## Duke Mathematical Journal

### A kinetic construction of global solutions of first order quasilinear equations

#### Article information

Source
Duke Math. J., Volume 50, Number 2 (1983), 505-515.

Dates
First available in Project Euclid: 20 February 2004

https://projecteuclid.org/euclid.dmj/1077303206

Digital Object Identifier
doi:10.1215/S0012-7094-83-05022-6

Mathematical Reviews number (MathSciNet)
MR705037

Zentralblatt MATH identifier
0519.35053

Subjects
Primary: 35A99: None of the above, but in this section
Secondary: 35L65: Conservation laws

#### Citation

Giga, Yoshikazu; Miyakawa, Tetsuro. A kinetic construction of global solutions of first order quasilinear equations. Duke Math. J. 50 (1983), no. 2, 505--515. doi:10.1215/S0012-7094-83-05022-6. https://projecteuclid.org/euclid.dmj/1077303206

#### References

• [1] E. Conway and J. Smoller, Clobal solutions of the Cauchy problem for quasi-linear first-order equations in several space variables, Comm. Pure Appl. Math. 19 (1966), 95–105.
• [2] M. G. Crandall, The semigroup approach to first order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108–132.
• [3] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21.
• [4] A. Douglis, Layering methods for nonlinear partial differential equations of first order, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 3, 141–227.
• [5] E. Giusti, Minimal surfaces and functions of bounded variation, Department of Pure Mathematics, Australian National University, Canberra, 1977.
• [6] E. Hopf, The partial differential equation $u\sb t+uu\sb x=\mu u\sb xx$, Comm. Pure Appl. Math. 3 (1950), 201–230.
• [7] A. Kaniel, A kinetic model for a mono-atomic gas, Preprint.
• [8] S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb. 10 (1970), 217–243.
• [9] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566.
• [10] P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973.
• [11] S. Ôharu and T. Takahashi, A convergence theorem of nonlinear semigroups and its application to first order quasilinear equations, J. Math. Soc. Japan 26 (1974), 124–160.
• [12] O. A. Oleĭ nik, Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl. (2) 26 (1963), 95–172.
• [13] B. K. Quinn, Solutions with shocks: An example of an $L\sb1$-contractive semigroup, Comm. Pure Appl. Math. 24 (1971), 125–132.