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• 1. Acedo, L.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow2002In: J. Statist. Phys, Vol. 109:5-6Article in journal (Refereed)
• 2.
Karlstad University, Faculty of Technology and Science, Department of Mathematics. Osaka Kyoiku Univ, Dept Math, Kashiwara, Osaka 5828582, Japan.. Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
Advances in Operator Cauchy-Schwarz Inequalities and their Reverses2015In: Annals of Functional Analysis, ISSN 2008-8752, E-ISSN 2008-8752, Vol. 6, no 3, p. 275-295Article in journal (Refereed)

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.
Karlstad University, Faculty of Technology and Science, Avdelningen för matematik.
Embedding Theorems for Mixed Norm Spaces and Applications2010Doctoral thesis, monograph (Other academic)

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.
Karlstad University, Faculty of Technology and Science, Avdelningen för matematik.
Embedding Theorems for Mixed Norm Spaces and Applications2008Licentiate thesis, monograph (Other scientific)

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.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
On Fournier-Gagliardo mixed norm spaces2011In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, Vol. 36, p. 493-508Article in journal (Refereed)

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.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Genus 2 Curves with Quaternionic Multiplication2008In: Canadian Journal of Mathematics, Vol. 60 (4)Article in journal (Refereed)

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
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Primes of superspecial reduction for QM abelian surfaces2008In: Bulletin of the London Mathematical Society, Vol. 40Article in journal (Refereed)
• 8.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
On the structure of some irrotational vector fields2005In: Gen. Math. 13Article in journal (Refereed)
• 9.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Blaschke produkt generated covering surfaces2009In: Math.Bohem., ISSN 0862-7959, Vol. 134, no 2, p. 173-182Article in journal (Refereed)
• 10.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Blaschke self-mappings of the real projective plane2007Conference paper (Refereed)
• 11.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Measure and integration on nonorientable Klein surfaces2002In: Int. J.Pure Appl.Math 1Article in journal (Refereed)
• 12.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Nonorientable Compactifications of Riemann SurfacesI2003In: n the volume "Progress in Analysis", Worlds ScientificArticle in journal (Refereed)
• 13.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
The geometry of Blaschke products mappings2009Conference paper (Refereed)
• 14.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Vector Fields on Nonorientable Surfaces2003In: International Journal of Mathematics and Mathematical SciencesArticle in journal (Refereed)
• 15.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Dynamics of Dianalytic Transformations of Klein Surfaces2004In: Mathematica BohemicaArticle in journal (Refereed)
• 16.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
The Normal Derivative to the Border of the Möbius Strip2002In: Revue Roumaine de Mathématiques Pures et AppliqueesArticle in journal (Refereed)
• 17.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Cousin-I spaces and domains of holomorphy2009In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 96, no 1, p. 51-60Article in journal (Refereed)
• 18.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Multidimensional integral inequalities with homogeneous weights2005In: Gen. Math. Vol. 13, No. 1 (2005)Article in journal (Refereed)
• 19.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Rumänien. Kroatien.
New characterizations of asymptotic stability for evolution families on Banach spaces2004In: Electronic Journal of Differential Equations, ISSN 1550-6150, E-ISSN 1072-6691, p. 1-9, article id 38Article in journal (Refereed)
• 20.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Duality theorems over the cone of monotone functions and sequences in higher dimensions2002In: J. of Inequal. & and Appl., 2002, Vol.7 (1), 79-108Article in journal (Refereed)
• 21.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
A Sawyer duality principle for radialy monotone functions in R^n2005In: JIPAM 6, (2005), electronic, 1-13Article in journal (Refereed)
• 22.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Mixed norm and multidimensional Lorentz spaces2006In: POsitivity, 10 (2006), no. 3Article in journal (Refereed)
• 23.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Karlstad University, Faculty of Technology and Science, Department of Mathematics. Department of Applied Mathematics and Analysis, University of Barcelona, Spain.
Sharp constants related to the triangle inequality in Lorentz spaces2009In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 361, no 10, p. 5555-5574Article in journal (Refereed)
• 24.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Matriceal Lebesgue spaces and Hölder inequality2005In: JFSA, 3(3)(2005)Article in journal (Refereed)
• 25.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Approximation of infinite marices by matricial Haar polynomials2005In: Ark.Mat.,43(2005), 251-269Article in journal (Refereed)
• 26.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
A new variational characterization of Sobolev spaces2015In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 25, no 4, p. 2185-2195Article in journal (Refereed)

