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Null Lagrangian Measures in Subspaces, Compensated Compactness and Conservation Laws

  • Andrew LorentEmail author
  • Guanying Peng
Article
  • 6 Downloads

Abstract

Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the study of hyperbolic conservation laws. One of the simplest formulations of a compensated compactness problem is to ask for conditions on a compact set \({\mathcal {K}}\subset M^{m\times n}\) such that
$$\begin{aligned}&\lim _{j\rightarrow \infty } \Vert \mathrm {dist}(Du_j,{\mathcal {K}})\Vert _{L^p}= 0 \text { and } \sup _{j}\Vert u_j\Vert _{W^{1,p}}<\infty \nonumber \\&\quad \Rightarrow \{Du_{j}\}_{j}\text { is precompact in } L^p. \end{aligned}$$
(1)
Let \(M_1,M_2,\dots , M_q\) denote the set of all minors of \(M^{m\times n}\). A sufficient condition for (1) is that any probability measure \(\mu \) supported on \({\mathcal {K}}\) satisfying
$$\begin{aligned} \int M_k(X) \hbox {d}\mu (X)=M_k\left( \int X \hbox {d}\mu (X)\right) \text { for all } k \end{aligned}$$
(2)
is a Dirac measure. We call measures that satisfy (2) Null Lagrangian Measures and following [21], we denote the set of Null Lagrangian Measures supported on \({\mathcal {K}}\) by \({\mathcal {M}}^{pc}({\mathcal {K}})\). For general mn, a necessary and sufficient condition for triviality of \({\mathcal {M}}^{pc}({\mathcal {K}})\) was an open question even in the case where \({\mathcal {K}}\) is a linear subspace of \(M^{m\times n}\). We answer this question and provide a necessary and sufficient condition for any linear subspace \({\mathcal {K}}\subset M^{m\times n}\). The ideas also allow us to show that for any \(d\in \left\{ 1,2,3\right\} \), d-dimensional subspaces \({\mathcal {K}}\subset M^{m\times n}\) support non-trivial Null Lagrangian Measures if and only if \({\mathcal {K}}\) has Rank-1 connections. This is known to be false for \(d\ge 4\) from [5]. Further using the ideas developed we are able to answer a question of Kirchheim et al. [18]. Let \(P_1(u,v):=\left( \begin{matrix} u &{} v\\ a(v) &{} u\\ ua(v) &{} \frac{1}{2}u^2+ F(v)\\ \end{matrix}\right) \) and \({\mathcal {K}}_1 := \left\{ P_1(u,v): u,v\in \mathrm {I\!R}\right\} \) for some function a and its primitive F. The set \({\mathcal {K}}_1\) arises in the study of entropy solutions to the \(2\times 2\) system of conservation laws
$$\begin{aligned} u_t=a(v)_x \quad \text { and }\quad v_t=u_x. \end{aligned}$$
In [18], the authors asked what are the conditions on the function a such that \({\mathcal {M}}^{pc}({\mathcal {K}}_1\cap U)\) consists of Dirac measures, where U is an open neighborhood of an arbitrary matrix in \({\mathcal {K}}_1\). Given \(\alpha =(\alpha _1,\alpha _2)\in \mathrm {I\!R}^2\), if \(a'(\alpha _2)>0\) then we construct non-trivial measures in \({\mathcal {M}}^{pc}({\mathcal {K}}_1\cap B_{\delta }\left( P_1(\alpha )\right) )\) for any \(\delta >0\). On the other hand if \(a'(\alpha _2)<0\) then for sufficiently small \(\delta >0\), we show that \({\mathcal {M}}^{pc}({\mathcal {K}}_1\cap B_{\delta }\left( P_1(\alpha )\right) )\) consists of Dirac measures.

Notes

Acknowledgements

Andrew Lorent is very grateful to V. Šverák for many very helpful conversations during a two-week visit to Minnesota in November of 2016. These conversations essentially introduced us to this topic and led to some initial ideas for Theorem 2. Both authors are very grateful for a great deal of very helpful correspondence since then. We would also like to thank S. Müller for providing us with a proof of Lemma 26. The proof provided is considerably simpler and more elegant than our original proof. Guanying Peng would like to thank Y. Shi for helpful discussions on elementary algebraic geometry. Andrew Lorent acknowledes the support of the Simons Foundation, Grant #426900.

