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Mathematics Without Boundaries pp 415–481Cite as

Extended Crystal PDEs

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Abstract

In this paper we show that between partial differential equations (PDEs) and crystallographic groups there is an unforeseen relation. In fact we prove that integral bordism groups of PDEs can be considered extensions of crystallographic subgroups. In this respect we can consider PDEs as extended crystals. Then an algebraic topological obstruction (crystal obstruction), characterizing existence of global smooth solutions for smooth boundary value problems, is obtained. Applications of this new theory to the Ricci flow equation and Navier–Stokes equation are given, which solve some well-known fundamental problems. These results are also extended to singular PDEs (introducing extended crystal singular PDEs). An application to singular magnetohydrodynamics partial differential equations (MHD-PDEs) is given following some our previous results on such equations, and showing existence of (finite times stable smooth) global solutions crossing the critical nuclear energy production zone.

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Notes

  1. 1.

    Recall that given two groups A and B and an homomorphism \(\alpha: B\to Aut(A)\), the semidirect product is a group denoted by \(A\rtimes_\alpha B\), that is the Cartesian product \(A\times B\), with product given by \((a,b).(\bar a,\bar b)=(a.\alpha(b)(\bar a),b.\bar b)\). The semidirect product reduces to the direct product, i.e., \(A\rtimes_\alpha B=A\times B\equiv A\oplus B\), when \(\alpha(b)=1\), for all \(b\in B\). In the following we will omit the symbol α.

  2. 2.

    A discrete subgroup \(H\subset G\) of a topological group G, is cocompact if there is a compact subset \(K\!\!\subset \! G\) such that \(H\!K=G\).

  3. 3.

    Usually one denotes such an extension simply with \(G(d)/{\bf T}\), whether we are not interested to emphasize the notation for \(G\cong G(d)/{\bf T}\).

  4. 4.

    \(\mathbb{Z}G\) is the free \(\mathbb{Z}\)-module generated by the elements of G. The multiplication in G extends uniquely to a \(\mathbb{Z}\)-bilinear product \(\mathbb{Z}G\times\mathbb{Z}G\to \mathbb{Z}G\) so that \(\mathbb{Z}G\) becomes a ring called the integral group ring of G. A G -module A, is just a (left)\(\mathbb{Z}G\)-module on the Abelian group A. This action can be identified by a group homomorphism \(G\to Hom_{group}(A)\).

  5. 5.

    The lattice subgroups of a group G is a lattice under inclusion. The identity \(1\in G\) identifies the minimum \(\{1\}<G\) and the maximum is just G. In the following we will denote also by e = 1 the identity of a group G. The subgroup \({\bf T}\subset G(d)\) is called also the Bieberbach lattice of G(d).

  6. 6.

    For example if \({}^{}E\) is identified by means of an infinite graph, (see, e.g., [100]), then the unit cell is the corresponding fundamental finite graph. Of course d-dimensional chains can be associated to such graphs, so that also the corresponding unit cells can be identified with compact d-chains.

  7. 7.

    Note that a space group is characterized other than by translational and point symmetry, also by the metric-parameters characterizing the unit cell. Thus the number of space groups is necessarily infinite.

  8. 8.

    See also Refs. [1, 4, 6, 7, 32, 45, 91, 95, 97], and works by M. Gromov [25] and E. Ruh [95] on the almost flat manifolds, that are related to such crystallographic groups.

  9. 9.

    Let A, B and C groups and \(B A\), \(B C\), the amalgamated free product \(A_BC\) is generated by the elements of A and C with the common elements from B identified.

  10. 10.

    Let us recall that the index of a subgroup \(H\subset G\), denoted \([G:H]\), is the number of left cosets \(aH=\{ah: h\in H\}\), (resp. right cosets \(Ha=\{ha: h\in H\}\)), of H. For a finite group G one has the following formula: (Lagrange’s formula) \([G:H]=\frac{o(G)}{o(H)}\), where o(G), resp. o(H), is the order of G, resp. H. If aH = Ha, for any \(a\in G\), then H is said to be a normal subgroup. Every subgroup of index 2 is normal, and its cosets are the subgroup and its complement.

  11. 11.

    See also the recent work by Bernstein and Schwarzman on the complex crystallographic groups [7].

  12. 12.

    For general information on bordism groups and related problems in differential topology, see, e.g., Refs. [31, 39, 41, 52, 90, 94, 98, 99, 102–105, 107, 108].

  13. 13.

    A partition \((i_1,\cdots,i_r)\) of n is nondyadic if none of the \(i_\beta\) are of the form \(2^s-1\).

