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On the computation of measure-valued solutions

Published online by Cambridge University Press:  23 May 2016

Ulrik S. Fjordholm ,
Siddhartha Mishra  and
Eitan Tadmor
Ulrik S. Fjordholm
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, N-7491, Norway E-mail: ulrik.fjordholm@math.ntnu.no
Siddhartha Mishra
Affiliation:
Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, Zürich, Switzerland E-mail: smishra@sam.math.ethz.ch
Eitan Tadmor
Affiliation:
Center of Scientific Computation and Mathematical Modeling (CSCAMM), Department of Mathematics, Institute for Physical Sciences and Technology (IPST), University of Maryland, MD 20742-4015, USA E-mail: tadmor@cscamm.umd.edu
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Abstract

A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm of measure-valued solutions for these problems.

We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.

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Type
Research Article
Information
Acta Numerica , Volume 25 , 01 May 2016 , pp. 567 - 679
Copyright
© Cambridge University Press, 2016 

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References

REFERENCES1

Abgrall, R. and Congedo, P. M. (2013), ‘A semi-intrusive deterministic approach to uncertainty quantification in non-linear fluid flow problems’, J. Comput. Phys. 235, 828–845.Google Scholar
Alibert, J. J. and Bouchitté, G. (1997), ‘Non-uniform integrability and generalized Young measures’, J. Convex Analysis 4, 129–147.Google Scholar
Ambrosio, L., Gigli, N. and Savaré, G. (2005), Gradient Flows, Birkhäuser.Google Scholar
Avrin, J. and Xiao, C. (2014), ‘Convergence results for a class of spectrally hyperviscous models of 3-D turbulent flow’, J. Math. Anal. Appl. 409, 742–751.Google Scholar
Ball, J. (1989), A version of the fundamental theorem for Young measures. In PDEs and Continuum Models of Phase Transitions (Rascle, M., Serre, D. and Slemrod, M., eds), Vol. 344 of Lecture Notes in Physics, pp. 207–215.Google Scholar
Bardos, C. and Tadmor, E. (2015), ‘Stability and spectral convergence of Fourier method for nonlinear problems: On the shortcomings of the 2/3 de-aliasing method’, Numer. Math. 129, 749–782.Google Scholar
Bardos, C. and Titi, E. (2007), ‘Euler equations for incompressible ideal fluids’, Russian Math. Surveys 62, 409–451.Google Scholar
Bardos, C. and Titi, E. (2013), ‘Mathematics and turbulence: Where do we stand?’ J. Turbul. 14, 42–76.Google Scholar
Barth, T. J. (1999), Numerical methods for gas-dynamics systems on unstructured meshes. In An Introduction to Recent Developments in Theory and Numerics of Conservation Laws, (Kroner, D., Ohlberger, M. and Rohde, C., eds), Vol. 5 of Lecture Notes in Computational Science and Engineering, Springer, pp. 195–285.Google Scholar
Barth, T. J. (2013), Non-intrusive uncertainty propagation with error bounds for conservation laws containing discontinuities. In Uncertainty Quantification in Computational Fluid Dynamics, Vol. 92 of Lecture Notes in Computational Science and Engineering, Springer, pp. 1–58.Google Scholar
Beale, J. T., Kato, T. and Majda, A. (1984), ‘Remarks on the breakdown of smooth solutions for the 3D Euler equations’, Comm. Math. Phys. 94, 61–66.CrossRefGoogle Scholar
Bell, J. B., Colella, P. and Glaz, H. M. (1989), ‘A second-order projection method for the incompressible Navier–Stokes equations’, J. Comput. Phys. 85, 257–283.Google Scholar
Benzoni-Gavage, S. and Serre, D. (2007), Multidimensional Hyperbolic Partial Differential Equations: First-Order Systems and Applications, Oxford University Press.Google Scholar
Bianchini, S. and Bressan, A. (2005), ‘Vanishing viscosity solutions of nonlinear hyperbolic systems’, Ann. of Math. (2) 161, 223–342.Google Scholar
