# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2019032

## A moment approach for entropy solutions to nonlinear hyperbolic PDEs

 1 CNRS, LAAS, Université de Toulouse, 7 avenue du colonel Roche, F-31400 Toulouse, France 2 Applied Mathematics and Plasma Physics Group and Center for Nonlinear Studies, Theoretical Division, Los Alamos National Laboratory, NM 87545 Los Alamos, USA 3 Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 4, CZ-16206 Prague, Czechia 4 Institute of Mathematics of Toulouse (IMT), Université Paul Sabatier, 118 Route de Narbonne, F-31400, Toulouse, France

* Corresponding author: Swann Marx

Received  July 2018 Revised  February 2019 Published  April 2019

Fund Project: This work was partly funded by the ERC Advanced Grant Taming and by project 16-19526S of the Grant Agency of the Czech Republic. Part of the research of the second author was also supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under project numbers 20180468ER and 20170508DR

We propose to solve hyperbolic partial differential equations (PDEs) with polynomial flux using a convex optimization strategy.This approach is based on a very weak notion of solution of the nonlinear equation,namely the measure-valued (mv) solution,satisfying a linear equation in the space of Borel measures.The aim of this paper is,first,to provide the conditions that ensure the equivalence between the two formulations and,second,to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex,finite-dimensional,semidefinite programming problems.This result is then illustrated on the celebrated Burgers equation.We also compare our results with an existing numerical scheme,namely the Godunov scheme.

Citation: Swann Marx, Tillmann Weisser, Didier Henrion, Jean Bernard Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2019032
##### References:
 [1] Y. Brenier, Solution by convex minimization of the Cauchy problem for hyperbolic systems of conservation laws with convex entropy, arXiv: 1710.03754, 2017. [2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. [3] S. I. Chernyshenko, P. Goulart, D. Huang and A. Papachristodoulou, Polynomial sum of squares in fluid dynamics: A review with a look ahead, Phil. Trans. R. Soc. A, 372 (2014), 20130350, 18pp. doi: 10.1098/rsta.2013.0350. [4] M. Claeys and R. Sepulchre, Reconstructing Trajectories from the Moments of Occupation Measures, Proc. IEEE Conf. on Decision and Control, 2014. [5] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6. [6] J. Dahl, Extending the Conic Optimizer in MOSEK with Semidefinite Cones, Proc. Intl. Symp. Math. Prog., Berlin, 2012. [7] C. DeLellis, F. Otto and M. Westdickenberg, Minimal entropy conditions for Burgers equation, Quarterly of Applied Mathematics, 62 (2004), 687-700. doi: 10.1090/qam/2104269. [8] S. Demoulini, D. M. A. Stuart and A. E. Tzavaras, Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics, Archive for Rational Mechanics and Analysis, 205 (2012), 927-961. doi: 10.1007/s00205-012-0523-6. [9] B. Després and F. Lagoutière, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, Journal of Scientific Computing, 16 (2001), 479-524. doi: 10.1023/A:1013298408777. [10] R. J. DiPerna, Measure-valued solutions to conservation laws, Archive for Rational Mechanics and Analysis, 88 (1985), 223-270. doi: 10.1007/BF00752112. [11] L. C. Evans, Partial Differential Equations, American Mathematical Society, 2010. doi: 10.1090/gsm/019. [12] H. O. Fattorini, Infinite Dimensional Optimization and Control Theory,, Cambridge University Press, 1999. doi: 10.1017/CBO9780511574795. [13] E. Feireisl, M. Lukáčová-Medvid'ová and H. Mizerová., Convergence of finite volume schemes for the Euler equations via dissipative measure-valued solutions, arXiv: 1803.08401, 2018. [14] U. S. Fjordholm, S. Mishra and E. Tadmor., On the computation of measured-valued solutions, Acta Numerica, 25 (2016), 567-679. [15] U. S. Fjordholm, R. Käppeli, S. Mishra and