## Smooth approximations of global in time solutions to scalar conservation laws.(English)Zbl 1172.35450

Summary: We construct global smooth approximate solution to a scalar conservation law with arbitrary smooth monotonic initial data. Different kinds of singularities interactions which arise during the evolution of the initial data are described as well. In order to solve the problem, we use and further develop the weak asymptotic method, recently introduced technique for investigating nonlinear waves interactions.

### MSC:

 35L65 Hyperbolic conservation laws 35A35 Theoretical approximation in context of PDEs 35L45 Initial value problems for first-order hyperbolic systems
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### References:

 [1] S. N. Kru\vzkov, “First order quasilinear equations in several independent variables,” Mathematics of the USSR-Sbornik, vol. 10, no. 2, pp. 217-243, 1970. · Zbl 0215.16203 [2] A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, vol. 102 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1992. · Zbl 0754.34002 [3] S. V. Zakharov and A. M. Il’in, “From a weak discontinuity to a gradient catastrophe,” Matematicheskiĭ Sbornik, vol. 192, no. 10, pp. 3-18, 2001. · Zbl 1028.35015 [4] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 2000. · Zbl 0940.35002 [5] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, vol. 20 of Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, Oxford, UK, 2000. · Zbl 0997.35002 [6] V. Bojkovic, V. Danilov, and D. Mitrovic, “Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process,” http://www.math.ntnu.no/conservation/. · Zbl 1189.35178 [7] V. G. Danilov, “Generalized solutions describing singularity interaction,” International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 8, pp. 481-494, 2002. · Zbl 1011.35093 [8] V. G. Danilov and V. M. Shelkovich, “Delta-shock wave type solution of hyperbolic systems of conservation laws,” Quarterly of Applied Mathematics, vol. 63, no. 3, pp. 401-427, 2005. [9] V. G. Danilov and V. M. Shelkovich, “Dynamics of propagation and interaction of \delta -shock waves in conservation law systems,” Journal of Differential Equations, vol. 211, no. 2, pp. 333-381, 2005. · Zbl 1072.35121 [10] V. G. Danilov and D. Mitrovic, “Weak asymptotics of shock wave formation process,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 4, pp. 613-635, 2005. · Zbl 1079.35067 [11] V. G. Danilov and D. Mitrovic, “Delta shock wave formation in the case of triangular hyperbolic system of conservation laws,” Journal of Differential Equations, vol. 245, no. 12, pp. 3704-3734, 2008. · Zbl 1192.35120 [12] D. Mitrovic and J. Susic, “Global solution to a Hopf equation and its application to non-strictly hyperbolic systems of conservation laws,” Electronic Journal of Differential Equations, vol. 2007, no. 114, pp. 1-12, 2007. · Zbl 1138.35367 [13] E. Yu. Panov and V. M. Shelkovich, “\delta $$^{\prime}$$-shock waves as a new type of solutions to systems of conservation laws,” Journal of Differential Equations, vol. 228, no. 1, pp. 49-86, 2006. · Zbl 1108.35116 [14] V. M. Shelkovich, “The Riemann problem admitting \delta -, \delta $$^{\prime}$$-shocks, and vacuum states (the vanishing viscosity approach),” Journal of Differential Equations, vol. 231, no. 2, pp. 459-500, 2006. · Zbl 1108.35117 [15] V. P. Maslov, Perturbations Theory and Asymptotic Methods, Izdat, Moscow, Russia, 1965. [16] V. P. Maslov, Perturbations Theory and Asymptotic Methods, Dunod, Paris, France, 1972. · Zbl 0247.47010 [17] V. P. Maslov, Perturbations Theory and Asymptotic Methods, Nauka, Moscow, Russia, 1988. · Zbl 0653.35002 [18] L. Hörmander, The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators, Springer, Berlin, Germany, 1994. [19] V. G. Danilov, G. A. Omel’yanov, and V. M. Shelkovich, “Weak asymptotics method and interaction of nonlinear waves,” in Asymptotic Methods for Wave and Quantum Problems, M. V. Karasev, Ed., vol. 208 of American Mathematical Society Translations Series 2, pp. 33-164, American Mathematical Society, Providence, RI, USA, 2003. · Zbl 1140.35382
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