H. Bijl, D. Lucor, S. Mishra and C. Schwab, eds (2014), Uncertainty Quantification in Computational Fluid Dynamics, Vol. 92 of Lecture Notes in Computational Science and Engineering, Springer.Google Scholar
Brenier, Y., De Lellis, C. and Székelyhidi, L. Jr (2011), ‘Weak–strong uniqueness for measure-valued solutions’, Comm. Math. Phys. 305, 351–361.CrossRefGoogle Scholar
Bressan, A. (2000), Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, Oxford University Press.Google Scholar
Bressan, A., Crasta, G. and Piccoli, B. (2000), Well-Posedness of the Cauchy Problem for n × n Systems of Conservation Laws, Vol. 694 of Memoirs of the American Mathematical Society, AMS.Google Scholar
Buckmaster, T., De Lellis, C. , Isett, P. and Székelyhidi, L. Jr (2015), ‘Anomalous dissipation for $1/5$-Hölder Euler flows’, Ann. Math. 182, 127–172.Google Scholar
Carrillo, J. A., Feireisl, E., Gwiazda, P. and Swierczewska-Gwiazda, A. (2015), Weak solutions for Euler systems with non-local interactions. arXiv:1512.03116Google Scholar
Carstensen, C. (2001), Numerical analysis of microstructure. In Theory and Numerics of Differential Equations (Durham 2000), Universitext, Springer, pp. 59–126.Google Scholar
Chemin, J.-Y. (1998), Perfect Incompressible Fluids, Clarendon Press, Oxford University Press.Google Scholar
Chen, G. Q. and Feldman, M. (2010), ‘Global solutions to shock reflections by large-angle wedges for potential flow’, Ann. of Math. 171, 1067–1182.Google Scholar
Chen, G. Q. and Glimm, J. (2012), ‘Kolmogorov’s theory of turbulence and inviscid limit of the Navier–Stokes equations in $\mathbb{R}^{3}$’, Comm. Math. Phys. 310, 267–283.Google Scholar
Chiodaroli, E., De Lellis, C. and Kreml, O. (2015), ‘Global ill-posedness of the isentropic system of gas dynamics’, Comm. Pure Appl. Math. 68, 1157–1190.Google Scholar
Chorin, A. (1968), ‘Numerical solution of the Navier–Stokes equations’, Math. Comp. 22, 745–762.Google Scholar
Chorin, A. and Marsden, J. (1993), A Mathematical Introduction to Fluid Mechanics, third edition, Springer.Google Scholar
Collins, C. and Luskin, M. (1989), The computation of the austenitic–martensitic phase transition. In Partial Differential Equations and Continuum Models of Phase Transitions, Vol. 344 of Lecture Notes in Physics, Springer, pp. 34–50.Google Scholar
Constantin, P. (2007), ‘On the Euler equations of incompressible fluids’, Bull. Amer. Math. Soc. 44, 603–621.Google Scholar
Constantin, P., Fefferman, C. and Majda, A. (1996), ‘Geometric constraints on potential singular solutions for the 3-D Euler equations’, Comm. Part. Diff. Equations 21, 559–571.Google Scholar
Crandall, M. G. and Majda, A. (1980), ‘Monotone difference approximations for scalar conservation laws’, Math. Comput. 34, 1–21.CrossRefGoogle Scholar
Dacarogna, B. (1989), Direct Methods in the Calculus of Variations, Vol. 78 of Applied Mathematical Sciences, Springer.Google Scholar
Dafermos, C. M. (2010), Hyperbolic Conservation Laws in Continuum Physics, Vol.325 of Grundlehren der Mathematischen Wissenschaften, Springer.Google Scholar
De Lellis, C. and Székelyhidi, L. Jr (2009), ‘The Euler equations as a differential inclusion’, Ann. of Math. (2) 170, 1417–1436.Google Scholar
De Lellis, C. and Székelyhidi, L. Jr (2010), ‘On the admissibility criteria for the weak solutions of Euler equations’, Arch. Rational Mech. Anal. 195, 225–260.Google Scholar
De Simone, A. (1993), ‘Energy minimizers for large ferromagnetic bodies’, Arch. Rational Mech. Anal. 125, 99–143.Google Scholar
Delort, J. M. (1991), ‘Existence de nappes de tourbillon en dimension deux’, J. Amer. Math. Soc. 4, 553–586.Google Scholar
Demoulini, S., Stuart, D. M. A. and Tzavaras, A. E. (2012), ‘Weak–strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics’, Arch. Rational Mech. Anal. 205, 927–961.Google Scholar
Després, B., Poëtte, G. and Lucor, D. (2009), ‘Uncertainty quantification for systems of conservation laws’, J. Comput. Phys. 228, 2443–2467.Google Scholar
