department of Mathematics and Computer Science, Bucharest, Romania. Technical University of Civil Engineering Bucharest, Department of Mathematics and Computer Science, Bucharest, Romania. Luleå University of Technology, Department of Mathematics.
Optimal estimates in Lorentz spaces of sequences with an increasing weight2013Inngår i: Proceedings of the Romanian Academy. Series A Mathematics, Physics, Technical Sciences, Information Science, ISSN 1454-9069, Vol. 14, nr 1, s. 20-27Artikkel i tidsskrift (Fagfellevurdert)
• 29.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Integral inequalities for concave functions2006Inngår i: Publ. Math. Debrecen, 68(2006), no. 1-2Artikkel i tidsskrift (Fagfellevurdert)
• 30.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Strong and weak weighted norm inequalities for the geometric fractional maximal operator2012Inngår i: Bulletin of the Australian Mathematical Society, ISSN 0004-9727, E-ISSN 1755-1633, Vol. 86, nr 2, s. 205-215Artikkel i tidsskrift (Fagfellevurdert)
• 31.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Department of Mathematics, Luleå University of Technology. epartment of Mathematics, "Lucian Blaga" University of Sibiu, Romania .
Some multiplicative inequalities for inner products and of the Carlson type2008Inngår i: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242XArtikkel i tidsskrift (Fagfellevurdert)

We prove a multiplicative inequality for inner products, which enables us to deduce improvements of inequalities of the Carlson type for complex functions and sequences, and also other known inequalities.

• 32.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
On weighted multidimensional integral inequalities for mixed monotone functions2000Inngår i: Bull. Math. Soc. Sci. Math. Roumanie (N.S), ISSN 1220-3874, Vol. 43, nr 91, s. 39-45Artikkel i tidsskrift (Fagfellevurdert)
• 33.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Sharp weighted multidimensional integral inequalities for monotone functions2000Inngår i: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 210, nr 1, s. 43-58Artikkel i tidsskrift (Fagfellevurdert)
• 34.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Equivalence of Modular Inequalities of Hardy type on non-negative respective non-increasing functions2008Konferansepaper (Fagfellevurdert)
• 35.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
A matriceal analogue of Fejer's theory2003Inngår i: Math NachrArtikkel i tidsskrift (Fagfellevurdert)
• 36.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Weighted multiplicative integral inequalities2006Inngår i: Journal of inequal pure, ISSN 1443-5756, Vol. 7, nr 5, s. 169-175Artikkel i tidsskrift (Fagfellevurdert)
• 37.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Department of Applied Mathematics and Analysis, University of Barcelona, , Spain .
Sharp constants between equivalent norms in weighted Lorentz spaces2010Inngår i: Journal of the Australian Mathematical Society, ISSN 1446-7887, E-ISSN 1446-8107, Vol. 88, nr 1, s. 19-27Artikkel i tidsskrift (Fagfellevurdert)

For an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

• 38.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
On weighted multidimensional embeddings for monotone functions2001Inngår i: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 88, nr 2, s. 303-319Artikkel i tidsskrift (Fagfellevurdert)
• 39.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Boundary Layers and Shock Profiles for the Discrete Boltzmann Equation for Mixtures2012Inngår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 5, nr 1, s. 1-19Artikkel i tidsskrift (Fagfellevurdert)

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 equation, existence results for half-space problems for the weakly non-linear discrete Boltzmann equation, and existence results for shock profile solutions of the discrete Boltzmann equation. A characteristic number, corresponding to the speed of sound in the continuous case, is calculated for a class of symmetric models. Some explicit calculations are also made for a simplified 6+4 -velocity model.

• 40.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Boundary layers for the nonlinear discrete Boltzmann equation: Condensing vapor flow in the presence of a non-condensable gas2012Inngår i: Proceedings of 28th International Symposium on Rarefied Gas Dynamics 2012 / [ed] Michel Mareschal, Andrés Santos, Melville, New York: American Institute of Physics (AIP), 2012, 1, s. 223-230Konferansepaper (Fagfellevurdert)

