# Layering methods for nonlinear partial differential equations of first order

Annales de l'institut Fourier (1972)

- Volume: 22, Issue: 3, page 141-227
- ISSN: 0373-0956

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topDouglis, Avron. "Layering methods for nonlinear partial differential equations of first order." Annales de l'institut Fourier 22.3 (1972): 141-227. <http://eudml.org/doc/74087>.

@article{Douglis1972,

abstract = {This paper is concerned with generalized, discontinuous solutions of initial value problems for nonlinear first order partial differential equations. “Layering” is a method of approximating an arbitrary generalized solution by dividing its domain, say a half-space $t\ge 0$, into thin layers $(i-\ell ) h \le t\le ih$, $i=1,2,\ldots (h>0)$, and using a strict solution $u_i$ in the $i$-th layer. On the interface $t = (i-\ell )h$, $u_t$ is required to reduce to a smooth function approximating the values on that plane of $u_\{i-\ell \}$. The resulting stratified configuration of strict solutions of the equation is called a “layered solution” . Under appropriate conditions, any generalized solution can be realized as the limit of a sequence of layered solutions for which smoothing is made finer and finer and $h\rightarrow 0$; the estimates needed to prove this pertain solely to strict solutions of the equation concerned. Layering was first used by N.N. Kuznetsov in connection with conservation laws and with initial data of bounded variation (in a multi-dimensional sense). These matters are also discussed here, the method extended to the case of bounded, measurable initial data, and a large class of possible smoothing operations discussed. In addition, the method is adapted to equations of Hamilton-Jacobi type.},

author = {Douglis, Avron},

journal = {Annales de l'institut Fourier},

language = {eng},

number = {3},

pages = {141-227},

publisher = {Association des Annales de l'Institut Fourier},

title = {Layering methods for nonlinear partial differential equations of first order},

url = {http://eudml.org/doc/74087},

volume = {22},

year = {1972},

}

TY - JOUR

AU - Douglis, Avron

TI - Layering methods for nonlinear partial differential equations of first order

JO - Annales de l'institut Fourier

PY - 1972

PB - Association des Annales de l'Institut Fourier

VL - 22

IS - 3

SP - 141

EP - 227

AB - This paper is concerned with generalized, discontinuous solutions of initial value problems for nonlinear first order partial differential equations. “Layering” is a method of approximating an arbitrary generalized solution by dividing its domain, say a half-space $t\ge 0$, into thin layers $(i-\ell ) h \le t\le ih$, $i=1,2,\ldots (h>0)$, and using a strict solution $u_i$ in the $i$-th layer. On the interface $t = (i-\ell )h$, $u_t$ is required to reduce to a smooth function approximating the values on that plane of $u_{i-\ell }$. The resulting stratified configuration of strict solutions of the equation is called a “layered solution” . Under appropriate conditions, any generalized solution can be realized as the limit of a sequence of layered solutions for which smoothing is made finer and finer and $h\rightarrow 0$; the estimates needed to prove this pertain solely to strict solutions of the equation concerned. Layering was first used by N.N. Kuznetsov in connection with conservation laws and with initial data of bounded variation (in a multi-dimensional sense). These matters are also discussed here, the method extended to the case of bounded, measurable initial data, and a large class of possible smoothing operations discussed. In addition, the method is adapted to equations of Hamilton-Jacobi type.