DiPerna, R. J. (1985), ‘Measure-valued solutions to conservation laws’, Arch. Rational Mech. Anal. 88, 223–270.Google Scholar
DiPerna, R. J. and Majda, A. (1987a), ‘Oscillations and concentrations in weak solutions of the incompressible fluid equations’, Comm. Math. Phys. 108, 667–689.Google Scholar
DiPerna, R. J. and Majda, A. (1987b), ‘Concentrations in regularizations for 2-D incompressible flow’, Comm. Pure Appl. Math. 40, 301–345.Google Scholar
E, W. and Kohn, R. (1991), ‘The initial-value problem for measure-valued solutions of a canonical $2\times 2$ system with linearly degenerate fields’, Comm. Pure Appl. Math. 44, 981–1000.Google Scholar
Fjordholm, U. S. (2013), High-order accurate entropy stable numerical schemes for hyperbolic conservation laws. ETH Zürich dissertation 21025.Google Scholar
Fjordholm, U. S., Käppeli, R., Mishra, S. and Tadmor, E. (2016a), ‘Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws’ J. FoCM, to appear. doi:10.1007/s10208-015-9299-zGoogle Scholar
Fjordholm, U. S., Lanthaler, S. and Mishra, S. (2016b), Statistical solutions of hyperbolic conservation laws I: Theory. In preparation.Google Scholar
Fjordholm, U. S., Lye, K. O. and Mishra, S. (2016c), Computation of statistical solutions for systems of conservation laws. In preparation.Google Scholar
Fjordholm, U. S., Mishra, S. and Tadmor, E. (2012), ‘Arbitrarily high-order order accurate essentially non-oscillatory entropy stable schemes for systems of conservation laws’, SIAM J. Numer. Anal 50, 544–573.Google Scholar
Fjordholm, U. S., Mishra, S. and Tadmor, E. (2013), ‘ENO reconstruction and ENO interpolation are stable’, FoCM 13, 139–159.Google Scholar
Foiaş, C. (1972), ‘Statistical study of Navier–Stokes equations I’, Rend. Sem. Mat. Univ. Padova 48, 219–348.Google Scholar
Foiaş, C. (1973), ‘Statistical study of Navier–Stokes equations II’, Rend. Sem. Mat. Univ. Padova 49, 9–123.Google Scholar
Foiaş, C., Manley, O., Rosa, R. and Temam, R. (2001), Navier–Stokes Equations and Turbulence, Cambridge University Press.Google Scholar
Folland, G. B. (1999), Real Analysis, Wiley.Google Scholar
Frid, H. and Liu, I.-S. (1995), ‘Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type’, Z. Angew. Math. Phys. 46, 913–931.Google Scholar
Frid, H. and Liu, I.-S. (1998), ‘Oscillation waves in Riemann problems for phase transitions’, Quart. Appl. Math. 56, 115–135.Google Scholar
Fuchs, F., McMurry, A., Mishra, S., Risebro, N. H. and Waagan, K. (2011), ‘Approximate Riemann solvers and robust high-order finite volume schemes for multi-dimensional ideal MHD equations’, Comm. Comput. Phys 9, 324–362.Google Scholar
Gérard, P. (1991), ‘Microlocal defect measures’, Comm. Part. Diff. Equations 16, 1761–1794.Google Scholar
Ghanem, R. G. and Spanos, P. D. (1991), Stochastic Finite Elements: A Spectral Approach, Springer.Google Scholar
R. Ghanem, D. Higdon and H. Owhadi, eds (2016), Handbook of Uncertainty Quantification, Springer.CrossRefGoogle Scholar
Giles, M. (2008), ‘Multilevel Monte Carlo Path Simulation’, Oper. Res. 56, 607–617.Google Scholar
Glimm, J. (1965), ‘Solutions in the large for nonlinear hyperbolic systems of equations’, Comm. Pure Appl. Math. 18, 697–715.Google Scholar
Glimm, J., Grove, J. and Zhang, Y. (1999), Numerical calculation of Rayleigh–Taylor and Richtmyer–Meshkov instabilities for three dimensional axisymmetric flows in cylindrical and spherical geometries. Report LA-UR99-6796, Los Alamos Laboratory.Google Scholar
Godlewski, E. and Raviart, P.-A. (1991), Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer.Google Scholar
Godunov, S. K. (1961), ‘An interesting class of quasilinear systems’, Dokl. Acad. Nauk. SSSR 139, 521–523.Google Scholar
Goodman, J., Kohn, R. and Reyna, L. (1986), ‘Numerical study of a relaxed variational problem from optimal design’, Comput. Meth. Appl. Mech. Engrg 57, 107–127.Google Scholar
