| 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:
|
http://hdl.handle.net/10338.dmlcz/118813 |
| . |
| 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 pipe.to appear. |
| Reference:
|
[Si] Simader Ch.: The Weak Dirichlet and Neumann Problem for the Laplacian in $L^q$ for Bounded and Exterior Domains. Applications.in 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 Domains.in 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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