E. Tadmor, Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws, Foundations of Computational Mathematics, 17 (2017), 763-827. doi: 10.1007/s10208-015-9299-z. [16] U. S. Fjordholm, K. Lye and S. Mishra, Numerical approximation of statistical solutions of scalar conservation laws, SIAM Journal on Numerical Analysis, 56 (2018), 2989-3009. doi: 10.1137/17M1154874. [17] S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 89 (1959), 271-306. [18] D. Goluskin, G. Fantuzzi., Bounds on mean energy in the Kuramoto-Sivashinsky equation computed using semidefinite programming., arXiv: 1802.08240, 2018. [19] L. Gosse and E. Zuazua, Filtered gradient algorithms for inverse design problems of one-dimensional Burgers equation, Innovative Algorithms and Analysis, 197-227, Springer INdAM Ser., 16, Springer, Cham, 2017. [20] D. Handelman, Representing polynomials by positive linear functions on compact convex polyhedra, Pacific J. Math., 132 (1988), 35-62. [21] D. Henrion and M. Korda, Convex computation of the region of attraction of polynomial control systems, IEEE Trans. Autom. Control, 59 (2014), 297-312. doi: 10.1109/TAC.2013.2283095. [22] D. Henrion, J. B. Lasserre and J. Löfberg, Gloptipoly 3: Moments, optimization and semidefinite programming, Optimization Methods & Software, 24 (2009), 761-779. doi: 10.1080/10556780802699201. [23] D. Henrion and E. Pauwels, Linear conic optimization for nonlinear optimal control, Advances and Trends in Optimization with Engineering Applications, 121-133, MOS-SIAM Ser. Optim., 24, SIAM, Philadelphia, PA, 2017. doi: 10.1137/1.9781611974683.ch10. [24] M. Korda, D. Henrion and J. B. Lasserre, Moments and convex optimization for analysis and control of nonlinear partial differential equations, arXiv: 1804.07565, 2018. [25] S. N. Kružkov, First order quasilinear equations in several independent variables, Mathematics of the USSR-Sbornik, 10 (1970), 217-243. [26] J. B. Lasserre, Moments, Positive Polynomials and Their Applications,, Imperial College Press, 2010. [27] J. B. Lasserre, D. Henrion, C. Prieur and E. Trélat, Nonlinear optimal control via occupation measures and LMI relaxations, SIAM Journal on Control and Optimization, 47 (2008), 1643-1666. doi: 10.1137/070685051. [28] P. Lax, Shock waves and entropy, Contributions to nonlinear Functional Analysis, 603-634, Academic Press, New York, 1971. [29] P. G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, Springer Science & Business Media, 2002. doi: 10.1007/978-3-0348-8150-0. [30] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge university press, 2002. doi: 10.1017/CBO9780511791253. [31] V. Magron and C. Prieur, Optimal Control of Linear PDEs using Occupation Measures and SDP Relaxations, IMA Journal of Mathematical Control and Information, 2018. [32] J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-valued Solutions to Evolutionary PDEs,, CRC Press, 1996. doi: 10.1007/978-1-4899-6824-1. [33] M. Mevissen, J. B. Lasserre and D. Henrion, Moment and SDP relaxation techniques for smooth approximations of problems involving nonlinear differential equations, Proc. IFAC World Congress on Automatic Control, 2011. [34] E. Y. Panov, Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Mathematical Notes, 55 (1994), 517-525. doi: 10.1007/BF02110380. [35] M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana University Mathematics Journal, 42 (1993), 969-984. doi: 10.1512/iumj.1993.42.42045. [36] J. Rubio, The global control of shock waves, Nonlinear Theory of Generalized Functions, 355-369, Erwin Schrödinger Institute, Vienna, 1997. [37] D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves,, Cambridge University Press, 1999. doi: 10.1017/CBO9780511612374. [38] L. Tartar, The compensated compactness method applied to systems of conservation laws, Systems of Nonlinear Partial Differential Equations, 111 (1983), 263-285. [39] G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, 2011.