Gottlieb, D., Hussaini, M. Y. and Orszag, S. (1984), Theory and application of spectral methods. In Spectral Methods for Partial Differential Equations (Voigt, R. G. et al. , ed.), SIAM, pp. 1–54.Google Scholar
Gwiazda, P. (2005), ‘On measure-valued solutions to a two-dimensional gravity-driven avalanche flow model’, Math. Methods Appl. Sci. 28, 2201–2223.Google Scholar
Gwiazda, P., Swierczewska-Gwiazda, A. and Wiedemann, E. (2015), ‘Weak–strong uniqueness for measure-valued solutions of some compressible fluid models’, Nonlinearity 28, 3873–3890.Google Scholar
Harten, A., Engquist, B., Osher, S. and Chakravarty, S. R. (1987), ‘Uniformly high order accurate essentially non-oscillatory schemes III’, J. Comput. Phys. 71, 231–303.Google Scholar
Hiltebrand, A. and Mishra, S. (2014), ‘Entropy stable shock capturing streamline diffusion space-time discontinuous Galerkin (DG) methods for systems of conservation laws’, Numer. Math. 126, 103–151.Google Scholar
Hou, T. Y. (2008), ‘Blow-up or no blow-up? The interplay between theory and numerics’, Physica D 237, 1937–1944.Google Scholar
Ismail, F. and Roe, P. L. (2009), ‘Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks’, J. Comput. Phys. 228, 5410–5436.Google Scholar
Jaffre, J., Johnson, C. and Szepessy, A. (1995), ‘Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws’, Math. Model. Meth. Appl. Sci. 5, 367–386.Google Scholar
Johnson, C. and Szepessy, A. (1987), ‘On the convergence of a finite element method for a nonlinear hyperbolic conservation law’, Math. Comput. 49(180), 427–444.Google Scholar
Käppeli, R., Whitehouse, S. C., Scheidegger, S., Pen, U.-L. and Liebendörfer, M. (2011), ‘FISH: A three-dimensional parallel magnetohydrodynamics code for astrophysical applications’, Astrophys. J. (supplement) 195, 20.Google Scholar
Karamanos, G. S. and Karniadakis, G. E. (2000), ‘A spectral vanishing viscosity method for large-eddy simulations’, J. Comput. Phys. 163, 22–50.Google Scholar
Kato, T. (1975), ‘The Cauchy problem for quasi-linear symmetric hyperbolic systems’, Arch. Rational Mech. Anal. 58, 181–205.Google Scholar
Kinderlehrer, D. and Pedregal, P. (1991), ‘Characterizations of gradient Young measures’, Arch. Rational Mech. Anal. 115, 329–365.Google Scholar
Krasny, R. (1986a), ‘A study of singularity formation in a vortex sheet by the point vortex approximation’, J. Fluid Mech. 167, 65–93.Google Scholar
Krasny, R. (1986b), ‘Desingularization of periodic vortex sheet roll-up’, J. Comput. Phys. 65, 292–313.Google Scholar
Kröner, D. and Zajaczkowski, W. M. (1996), ‘Measure-valued solutions of the Euler equations for ideal compressible polytropic fluids’, Math. Methods Appl. Sci. 19, 235–252.Google Scholar
Kružkov, S. N. (1970), ‘First order quasilinear equations in several independent variables’, USSR Math. Sbornik 10, 217–243.Google Scholar
Landau, L. D. and Lipschitz, E. M. (1987), Fluid Mechanics, second edition, Elsevier Butterworth Heinemann.Google Scholar
Lanthaler, S. and Mishra, S. (2015), ‘Computation of measure valued solutions for the incompressible Euler equations’, Math. Mod. Meth. Appl. Sci. (M3AS) 25, 2043–2088.Google Scholar
Laumer, S. (2014), Finite difference approach for the measure valued vanishing dispersion limit of Burgers’ equation. Master’s thesis, ETH Zürich.Google Scholar
Lax, P. D. (1957), ‘Hyperbolic systems of conservation laws II’, Comm. Pure Appl. Math. 10, 537–566.Google Scholar
Lax, P. D. (1971), Shock waves and entropy. In Contributions to Nonlinear Functional Analysis (Zarantonello, E., ed.), Academic Press, pp. 603–634.Google Scholar
Lax, P. D. and Levermore, C. D. (1983a), ‘The small dispersion limit for the KdV equation I’, Comm. Pure. Appl. Math. 36, 253–290.Google Scholar
Lax, P. D. and Levermore, C. D. (1983b), ‘The small dispersion limit for the KdV equation II’, Comm. Pure. Appl. Math. 36, 571–594.Google Scholar