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.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
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 spaces2012In: Journal of Function Spaces and Applications, ISSN 0972-6802, E-ISSN 1758-4965Article in journal (Refereed)
• 28.
Karlstad University, Faculty of Technology and Science, Department of Mathematics. Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
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 weight2013In: Proceedings of the Romanian Academy. Series A Mathematics, Physics, Technical Sciences, Information Science, ISSN 1454-9069, Vol. 14, no 1, p. 20-27Article in journal (Refereed)
• 29.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Integral inequalities for concave functions2006In: Publ. Math. Debrecen, 68(2006), no. 1-2Article in journal (Refereed)
• 30.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Strong and weak weighted norm inequalities for the geometric fractional maximal operator2012In: Bulletin of the Australian Mathematical Society, ISSN 0004-9727, E-ISSN 1755-1633, Vol. 86, no 2, p. 205-215Article in journal (Refereed)
• 31.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
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 type2008In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242XArticle in journal (Refereed)

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.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
On weighted multidimensional integral inequalities for mixed monotone functions2000In: Bull. Math. Soc. Sci. Math. Roumanie (N.S), ISSN 1220-3874, Vol. 43, no 91, p. 39-45Article in journal (Refereed)
• 33.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Sharp weighted multidimensional integral inequalities for monotone functions2000In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 210, no 1, p. 43-58Article in journal (Refereed)
• 34.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Equivalence of Modular Inequalities of Hardy type on non-negative respective non-increasing functions2008Conference paper (Refereed)
• 35.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
A matriceal analogue of Fejer's theory2003In: Math NachrArticle in journal (Refereed)
• 36.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Weighted multiplicative integral inequalities2006In: Journal of inequal pure, ISSN 1443-5756, Vol. 7, no 5, p. 169-175Article in journal (Refereed)
• 37.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Department of Applied Mathematics and Analysis, University of Barcelona, , Spain .
Sharp constants between equivalent norms in weighted Lorentz spaces2010In: Journal of the Australian Mathematical Society, ISSN 1446-7887, E-ISSN 1446-8107, Vol. 88, no 1, p. 19-27Article in journal (Refereed)

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.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
On weighted multidimensional embeddings for monotone functions2001In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 88, no 2, p. 303-319Article in journal (Refereed)
• 39.
Umeå universitet.
Karlstad University, Faculty of Technology and Science, Department of Mathematics. Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science. Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science. Umeå universitet.
Evaluation of a large scale professional development program2017In: Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education / [ed] Kaur, B., Ho, W.K., Toh, T.L., & Choy, B.H., Singapore: PME , 2017, p. 153-160Conference paper (Refereed)

This paper reports on an evaluation of the professional development program Boost for Mathematics in Sweden. 200 mathematics lessons were visited, and the teachers were interviewed after each lesson.The analysis used an analytical framework based on Lithner et al. (2010).The findings indicate that the PD-program has had a significant impact on the teachers’ knowledge about the mathematical competencies as they are presented in the national curriculum documents, and that the teaching practice has improved and give the students better possibilities todevelop the competencies.

• 40.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Boundary Layers and Shock Profiles for the Discrete Boltzmann Equation for Mixtures2012In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 5, no 1, p. 1-19Article in journal (Refereed)

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.

• 41.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Boundary layers for the nonlinear discrete Boltzmann equation: Condensing vapor flow in the presence of a non-condensable gas2012In: 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, p. 223-230Conference paper (Refereed)

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).

• 42.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Discrete Velocity Models and Half-Space Problems2003Licentiate thesis, monograph (Other academic)

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

• 43.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Half-Space and Shock-Wave Problems for Discrete Velocity Models of the Boltzmann Equation2005Doctoral thesis, monograph (Other academic)

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.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Half-Space Problem for the Discrete Boltzmann Equation: Condensing Vapor Flow in the Presence of a Non-condensable Gas2012In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 147, no 6, p. 1156-1181Article in journal (Refereed)

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.

• 45.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
On half-space problems for the discrete Boltzmann equation2010In: Il Nuovo Cimento C, ISSN 2037-4909, 1826-9885, Vol. 33, no 1, p. 47-54Article in journal (Refereed)

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

• 46.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
On half-space problems for the linearized discrete Boltzmann equation2008In: Rivista di Matematica della Universita' di Parma, Vol. (7)9, p. 73-124Article in journal (Refereed)

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.

• 47.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
On half-space problems for the weakly non-linear discrete Boltzmann equation2010In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 3, no 2, p. 195-222Article in journal (Refereed)

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

• 48.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Weak Shock Wave Solutions for the Discrete Boltzmann Equation2007In: 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, p. 173-178Conference paper (Refereed)

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

• 49.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Weak shock waves for the general discrete velocity model of the Boltzmann equation2007In: Communications in Mathematical Sciences, ISSN 1539-6746, Vol. 5, no 4, p. 815-832Article in journal (Refereed)

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

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Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Boltzmann Equation and Hydrodynamics Beyond the Navier-Stokes Level (Harold Grad Lecture)2007Conference paper (Refereed)
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