show all references

##### References:
 [1] Y. Brenier, Solution by convex minimization of the Cauchy problem for hyperbolic systems of conservation laws with convex entropy, arXiv: 1710.03754, 2017. [2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. [3] S. I. Chernyshenko, P. Goulart, D. Huang and A. Papachristodoulou, Polynomial sum of squares in fluid dynamics: A review with a look ahead, Phil. Trans. R. Soc. A, 372 (2014), 20130350, 18pp. doi: 10.1098/rsta.2013.0350. [4] M. Claeys and R. Sepulchre, Reconstructing Trajectories from the Moments of Occupation Measures, Proc. IEEE Conf. on Decision and Control, 2014. [5] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6. [6] J. Dahl, Extending the Conic Optimizer in MOSEK with Semidefinite Cones, Proc. Intl. Symp. Math. Prog., Berlin, 2012. [7] C. DeLellis, F. Otto and M. Westdickenberg, Minimal entropy conditions for Burgers equation, Quarterly of Applied Mathematics, 62 (2004), 687-700. doi: 10.1090/qam/2104269. [8] S. Demoulini, D. M. A. Stuart and A. E. Tzavaras, Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics, Archive for Rational Mechanics and Analysis, 205 (2012), 927-961. doi: 10.1007/s00205-012-0523-6. [9] B. Després and F. Lagoutière, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, Journal of Scientific Computing, 16 (2001), 479-524. doi: 10.1023/A:1013298408777. [10] R. J. DiPerna, Measure-valued solutions to conservation laws, Archive for Rational Mechanics and Analysis, 88 (1985), 223-270. doi: 10.1007/BF00752112. [11] L. C. Evans, Partial Differential Equations, American Mathematical Society, 2010. doi: 10.1090/gsm/019. [12] H. O. Fattorini, Infinite Dimensional Optimization and Control Theory,, Cambridge University Press, 1999. doi: 10.1017/CBO9780511574795. [13] E. Feireisl, M. Lukáčová-Medvid'ová and H. Mizerová., Convergence of finite volume schemes for the Euler equations via dissipative measure-valued solutions, arXiv: 1803.08401, 2018. [14] U. S. Fjordholm, S. Mishra and E. Tadmor., On the computation of measured-valued solutions, Acta Numerica, 25 (2016), 567-679. [15] U. S. Fjordholm, R. Käppeli, S. Mishra and E. Tadmor, Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws, Foundations of Computational Mathematics, 17 (2017), 763-827. doi: 10.1007/s10208-015-9299-z. [16] U. S. Fjordholm, K. Lye and S. Mishra, Numerical approximation of statistical solutions of scalar conservation laws, SIAM Journal on Numerical Analysis, 56 (2018), 2989-3009. doi: 10.1137/17M1154874. [17] S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 89 (1959), 271-306. [18] D. Goluskin, G. Fantuzzi., Bounds on mean energy in the Kuramoto-Sivashinsky equation computed using semidefinite programming., arXiv: 1802.08240, 2018. [19] L. Gosse and E. Zuazua, Filtered gradient algorithms for inverse design problems of one-dimensional Burgers equation, Innovative Algorithms and Analysis, 197-227, Springer INdAM Ser., 16, Springer, Cham, 2017. [20] D. Handelman, Representing polynomials by positive linear functions on compact convex polyhedra, Pacific J. Math., 132 (1988), 35-62. [21] D. Henrion and M. Korda, Convex computation of the region of attraction of polynomial control systems, IEEE Trans. Autom. Control, 59 (2014), 297-312. doi: 10.1109/TAC.2013.2283095. [22] D. Henrion, J. B. Lasserre and J. Löfberg, Gloptipoly 3: Moments, optimization and semidefinite programming, Optimization Methods & Software, 24 (2009), 761-779. doi: 10.1080/10556780802699201. [23] D. Henrion and E. Pauwels, Linear conic optimization for nonlinear optimal control, Advances and Trends in Optimization with Engineering Applications, 121-133, MOS-SIAM Ser. Optim., 24, SIAM, Philadelphia, PA, 2017. doi: 10.1137/1.9781611974683.ch10. [24] M. Korda, D. Henrion and J. B. Lasserre, Moments and convex optimization for analysis and control of nonlinear partial differential equations, arXiv: 1804.07565, 2018. [25] S. N. Kružkov, First order quasilinear equations in several independent variables, Mathematics of the USSR-Sbornik, 10 (1970), 217-243. [26] J. B. Lasserre, Moments, Positive Polynomials and Their Applications,, Imperial College Press, 2010. [27] J. B. Lasserre, D. Henrion, C. Prieur and E. Trélat, Nonlinear optimal control via occupation measures and LMI relaxations, SIAM Journal on Control and Optimization, 47 (2008), 1643-1666. doi: 10.1137/070685051. [28] P. Lax, Shock waves and entropy, Contributions to nonlinear Functional Analysis, 603-634, Academic Press, New York, 1971. [29] P. G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, Springer Science & Business Media, 2002. doi: 10.1007/978-3-0348-8150-0. [30] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge university press, 2002. doi: 10.1017/CBO9780511791253. [31] V. Magron and C. Prieur, Optimal Control of Linear PDEs using Occupation Measures and SDP Relaxations, IMA Journal of Mathematical Control and Information, 2018. [32] J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-valued Solutions to Evolutionary PDEs,, CRC Press, 1996. doi: 10.1007/978-1-4899-6824-1. [33] M. Mevissen, J. B. Lasserre and D. Henrion, Moment and SDP relaxation techniques for smooth approximations of problems involving nonlinear differential equations, Proc. IFAC World Congress on Automatic Control, 2011. [34] E. Y. Panov, Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Mathematical Notes, 55 (1994), 517-525. doi: 10.1007/BF02110380. [35] M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana University Mathematics Journal, 42 (1993), 969-984. doi: 10.1512/iumj.1993.42.42045. [36] J. Rubio, The global control of shock waves, Nonlinear Theory of Generalized Functions, 355-369, Erwin Schrödinger Institute, Vienna, 1997. [37] D. Serre, Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves,, Cambridge University Press, 1999. doi: 10.1017/CBO9780511612374. [38] L. Tartar, The compensated compactness method applied to systems of conservation laws, Systems of Nonlinear Partial Differential Equations, 111 (1983), 263-285. [39] G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, 2011.
Approximation of the solution $y(t,x)$ obtained with our GMP approach, in the case of a shock
Approximation of the solution $y(t,x)$ obtained with our GMP approach, in the case of a rarefaction wave
Approximation of $y(0.75,x)$ with Godunov and GMP
 $x$ 0.185 0.1855 0.186 0.1865 0.187 0.1875 0.188 0.1885 Godunov 0.9999 0.9991 0.9936 0.958 0.7647 0.2724 0.0123 0 GMP 1 1 1 1 1 0 0 0
 $x$ 0.185 0.1855 0.186 0.1865 0.187 0.1875 0.188 0.1885 Godunov 0.9999 0.9991 0.9936 0.958 0.7647 0.2724 0.0123 0 GMP 1 1 1 1 1 0 0 0
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