Lax, P. D. and Levermore, C. D. (1983c), ‘The small dispersion limit for the KdV equation III’, Comm. Pure. Appl. Math. 36, 809–829.Google Scholar
LeFloch, P. G., Mercier, J. M. and Rohde, C. (2002), ‘Fully discrete entropy conservative schemes of arbitrary order’, SIAM J. Numer. Anal. 40, 1968–1992.Google Scholar
Leonardi, F. and Mishra, S. (2016), A projection-finite difference method for computing measure-valued solutions of the incompressible Euler equations. In preparation.Google Scholar
LeVeque, R. J. (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge University Press.CrossRefGoogle Scholar
Lichtenstein, L. (1925), ‘Über einige Existenzprobleme der Hydrodynamik homogener, unzusammendrückbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsätze’, Math. Zeit. Phys. 23, 89–154.Google Scholar
Lim, H., Yu, Y., Glimm, J., Li, X. L. and Sharp, D. H. (2008), ‘Chaos, transport and mesh convergence for fluid mixing’, Act. Math. Appl. Sinica 24, 355–368.Google Scholar
Lions, P.-L. (1996), Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible Models, Oxford University Press.Google Scholar
Lopes Filho, M., Nussenzveig, H. J. and Tadmor, E. (2000), ‘Approximate solutions of the incompressible Euler equations with no concentrations’, Ann. de l’Institut Henri Poincaré (C) 17, 371–412.Google Scholar
Luskin, M. (1996), On the computation of crystalline microstructure. In Acta Numerica,Vol. 5, Cambridge University Press, pp. 191–257.Google Scholar
Lye, K. O. and Mishra, S. (2016), Multi-level Monte Carlo methods for computing measure-valued solutions of hyperbolic conservation laws. In preparation.Google Scholar
Majda, A. (1993), ‘Remarks on weak solutions for vortex sheets with a distinguished sign’, Indiana Univ. Math. J. 42, 921–939.Google Scholar
Majda, A. and Bertozzi, A. (2002), Vorticity and Incompressible Flow, Cambridge University Press.Google Scholar
Málek, J., Nečas, J., Rokyta, M. and Ružička, M. (1996), Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman & Hall.Google Scholar
Marchioro, C. and Pulvirenti, M. (1994), Mathematical Theory of Incompressible Nonviscous Fluids, Vol. 96 of Applied Mathematical Sciences, Springer.Google Scholar
Mclaughlin, D. W. and Strain, J. A. (1994), ‘Computing the weak limit of KdV’, Comm. Pure Appl. Math. 47, 1319–1364.Google Scholar
Mishra, S. and Risebro, N. H. (2016), Computation of measure-valued solutions for a three phase flow model. In preparation.Google Scholar
Mishra, S. and Schwab, C. (2012), ‘Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data’, Math. Comput. 81(180), 1979–2018.Google Scholar
Mishra, S., Risebro, N. H., Schwab, C. and Tokareva, S. (2016), ‘Numerical solution of scalar conservation laws with random flux functions’, J. Uncertainty Quantification, to appear. Research report 2012-35, SAM ETH Zürich.Google Scholar
Mishra, S., Schwab, C. and Šukys, J. (2012a), ‘Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions’, J. Comput. Phys 231, 3365–3388.Google Scholar
Mishra, S., Schwab, C. and Šukys, J. (2012b), ‘Multi-level Monte Carlo finite volume methods for shallow water equations with uncertain topography in multi-dimensions’, SIAM J. Sci. Comput. 34, B761–B784.Google Scholar
Mishra, S., Schwab, C. and Šukys, J. (2013), Monte Carlo and multi-level Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws. In Uncertainty Quantification in Computational Fluid Dynamics, Vol. 92 of Lecture Notes in Computational Science and Engineering, Springer, pp. 225–294.Google Scholar
Mishra, S., Schwab, C. and Šukys, J. (2016), ‘Multi-level Monte Carlo finite volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium’, J. Comput. Phys. 312, 192–217.Google Scholar
Murat, F. (1978), ‘Compacité par compensation’, Ann. Sc. Norm. Sup. Pisa 5, 489–507.Google Scholar
Murat, F. (1979), Compacité par compensation II. In Proc. Int. Meeting ‘Recent Methods in Nonlinear Analysis (Giorgi, E. De, Magenes, E. and Mosco, U., eds), Pitagora Editrice, pp. 245–256.Google Scholar