  14. 14.

    Pontrjagin numbers are determined by means of homonymous characteristic classes belonging to \(H^\bullet(BG,\mathbb{Z})\), where BG is the classifying space for G-bundles, with \(G=S_p(n)\). See, e.g., Refs. [39, 41, 94, 99, 102–104, 107, 108].

  15. 15.

    Let us recall that weak solutions, are solutions V, where the set \(\Sigma(V)\) of singular points of V, contains also discontinuity points, \(q,q'\in V\), with \(\pi_{k,0}(q)=\pi_{k,0}(q')=a\in W\), or \(\pi_{k}(q)=\pi_{k}(q')=p\in M\). We denote such a set by \(\Sigma(V)_S\subset\Sigma(V)\), and, in such cases we shall talk more precisely of singular boundary of V, like \((\partial V)_S=\partial V\setminus\Sigma(V)_S\). However, for abuse of notation we shall denote \((\partial V)_S\), (resp. \(\Sigma(V)_S\)), simply by \((\partial V)\), (resp. \(\Sigma(V)\)), also if no confusion can arise. Solutions with such singular points are of great importance and must be included in a geometric theory of PDEs, too [62]. Let us also emphasize that singular solutions can be identified with integral n-chains in E k , and in this category can be considered also fractal solutions, i.e., solutions with sectional fractal or multifractal geometry. (For fractal geometry see, e.g., [20, 35, 40].)

  16. 16.

    The bording solutions considered for the bordism groups are singular solutions if the symbols g k and \(g_{k+1}\) are different from zero, and for singular weak solutions in the general case. Here we have denoted \(\underline{\Omega}_p(X)\) the p-bordism group of a manifold X. For information on such structure of the algebraic topology see, e.g., [31, 52, 90, 94, 99, 101–104, 107, 108].

  17. 17.

    Let us remark that sequences (29) and (31) do not necessitate to be exact but are always partially exact. In fact, even if it does not necessitate that \(d_\circ e_*=0\) and \(f_*\circ g_*=0\), (hence neither \({\hbox{\rm im}}\,(e_*)={\rm ker}(d_*)\) and \({\hbox{\rm im}}\,(g_*)={\rm ker}(f_*)\)), one has that \(e_*\) and \(g_*\) are monomorphisms and \(d_*\) and \(f_*\) are epimorphisms. This is enough for our proof. Let us recall also that \({\bf H}_{n-1}(E_k)\equiv Map(\Omega_{n-1}^{E_k};\mathbb{R})\) is an Hopf algebra in extended sense, i.e. it contains the Hopf algebra \(\mathbb{R}^{\Omega_{n-1}}\) as a subalgebra. (See also [56].)

  18. 18.

    Singular solutions, like those described in this example, are very important in many physical applications too, since they represent complex phenomena related to perturbations of some fixed dynamic background. For example, for suitable values of the parameters a and b in (35), manifolds \(Y[\mu,a,b]\) intersect Y S along common characteristic lines, and the singular solutions V are piecewise \(\mathbb{Z}_2\)-manifolds. (For complementary information on such singular manifolds and singular solutions of PDEs, see also [13, 42, 56, 72].)

  19. 19.

    This function \(\widetilde{u}(t,x)\), is well defined and limited in all \((t,x)\in\mathbb{R}^+\times[0,1]\), since the function h(x) is smooth in [0,1] and with \(|\frac{d^{2n}h}{dx^{2n}}(x)|\le C\in\mathbb{R}\), \(\forall x\in[0,1]\). In fact one has \(\sum_{n\ge 0}t^n\widetilde{a}_n\le C\sum_{n\ge 0}\frac{t^n}{n!}\equiv C\sum_{n\ge 0}b_n\). The last series converges, with convergence radius \(r=\infty\) since \(\lim_{n\to\infty}\left|\frac{b_{n+1}}{a_n}\right|=\lim_{n\to\infty}\frac{t}{n+1}=0=\frac{1}{r}\). Therefore the series \(\sum_{n\ge 0}t^n \widetilde{a}_n(x)\) is convergent as it is absolutely convergent.

  20. 20.

    An extended 0-crystal PDE \(E_k\subset J^k_n(W)\) does not necessitate to be a 0-crystal PDE. In fact E k is an extended 0-crystal PDE if \(\Omega_{n-1,w}^{E_k}=0\). This does not necessarily imply that \(\Omega_{n-1}^{E_k}=0\).

  21. 21.