Half-space problems for the Boltzmann equation are of great importance in the study of the asymptotic behaviorof 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 fluid-dynamic-type equations in a neighborhood of the boundary. Here we consider a half-space problem of condensation for apure vapor in the presence of a non-condensable gas by using discrete velocity models (DVMs) of the Boltzmann equation. The Boltzmann equation can be approximated by DVMs up to any order, and these DVMs can be applied for numerical methods,but also for mathematical studies to bring deeper understanding and new ideas. For one-dimensional half-space problems,the discrete Boltzmann equation (the general DVM) reduces to a system of ODEs. We obtain that the number of parametersto be specified in the boundary conditions depends on whether the condensing vapor flow is subsonic or supersonic. Thisbehavior has earlier been found numerically. We want to stress that our results are valid for any finite number of velocities.This is an extension of known results for single-component gases (and for binary mixtures of two vapors) to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian, with a flow velocity towards thecondensed phase, at infinity, while the non-condensable gas tends to zero at infinity. Steady condensation of the vapor takes place at the condensed phase, which is held at a constant temperature. We assume that the vapor is completely absorbed, that the non-condensable gas is diffusively reflected at the condensed phase, and that vapor molecules leaving the condensed phase are distributed according to a given distribution. The conditions, on the given distribution at the condensed phase, needed for the existence of a unique solution of the problem are investigated, assuming that the given distribution at the condensed phase is sufficiently close to the Maxwellian at infinity and that the total mass of the non-condensable gas is sufficiently small. Exact solutions and solvability conditions are found for a specific simplified discrete velocity model (with few velocities).

• 41.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Discrete Velocity Models and Half-Space Problems2003Licentiatavhandling, monografi (Annet vitenskapelig)

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 made and dimensions of the corresponding stable, unstable and center manifolds are computed. These results are applied to typical half-space problems of rarefied gas dynamics. A classification of well-posed half-space problems for linearized DVMs is made. Exact solutions of a (simplified) linearized kinetic model of BGK type are found as limiting case of corresponding discrete models. The main results of the paper can be also used for moment approximations and other versions of discretizised kinetic equations

• 42.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Half-Space Problem for the Discrete Boltzmann Equation: Condensing Vapor Flow in the Presence of a Non-condensable Gas2012Inngår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 147, nr 6, s. 1156-1181Artikkel i tidsskrift (Fagfellevurdert)

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 assume that the vapor is completely absorbed and that the non-condensable gas is diffusively reflected at the condensed phase and that the vapor molecules leaving the condensed phase are distributed according to a given distribution. The conditions, on the given distribution, needed for the existence of a unique solution of the problem are investigated. We also find exact solvability conditions and solutions for a simplified six+four-velocity model, as the given distribution is a Maxwellian at rest, and study a simplified twelve+six-velocitymodel.

• 43.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
On Half-Space and Shock-Wave Problems for Discrete Velocity Models of the Boltzmann Equation2005Doktoravhandling, med artikler (Annet vitenskapelig)

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 classification of their singular points is made and dimensions of the corresponding stable, unstable and center manifolds are computed.

These results are applied to typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer. A classification of well-posed half-space problems for linearized DVMs is made. Exact solutions of a (simplified) linearized kinetic model of BGK type are found as a limiting case of the corresponding discrete models.

Existence of solutions of weakly non-linear half-space problems for general DVMs 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 depending on the data for the incoming particles. The conditions, on the data at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. Both implicit, in the non-degenerate cases, and sometimes, in both degenerate and non-degenerate cases, explicit conditions are found.

Shock-waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians) for DVMs. We give a constructive proof for the existence of solutions of the shock-wave problem for the general DVM. This is worked out for shock speeds close to a typical speed, corresponding to the sound speed in the continuous case. We clarify how close the shock speed must be for our theorem to hold, and present an iteration scheme for obtaining the solution.

The main results of the paper can be used for DVMs for mixtures as well as for DVMs for one species.

• 44.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
On half-space problems for the discrete Boltzmann equation2010Inngår i: Il Nuovo Cimento C, ISSN 2037-4909, 1826-9885, Vol. 33, nr 1, s. 47-54Artikkel i tidsskrift (Fagfellevurdert)

We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for the discrete Boltzmann equation (a general discrete velocity model, DVM, with an arbitrary finite number of velocities). Then the discrete Boltzmann equation reduces to a system of ODEs. The data for the outgoing particles at the boundary are assigned, possibly linearly depending on the data for the incoming particles. A classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete Boltzmann equation is made. In the non-linear case the solutions are assumed to tend to an assigned Maxwellian at infinity. The conditions on the data at the boundary needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. In the non-degenerate case (corresponding, in the continuous case, to the case when the Mach number at the Maxwellian at infinity is different of -1, 0 and 1) implicit conditions are found. Furthermore, under certain assumptions explicit conditions are found, both in the non-degenerate and degenerate cases. An application to axially symmetric models is also studied

• 45.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
On half-space problems for the linearized discrete Boltzmann equation2008Inngår i: Rivista di Matematica della Universita' di Parma, Vol. (7)9, s. 73-124Artikkel i tidsskrift (Fagfellevurdert)

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-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete Boltzmann equation is made. Applications for axially symmetric models are studied in more detail. Exact solutions of a (simplified) linearized kinetic model of BGK type are also found as a limiting case of the corresponding discrete models.