References

  1. 1.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical MonographsOxford University Press, New York 2000zbMATHGoogle Scholar
  2. 2.
    Astala, K.: Analytic aspects of quasiconformality. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. Extra Vol. II, 617–626, 1998Google Scholar
  3. 3.
    Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63(4), 337–403 (1976/77)Google Scholar
  4. 4.
    Ball, J.M.: Does rank-one convexity imply quasiconvexity? Metastability and Incompletely Posed Problems (Minneapolis, Minn., 1985), Vol. 3. The IMA Volumes in Mathematics and its Applications. Springer, New York, 17–32, 1987Google Scholar
  5. 5.
    Bhattacharya, K., Firoozye, N.B., James, R.D., Kohn, R.V.: Restrictions on microstructure. Proc. R. Soc. Edinb. Sect. A 124(5), 843–878, 1994MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge 2004CrossRefzbMATHGoogle Scholar
  7. 7.
    Buckmaster, T., De Lellis, C., Isett, P., Székelyhidi, L.: Anomalous dissipation for \(1/5\)-Hölder Euler flows. Ann. Math. (2) 182(1), 127–172, 2015MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    De Lellis, C., Székelyhidi, L.: The Euler equations as a differential inclusion. Ann. Math. (2) 170(3), 1417–1436, 2009MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    De Lellis, C., Székelyhidi, L.: Dissipative continuous Euler flows. Invent. Math. 193(2), 377–407, 2013ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    DiPerna, R.J.: Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82(1), 27–70, 1983MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    DiPerna, R.J.: Compensated compactness and general systems of conservation laws. Trans. Am. Math. Soc. 292(2), 383–420, 1985MathSciNetCrossRefGoogle Scholar
  12. 12.
    Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. Vol. 74. CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990Google Scholar
  13. 13.
    Isett, P.: Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time, vol. 196. Annals of Mathematics StudiesPrinceton University Press, Princeton 2017CrossRefzbMATHGoogle Scholar
  14. 14.
    Kinderlehrer, D., Pedregal, P.: Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115(4), 329–365, 1991MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kinderlehrer, D., Pedregal, P.: Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4(1), 59–90, 1994MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kirchheim, B.: Deformations with finitely many gradients and stability of quasiconvex hulls. C. R. Acad. Sci. Paris Sér. I Math. 332(3), 289–294, 2001ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kirchheim, B.: Rigidity and geometry of microstructures. Habilitation Thesis, University of Leipzig, 2003Google Scholar
  18. 18.
    Kirchheim, B., Müller, S., Šverák, V.: Studying nonlinear pde by geometry in matrix space. Geometric Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, 347–395, 2003Google Scholar
  19. 19.
    Lorent, A., Peng, G.: Null Lagrangian Measures in planes, compensated compactness and conservation laws (Previous version v2). Preprint arxiv:1801.02912
  20. 20.
    Milnor, J.: Singular Points of Complex Hypersurfaces, vol. 61. Annals of Mathematics StudiesPrinceton University Press, Princeton 1968zbMATHGoogle Scholar
  21. 21.
    Müller, S.: Variational models for microstructure and phase transitions. Calculus of variations and geometric evolution problems (Cetraro, 1996). Lecture Notes in Mathematics, 1713, Fond. CIME/CIME Found. Subser., Springer, Berlin, 85–210, 1999Google Scholar
  22. 22.
    Müller, S.: A sharp version of Zhang’s theorem on truncating sequences of gradients. Trans. Am. Math. Soc. 351(11), 4585–4597, 1999MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Müller, S.: Personal communicationGoogle Scholar
  24. 24.
    Müller, S., Šverák, V.: Attainment results for the two-well problem by convex integration. Geometric Analysis and the Calculus of Variations. International Press, Cambridge, 239–251, 1996Google Scholar
  25. 25.
    Müller, S., Šverák, V.: Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. Math. (2) 157(3), 715–742, 2003MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Müller, S., Sychev, M.A.: Optimal existence theorems for nonhomogeneous differential inclusions. J. Funct. Anal. 181(2), 447–475, 2001MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Piccione, P., Tausk, D.V.: A student’s guide to symplectic spaces, Grassmannians and Maslov index. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications] Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2008Google Scholar
  28. 28.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete MathematicsWiley, Chichester 1986zbMATHGoogle Scholar
  29. 29.
    Šverák, V.: On Tartar’s conjecture. Ann. Inst. H. Poincaré Anal. Non Linéaire 10, 405–412, 1993MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tartar, L.: Compensated compactness and applications to partial differential equations. Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV. Research Notes in Mathemathics, 39, Pitman, Boston, 136–212, 1979Google Scholar
  31. 31.
    Tartar, L.: The compensated compactness method applied to systems of conservation laws. Systems of nonlinear partial differential equations (Oxford, 1982) NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 111. Reidel, Dordrecht, 263–285, 1983Google Scholar
  32. 32.
    Whitney, H.: Elementary structure of real algebraic varieties. Ann. Math. Second Ser. 66(3), 545–556, 1957MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of CincinnatiCincinnatiUSA
  2. 2.Department of MathematicsUniversity of ArizonaTucsonUSA

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