# zbMATH — the first resource for mathematics

Abstract time-dependent transport equations. (English) Zbl 0657.45007
The following type of time dependent linear initial-boundary value problems is investigated: $\partial u/\partial t+\xi \cdot \partial u/\partial x+a\cdot \partial u/\partial \xi +hu+Au=f\quad for\quad x\in \Omega,\quad \xi \in S,\quad 0<t<T;$ u(x,$$\xi$$,0)$$=u_ 0(x,\xi)$$ for $$x\in \Omega$$, $$\xi\in S$$; $$u_-(x,\xi,t)=Ku_+(x,\xi,t)+g(x,\xi,t)$$ for $$(x,\xi)\in B_-$$, $$0<t<T$$. Here the solution u describing a particle density is to be found, on $$\Omega \times S\times (0,T)\subset {\mathbb{R}}^ n\times {\mathbb{R}}^ n\times {\mathbb{R}}$$. The internal sources f and the incident current g are known; $$u_{\pm}$$ denote the traces of u on the “incoming” and “outgoing” parts $$B_{\pm}$$ of the boundary, and K is an operator describing the boundary process; the vector field a, the function h and the operator A are also given. There are certain suitable conditions on the data, and in fact the main results are obtained for a more general form of the problem.
The problem is treated in $$L_ p$$-setting, $$1\leq p<\infty$$, and in the space of measures. It is shown that the problem is well-posed under suitable assumptions, among which $$\| K\| <1$$. One of the technical tools is a general Gauss-Green-identity. In the more delicate case of conservative boundary conditions $$(\| K\| =1)$$ the notion of solution has to be generalized. In the case that the data do not depend on time, the authors show how their results apply to the theory of $$C_ 0$$-semigroups.
Reviewer: J.Voigt

##### MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 45K05 Integro-partial differential equations 47D03 Groups and semigroups of linear operators 82C70 Transport processes in time-dependent statistical mechanics
Full Text:
##### References:
  Arsen’ev, A.A, The Cauchy problem for the linearized Boltzmann equation, U.S.S.R. comput. math. and math. phys., 5, 110-136, (1965) · Zbl 0157.57601  Asano, K, Local solutions to the initial and initial boundary value problem for the Boltzmann equation with an external force I, J. math. Kyoto univ., 24, 225-238, (1984) · Zbl 0571.76075  Bartolomäus, G; Wilhelm, J, Existence and uniqueness of the solution of the nonstationary Boltzmann equation for the electrons in a collision dominated plasma by means of operator semigroups, Ann. phys., 38, 211-220, (1981)  Bärwinkel, K; Schmidt, H.J, Fixed point properties of gas surface scattering operators, (), 150-166  Beals, R, Indefinite Sturm-Liouville problems and half-range completeness, J. differential equations, 56, 381-407, (1985) · Zbl 0512.34017  Belleni-Morante, A; Farano, R, Neutron transport in a slab with moving boundaries, SIAM J. appl. math., 31, 591-599, (1976) · Zbl 0336.35085  Cercignani, C, Existence, uniqueness, and convergence of the solutions of models in kinetic theory, J. math. phys., 9, 633-639, (1968) · Zbl 0162.59003  Cercignani, C, Scattering kernels for gas-surface interactions, Transport theory statist. phys., 2, 27-53, (1972)  Drange, H.B, On the Boltzmann equation with external forces, SIAM J. appl. math., 34, 577-592, (1978) · Zbl 0397.76068  Greenberg, W; van Der Mee, C.V.M; Zweifel, P.F, Generalized kinetic equations, Integral equations operator theory, 7, 60-95, (1984) · Zbl 0532.34043  Greiner, G, Spectral properties and asymptotic behavior of the linear transport equation, Math. Z., 185, 167-177, (1984) · Zbl 0567.45001  Hejtmanek, J, Time dependent linear transport equations, (), 1-59 · Zbl 0545.76079  Kaper, H.G; Lekkerkerker, C.G; Hejtmanek, J, Spectral methods in linear transport theory, (1982), Birkhäuser Basel · Zbl 0498.47001  Larsen, E.W; Zweifel, P.F, On the spectrum of the linear transport operator, J. math. phys., 15, 1987-1997, (1974)  Lehner, J; Wing, G.M, On the spectrum of an unsymmetric operator arising in the transport theory of neutrons, Comm. pure appl. math., 8, 217-234, (1955) · Zbl 0064.23004  Lumer, G; Phillips, R.S, Dissipative operators in a Banach space, Pacific J. math., 11, 679-698, (1961) · Zbl 0101.09503  Molinet, F.A, Existence, uniqueness and properties of the solutions of the bolzmann kinetic equation for a weakly ionized gas, I, J. math. phys., 18, 984-996, (1977) · Zbl 0367.76068  Narasimhan, R, Analysis on real and complex manifolds, (1968), North-Holland Amsterdam · Zbl 0188.25803  Palczewski, A, A time dependent linear Boltzmann operator as the generator of a semigroup, Bull. acad. polon. sci. Sér. sci. tech., 25, 233-237, (1977) · Zbl 0363.45009  Scharf, G, Functional-analytic discussion of the linearized Boltzmann equation, Helv. phys. acta, 40, 929-945, (1967) · Zbl 0154.46405  Schnute, J; Shinbrot, M, Kinetic theory and boundary conditions for fluids, Canad. J. math., 25, 1183-1215, (1973) · Zbl 0239.76089  Shikhov, S.B; Shkurpelov, A.A, The analysis of the time dependent kinetic equation for neutron transport in moderating and multiplicative media, (), 97-165, [in Russian]  Shizuta, Y, On the classical solutions of the Boltzmann equation, Comm. pure appl. math., 36, 705-754, (1983) · Zbl 0515.35002  Suhadolc, A; Vidav, I, Linearized Boltzmann equation in spaces of measures, Math. balkanica, 3, 514-529, (1973) · Zbl 0292.45014  Tabata, M, Estimations for the spectrum of the linearized Boltzmann equation with some external potential field, (1984), Osaka University, preprint  Truesdell, C; Muncaster, R.G, Fundamentals of Maxwell’s kinetic theory of a simple monatonic gas, (1980), Academic Press New York  Vidav, I, Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator, J. math. anal. appl., 22, 144-155, (1968) · Zbl 0155.19203  Voigt, J, Functional analytic treatment of the initial boundary value problem for collisionless gases, ()  Bardos, C, Problèmes aux limites pour LES équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation; application à l’équation de transport, Ann. sci. école norm. sup. (4), 3, 185-233, (1970) · Zbl 0202.36903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.