• 46.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
On half-space problems for the weakly non-linear discrete Boltzmann equation2010Inngår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 3, nr 2, s. 195-222Artikkel i tidsskrift (Fagfellevurdert)

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 depending on the data for the incoming particles. The conditions, on the data at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. In the non-degenerate case (corresponding, in the continuous case, to the case when the Mach number at infinity is different of -1, 0 and 1) implicit conditions are found. Furthermore, under certain assumptions explicit conditions are found, both in the non-degenerate and degenerate cases. Applications to axially symmetric models are studied in more detail

• 47.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Weak Shock Wave Solutions for the Discrete Boltzmann Equation2007Inngår i: Rarefied Gas Dynamics: 25th International Symposium on Rarefied Gas Dynamics, Saint-Petersburg, Russia, July 21-28, 2006 (M.S. Ivanov and A.K. Rebrov, eds), Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences , 2007, s. 173-178Konferansepaper (Fagfellevurdert)

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 (dynamical system). Then the shock waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians). In this work we give a constructive proof for the existence of solutions in the case of weak shocks. We assume that a given Maxwellian is approached at infinity, and consider shock speeds close to a typical speed , corresponding to the sound speed in the continuous case. The existence of a non-negative locally unique (up to a shift in the independent variable) bounded solution is proved by using contraction mapping arguments (after a suitable decomposition of the system). This solution is then shown to tend to a Maxwellian at minus infinity. Existence of weak shock wave solutions for DVMs was proved by Bose, Illner and Ukai in 1998 [1]. In their technical proof Bose et al. are following the lines of the pioneering work for the continuous Boltzmann equation by Caflisch and Nicolaenko [2]. In this work, we follow a more straightforward way, suiting the discrete case. Our approach is based on results by the authors on the main characteristics (dimensions of corresponding stable, unstable and center manifolds) for singular points [3] to general dynamical systems of the same type as in the shock wave problem for DVMs. Our proof is constructive, and it is also shown (at least implicitly) how close to the typical speed , the shock speed must be for our results to be valid. All results are mathematically rigorous. Our results are also applicable for DVMs for mixtures. ACKNOWLEDGEMENTS. The support by the Swedish Research Council grant 20035357 are gratefully acknowledged by both of the authors.REFERENCES[1] C. Bose, R. Illner, S. Ukai, Transp. Th. Stat. Phys., 27, 35-66 (1998) [2] R.E. Caflisch, B. Nicolaenko, Comm. Math. Phys., 86, 161-194 (1982)[3] A.V. Bobylev, N. Bernhoff, Lecture Notes on the Discretization of the Boltzmann Equation, World Scientific, 2003, pp. 203-222

• 48.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Weak shock waves for the general discrete velocity model of the Boltzmann equation2007Inngår i: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 5, nr 4, s. 815-832Artikkel i tidsskrift (Fagfellevurdert)

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 this paper we give a constructive proof for the existence of solutions in the case of weak shocks. We assume that a given Maxwellian is approached at infinity, and consider shock speeds close to a typical speed c, corresponding to the sound speed in the continuous case. The existence of a non-negative locally unique (up to a shift in the independent variable) bounded solution is proved by using contraction mapping arguments (after a suitable decomposition of the system). This solution is shown to tend to a Maxwellian at minus infinity. Existence of weak shock wave solutions for DVMs was proved by Bose, Illner and Ukai in 1998. In this paper, we give a constructive proof following a more straightforward way, suiting the discrete case. Our approach is based on earlier results by the authors on the main characteristics (dimensions of corresponding stable, unstable and center manifolds) for singular points to general dynamical systems of the same type as in the shock wave problem for DVMs. The same approach can also be applied for DVMs for mixtures

• 49.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Boltzmann Equation and Hydrodynamics Beyond the Navier-Stokes Level (Harold Grad Lecture)2007Konferansepaper (Fagfellevurdert)
• 50.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
Generalized Burnett Hydrodynamics2008Inngår i: J. Statist. Phys, Vol. 132:3Artikkel i tidsskrift (Fagfellevurdert)

Equations of hydrodynamics (derived from the Boltzmann equation) beyond the Navier-Stokes level are studied by a method proposed earlier by the author. The main question we consider is the following: What is the most natural replacement for classical (ill-posed) Burnett equations?

It is shown that, in some sense, it is a two-parameter set of Generalized Burnett Equations (GBEs) derived in this paper. Some equations of this class are even simpler than original Burnett equations. The region of stability in the space of parameters and other properties of GBEs are discussed.

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