Murat, F. (1981), ‘Compacité par compensation: Condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant’, Ann. Sc. Norm. Sup. Pisa 8, 69–102.Google Scholar
Nicolaides, R. A. and Walkington, N . J. (1993), ‘Computation of microstructure utilizing Young measure representations’, J. Intelligent Mat. Sys. Struct. 4, 457–462.Google Scholar
Panov, E. Y. (1993), ‘Strong measure-valued solutions of a first-order quasilinear equation’, Moscow Univ. Math. Bull. 48, 18–21.Google Scholar
Panov, E. Y. (1994), ‘On sequences of measure-valued solutions of a first-order quasilinear equation’, Mat. Sb. 185, 87–106.Google Scholar
Pasquetti, R. (2006), ‘Spectral vanishing viscosity method for large-eddy simulation of turbulent flow’, J. Sci. Comput. 27, 365–375.Google Scholar
Pedregal, P. (1996), ‘On the numerical analysis of non-convex variational problems’, Numer. Math. 74, 325–336.Google Scholar
Roubíček, T. (1997), Relaxation in Optimization Theory and Variational Calculus, Walter de Gruyter.Google Scholar
Roy, A. and Acharya, A. (2006), ‘Size effects and idealized dislocation microstructure at small scales: Predictions of a phenomenological model of mesoscopic field dislocation mechanics: Part II’, J. Mech. Phys. Solids 54, 1711–1743.Google Scholar
Scheffer, V. (1993), ‘An inviscid flow with compact support in space-time’, J. Geom. Anal. 3, 343–401.Google Scholar
Schochet, S. (1989), ‘Examples of measure-valued solutions’, Comm. Part. Diff. Equations 14, 545–575.Google Scholar
Schochet, S. (1995), ‘The weak vorticity formulation of the $2$-D Euler equations and concentration–cancellation’, Comm. Part. Diff. Equations 20, 1077–1104.Google Scholar
Schonbek, M. (1982), ‘Convergence of solutions to nonlinear dispersion equations’, Comm. Part. Diff. Equations 7, 959–1000.Google Scholar
Shnirelman, A. (2000), ‘Weak solutions with decreasing energy of the incompressible Euler equations’, Comm. Math. Phys. 210, 541–603.Google Scholar
Székelyhidi, L. (2011), ‘Weak solutions to the incompressible Euler equations with vortex sheet initial data’, CR Math. Acad. Sci. Paris 349, 1063–1066.Google Scholar
Székelyhidi, L. Jr and Wiedemann, E. (2012), ‘Young measures generated by ideal incompressible fluid flows’, Arch. Rational Mech. Anal. 206, 333–366.Google Scholar
Tadmor, E. (1987), ‘The numerical viscosity of entropy stable schemes for systems of conservation laws I’, Math. Comput. 49, 91–103.Google Scholar
Tadmor, E. (1989), ‘Convergence of spectral methods for nonlinear conservation laws’, SIAM J. Numer. Anal. 26, 30–44.Google Scholar
Tadmor, E. (2003), Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. In Acta Numerica, Vol. 12, Cambridge University Press, pp. 451–512.Google Scholar
Tadmor, E. (2012), ‘A review of numerical methods for nonlinear partial differential equations’, Bull. Amer. Math. Soc. 49, 507–554.Google Scholar
Tartar, L. (1979), Compensated compactness and applications to partial differential equations. In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Pitman, pp. 136–212.Google Scholar
Tartar, L. (1990), ‘H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations’, Proc. Roy. Soc. Edin. Sect. A 115, 193–230.Google Scholar
Tryoen, J., Le Maître, O., Ndjinga, M. and Ern, A. (2010), ‘Intrusive projection methods with upwinding for uncertain non-linear hyperbolic systems’, J. Comput. Phys. 229, 6485–6511.Google Scholar
Xiu, D. and Hesthaven, J. S. (2005), ‘High-order collocation methods for differential equations with random inputs’, SIAM J. Sci. Comput. 27, 1118–1139.Google Scholar
Young, L. C. (1969), Lectures on the Calculus of Variations and Optimal Control Theory, Saunders.Google Scholar
Yudovich, V. I. (1963), ‘Non-stationary flow of an ideal incompressible liquid’, Zh. Vych. Mat. 3, 1032–1066.Google Scholar