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Title: About steady transport equation I -- $L^p$-approach in domains with smooth boundaries (English)
Author: Novotný, Antonín
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 37
Issue: 1
Year: 1996
Pages: 43-89
Category: math
Summary: We investigate the steady transport equation $$ \lambda z+w\cdot \nabla z+az=f,\quad \lambda >0 $$ in various domains (bounded or unbounded) with smooth noncompact boundaries. The functions $w,\,a$ are supposed to be small in appropriate norms. The solution is studied in spaces of Sobolev type (classical Sobolev spaces, Sobolev spaces with weights, homogeneous Sobolev spaces, dual spaces to Sobolev spaces). The particular stress is put onto the problem to extend the results to as less regular vector fields $w,\,a$, as possible (conserving the requirement of smallness). The theory presented here is well adapted for applications in various problems of compressible fluid dynamics. (English)
Keyword: steady transport equation
Keyword: bounded
Keyword: unbounded
Keyword: exterior domains
Keyword: existence of solutions
Keyword: estimates
MSC: 35Q35
MSC: 76N10
MSC: 82C70
idZBL: Zbl 0852.35115
idMR: MR1396161
Date available: 2009-01-08T18:22:10Z
Last updated: 2012-04-30
Stable URL:
Reference: [A] Adams R.A.: Sobolev Spaces.Academic Press, 1976. Zbl 1098.46001, MR 0450957
Reference: [BV1] Beir ao da Veiga H.: Boundary value problems for a class of first order partial differential equations in Sobolev spaces and applications to the Euler flow.Rend. Sem. Mat. Univ. Padova 79 (1988), 247-273. MR 0964034
Reference: [BV2] Beir ao da Veiga H.: Existence results in Sobolev spaces for a stationary transport equation.Ricerche di Matematica, vol. in honour of Prof. C. Miranda, 1987.
Reference: [BV3] Beir ao da Veiga H.: An $L^p$-theory for the $n$-dimensional, stationary compressible NavierStokes equations and the incompressible limit for compressible fluids. The equilibrium solutions.Comment. Math. Phys. 109 (1987), 229-248. MR 0880415
Reference: [DL] Di Perna R.J., Lions P.L.: Ordinary differential equations, transport theory and Sobolev spaces.Invent. Math. 98 (1989), 511-547. MR 1022305
Reference: [Fi1] Fichera G.: Sulle equazioni differenziali lineari ellitico paraboliche del secondo ordine.Atti Acad. Naz. Lincei, Mem. Sc. Fis. Mat. Nat., Sez. I5 (1956), 1-30. MR 0089348
Reference: [Fi2] Fichera G.: On a unified theory of boundary value problem for elliptic-parabolic equations of second order in boundary problems.Diff. Eq., Univ. Wisconsin Press, Madison-Wisconsin, 1960, pp. 87-120. MR 0111931
Reference: [F] Fridrichs K.O.: Symmetric positive linear differential equations.Comm. Pure Appl. Math. 11 (1958), 333-418. MR 0100718
Reference: [G] Galdi G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Volume I: Linearized Stationary Problems.Springer Tracts in Natural Philosophy, 1994. MR 1284205
Reference: [GS] Galdi G.P., Simader Ch.: Existence, uniqueness and $L^q$-estimates for the Stokes problem in an exterior domain.Arch. Rational Mech. Anal. 112 (1990), 291-318. MR 1077262
Reference: [GNP] Galdi G.P., Novotný A., Padula M.: On the twodimensional steady-state problem of a viscous gas in an exterior domain.Pacific J. Math., in press.