LA - eng

UR - http://eudml.org/doc/74087

ER -

## References

top- [1] D.P. BALLOU, Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions, Trans. A.M.S. 152 (1970), 441-460. Zbl0207.40401MR55 #8573
- [2] J.D. COLE, On a quasilinear parabolic equation occuring in aerodynamics, Quart. Appl. Math. 9 (1951), 225-236, Zbl0043.09902MR13,178c
- [3] E.D. CONWAY and E. HOPF, Hamilton's theory and generalized solutions of the Hamilton-Jacobi equation, J. Math. Mech. 13 (1964), 939-986. Zbl0178.11002MR32 #243
- [4] E. CONWAY and J. SMOLLER, Global solutions of the Cauchy problem for quasilinear first order equations in several space variables, Comm. Pure Appl. Math. 19 (1966), 95-105. Zbl0138.34701MR33 #388
- [5] R. COURANT and D. HILBERT, Methods of Mathematical Physics, Vol. II. Interscience Publishers, 1962, 699-700. Zbl0099.29504
- [6] A. DOUGLIS, On calculating weak solutions of quasilinear, first order partial differential equations, Contributions to Differential Equations 1 (1963), 59-94. Zbl0196.50105MR28 #1787
- [7] A. DOUGLIS, Solutions in the large for multi-dimensional, non-linear partial differential equations of first order, Ann. Inst. Fourier de l'Univ. de Grenoble 15 (1965), 2-35. Zbl0137.29001MR33 #7686
- [8] H. FEDERER, An analytical characterization of distributions whose partial derivatives are representable by measures. (Abstract). Bull. A.M.S. 60 (1954), 339.
- [9] H. FEDERER, Geometric Measure Theory, Springer Verlag, 1969. Zbl0176.00801MR41 #1976
- [10] W.H. FLEMING, Functions with generalized gradient and generalized surfaces, Ann. Mat. Pura Appl., Ser. 4, 44 (1957), 93-104. Zbl0082.26701MR20 #2421
- [11] W.H. FLEMING, The Cauchy problem for a nonlinear first order partial differential equation, J. Diff. Eqns ; 5 (1969), 515-530. Zbl0172.13901MR38 #3579
- [12] E. de GIORGI, Su una teoria generale della misura (r — 1)- dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl., 36 (1954), 191-213. Zbl0055.28504MR15,945d
- [13] E. de GIORGI, Nuovi teoremi relativi alle misure (r — 1)- dimensionale in uno spazio ad r dimensioni, Richerche Mat., 4 (1955), 95-113. Zbl0066.29903MR17,596a
- [14] J. GLIMM, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math 18 (1965). 697-715. Zbl0141.28902MR33 #2976
- [15] J. GLIMM and P.D. LAX, Decay of solutions of systems of hyperbolic conservation laws. Report No. NYO-1480-115, AEC Computing and Applied Mathematics Center, Courant Inst. Math. Sciences, New York Univ, April, 1969, 111 pp. Zbl0146.33803
- [16] E. HOPF, The partial differential equation ut + uux = uuxx', Comm. Pure Appl. Math. 3 (1950), 201-230. Zbl0039.10403MR13,846c
- [17] E. HOPF, Generalized solutions of nonlinear equations of first order, J. Math. Mech., 14 (1965), 951-972. Zbl0168.35101MR32 #272
- [18] E. HOPF, On the right weak solution of the Cauchy problem for a quasilinear equation of first order, J. Math, Mech., 19 (1969), 483-487. Zbl0188.16102MR40 #4588
- [19] J.L. JOHNSON and J.A. SMOLLER, Global solutions for an extended class of hyperbolic systems of conservations laws, Arch. Rat. Mech. Anal. 32 (1969), 169-189. Zbl0167.10204MR38 #4822
- [20] K. KOJIMA, On the existence of discontinuous solutions of the Cauchy problem for quasi-linear first order equations, Proc. Jap. Acad. 42 (1966), 705-709. Zbl0166.36301MR35 #3225
- [21] D.B. KOTLOW, On the equations ut + ∇ . F(u) = 0 and ut + ∇ . F(u) = vΔu, Bull. A.M.S. 75 (1969), 1362-4. Zbl0185.34601
- [22] K. KRICKEBERG, Distributionen, Funktionen beschränkter Variation und Lebesguescher Inhalt nicht parametrischer Flächen, Ann. Mat. Pura Appl., Ser. 4, 44 (1957), 105-133. Zbl0082.26702MR20 #2420
- [23] S.N. KRUZHKOV, The Cauchy problem in the large for nonlinear equations and for certain quasilinear systems of the first order with several variables, Soviet Math, 5 (1964), 493-496. Zbl0138.34702
- [24] S.N. KRUZHKOV, Generalized solutions of first order nonlinear equations in several independent variables, Mat. Sb. 70 (112) (1966), 394-415. Zbl0152.09502
- [25] S.N. KRUZHKOV, Generalized solutions of the Cauchy problem in the large for nonlinear equations of first order. Soviet Math. Dokl. 10 (1969), 785-788. Zbl0202.37701
- [26] S.N. KRUZHKOV, First order quasilinear equations in several independent variables, Math of the USSR-Sbornik, 10 (1970), 217-243. Zbl0215.16203
- [27] N.N. KUZNETSOV, The weak solution of the Cauchy problem for a multi-dimensional quasi-linear equation. Math. Zametki 2 (1967), 401-410. Zbl0184.12901
- [28] P.D. LAX, Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math. 10 (1957), 537-556. Zbl0081.08803MR20 #176
- [29] I.P. NATANSON, Theory of Functions of a Real Variable, Vol. 1. (Translated from Russian by L.F. Boron, with editorial collaboration of E. Hewitt.) Revised edition, pp. 220-223. Ungar, New York, 1961.
- [30] M. RIESZ, Sur les ensembles compacts de fonctions sommables, Acta Sci. Math. Szeged 6 (1933), 136-142. Zbl0008.00702JFM59.0276.01
- [31] B.L. ROZHDESTVENSKII, A new method of solving the Cauchy problem in the large for quasilinear equations, Dokl. Akad. Nauk SSSR, 138 (1961), 309-312. Zbl0121.31703
- [32] J.A. SMOLLER, Contact discontinuities in quasi-linear hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 791-801. Zbl0195.39201MR42 #2175
- [1-1] S. AIZAWA and N. KIRUCHI, A mixed initial and boundary-value problem for the Hamilton-Jacobi equation in several space variables, Funkcial, Ekv. 9 (1966), 139-50. Zbl0162.41102MR35 #1938
- [1-2] P.A. ANDREYANOV, Cauchy Problem for a first order quasi-linear, equation in the class of locally integrable functions, Vestnik Moscow Univ., 26 (1971), 42-47. Zbl0226.35015
- [1-3] S.H. BENTON, JR., A general space-time boundary problem for the Hamilton-Jacobi equation, J. Diff. Eqns. 11 (1972), 425-435. Zbl0215.28503MR45 #7248
- [1-4] W.H. FLEMING, Nonlinear partial differential equation - Probabilistic and game theoretic methods. Problems in Nonlinear Analysis (C.I.M.E., IV Ciclo, Varenna, 1970), pp. 95-128. Edizioni Cremonese, Rome, 1971. Zbl0225.35020
- [1-5] B.K. QUINN, Solutions with shocks : an example of an L1 - contractive semigroup, Comm. Pure Appl. Math. 24 (1971), 125-132. Zbl0209.12401MR42 #6428
- [1-6] J.A. SMOLLER and C.C. CONLEY, Shock waves as limits of progressive wave solutions of higher order equations, II, Comm. Pure Appl. Math. 25 (1972), 133-146. Zbl0225.35067MR46 #5843

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