Reference: [H] Heywood J.: The Navier-Stokes equations: On the existence, regularity and decay of solutions.Indiana Univ. Math. J. 29 (1980), 639-681. Zbl 0494.35077, MR 0589434
Reference: [Hi] Hille E.: Methods in Classical and Functional Analysis.Addison-Wesley, 1972. Zbl 0223.46001, MR 0463863
Reference: [KN] Kohn J.J., Nirenberg L.: Elliptic-parabolic equations of second order.Comm. Pure Appl. Math. 20 (1967), 797-872. Zbl 0153.14503, MR 0234118
Reference: [L] Leray J.: Sur le mouvement d'un liquide visqueux emplissant l'espace.Acta Math. 63 (1934), 193-248. MR 1555394
Reference: [LP] Lax P.D., Philips R.S.: Symmetric positive linear differential equations.Comm. Pure Appl. Math. 11 (1958), 333-418. MR 0100718
Reference: [Mi] Mizohata S.: The theory of Partial Differential Equations.Cambridge Univ. Press, 1973. Zbl 0263.35001, MR 0599580
Reference: [N1] Novotný A.: Steady Flows of Viscous Compressible Fluids - $L^2$-approach -.Proc. EQUAM 92, Varenna, Eds. R. Salvi, J. Straškraba, Stab. Anal. Cont. Media, 1993.
Reference: [N2] Novotný A.: Steady flows of viscous compressible fluids in exterior domains under small perturbation of great potential forces.Math. Meth. Model. Appl. Sci. (M$^3$AS) 3.6 (1993), 725-757. MR 1245633
Reference: [N3] Novotný A.: A note about the steady compressible flows in $\Bbb R^3$, $\Bbb R_+^3$-$L^p$-approach.Preprint Univ. Toulon, 1993.
Reference: [N4] Novotný A.: About the steady transport equation II - Schauder estimates in domain with smooth boundaries.1994, to appear. MR 1472165
Reference: [NP1] Novotný A., Padula M.: $L^p$-approach to steady flows of viscous compressible fluids in exterior domains.Arch. Rational Mech. Anal. 126 (1994), 243-297. MR 1293786
Reference: [NP2] Novotný A., Padula M.: Existence and uniqueness of stationary solutions for viscous compressible heat-conductive fluid with large potential and small nonpotential external forces.Sib. Math. 34 (1993), 120-146. MR 1255466
Reference: [NP3] Novotný A., Padula M.: On physically reasonable solutions of steady compressible Navier-Stokes equations in 3-D exterior domains I $(v_\infty =0)$, II $(v_\infty \neq 0)$.to appear. MR 1411338
Reference: [NPe] Novotný A., Penel P.: An $L^p$-approach for steady flows of viscous compressible heat conductive gas.Math. Meth. Model. Appl. Sci. (M$^3$AS), in press.
Reference: [O] Oleinik O.A.: Linear equations of second order with nonnegative characteristic form.Amer. Math. Soc. Transl. 65 (1967), 167-199.
Reference: [OR] Oleinik O.A., Radekevic E.V.: Second Order Equations with Nonnegative Characteristic Form.Amer. Math. Soc. and Plenum Press, New York, 1973. MR 0470454
Reference: [P1] Padula M.: A representation formula for steady solutions of a compressible fluid moving at low speed.Transp. Theory and Stat. Phys. 21 (1992), 593-614. MR 1194463
Reference: [P2] Padula M.: On the exterior steady problem for the equations of a viscous isothermal gas.Comment. Math. Univ. Carolinae 34.2 (1993), 275-293. Zbl 0778.76087, MR 1241737
Reference: [PP] Padula M., Pileckas K.: Steady flows of a viscous ideal gas in domains with non compact boundaries: Existence and asymptotic behavior in a appear.
Reference: [Si] Simader Ch.: The Weak Dirichlet and Neumann Problem for the Laplacian in $L^q$ for Bounded and Exterior Domains. Nonlinear Analysis, Function Spaces and Applications, editors Krbec, Kufner, Opic, Rákosník, Leipzig, Teubner 4 (1990), 180-223. MR 1151436
Reference: [SiSo1] Simader Ch., Sohr. H.: The Weak Dirichlet Problem for $\Delta $ in $L^q$ in Bounded and Exterior Domains.Pitman Research Notes in Math., in press.
Reference: [SiSo2] Simader Ch., Sohr. H.: A New Approach to the Helmholtz Decomposition and the Neumann Problem in $L^q$-spaces for Bounded and Exterior Mathematical Topics Relating to Navier-Stokes Equations, vol. 11, Series of Advances in Mathematics for Applied Sciences, editor G.P. Galdi, World Scientific, 1992. MR 1190728


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