    In general a steady state solution of \((RF)\) is not admitted since this should imply that \((M,\gamma)\) is Ricci flat. Furthermore, regular solutions \(g_{ij}(t,x^k)=h(t)\bar g_{ij}(x^k)\), with separated time variable from space ones, imply that \((M,\gamma)\) is an Einstein manifold. In fact \(R_{ij}(g)=R_{ij}(h\bar g)=R_{ij}(\bar g)\). The Ricci flow equation becomes \(h_t(t)\bar g_{ij}(x^k)=\kappa R_{ij}(\bar g)\). Therefore, must be \(h_t(t)=\omega=\kappa R_{ij}(\bar g)/\bar g_{ij}(x^k)\), with \(\omega \in\mathbb{R}\). By imposing the initial condition \(g_{ij}(0,x^k)=\gamma_{ij}(x^k)\), we get that must be \(h(t)=\omega t+1\) and \(\bar g_{ij}(x^k)=\gamma_{ij}(x^k)\), hence \(R_{ij}(\gamma)=\gamma_{ij}\omega/\kappa\). Vice versa, if the Ricci flow equation is considered only for Einstein manifolds, then solutions with separated variables like above, are admitted. This means that in general, i.e., starting with any \((M,\gamma)\), we cannot assume solutions \(g_{ij}(t,x^k)\) with the above separated variables structure, even if these solutions “arrive” to S 3, that is just an Einstein manifold. The same results can be obtained by considering metrics \(g(t,x^k)\), obtained deforming \(\gamma(x^k)\), under a space–time flow \(\phi_\lambda\) of \(M\times\mathbb{R}\), i.e., \(\{t\circ\phi_\lambda=t+\lambda, x^k\circ\phi_\lambda=\phi_\lambda^k(t,x^i)\}_{1\le i,k\le 3}\), and with initial condition \(\{\phi_0^k=x^k\}_{k=1,2,3}\). In fact if we assume that \(\phi_\lambda^r=h(\lambda)\bar\phi^r(x^k)\), then the Ricci flow equation becomes as given in (45).

    $$\frac{\dot h^2(\lambda)}{h^2(\lambda)}=\kappa\frac{\bar\phi^r_i\bar\phi^r_i(R_{rs}(\gamma)\circ\phi_\lambda)} {\bar\phi^r_i\bar\phi^r_i(\gamma_{rs}(\gamma)\circ\phi_\lambda)}=\omega\in\mathbb{R}^{+} \Rightarrow\left\{\begin{array}{@{}ll@{}} {\rm(a)}& \!\!\dot h(\lambda)-\pm\sqrt{\omega}h(\lambda)=0\\\rm(b)& \!\!\bar\phi^r_i\bar\phi^r_i[R_{rs}(\gamma)-\frac{\omega}{\kappa}\gamma_{rs}]\circ\phi_\lambda=0\end{array} \right\}$$
    (45)

    The integration of the Eq. (45(a)) gives \(h(\lambda)=Ce^{\pm\lambda\sqrt{\omega}}\). Furthermore, if \((M,\gamma)\) is an Einstein manifold, i.e., there exists \(\mu\in\mathbb{R}\), such that \(R_{rs}(\gamma)=\mu\gamma_{rs}\), then taking \(\omega=\mu\kappa\), one has the following solution, uniquely identified by the initial condition, (up to rigid flows): \(\phi_\lambda=e^{\pm\lambda\sqrt{\mu\kappa}}x^r\). If \((M,\gamma)\) is not Einstein, or equivalently, assuming \(\omega\not=\kappa\mu\), we see that the solutions of Eq. (45(b)) are \(\bar\phi^r=a^r\in\mathbb{R}\), since the metric \(\bar\gamma_{rs}\equiv R_{rs}(\gamma)-\frac{\omega}{\kappa}\gamma_{rs}\) is not degenerate, hence must necessarily be \(\bar\phi^r_i=0\). But such a flow does not satisfy initial condition \(\{\phi_0^r=x^r\}_{r=1,2,3}\). In conclusion a metric \(g_{ij}(t,x^k)\), obtained deforming γ with a space-time flow \(\phi_\lambda\), where \(\phi_\lambda^r=h(\lambda)\bar\phi^r(x^k)\), is a solution of the Ricci flow equation iff \((M,\gamma)\) is Einstein, or Ricci flat. This last case corresponds to take \(\omega=0\) in Eq. (45) and has as solution the unique flow, up to rigid ones, \(\phi_\lambda^r=x^r\).

  22. 22.

    Generalized solutions of PDEs, called viscosity solutions were introduced by Pierre-Louis Lions and Michael Crandall in the paper [36]. Such mathematical objects do not necessitate to be solutions, but are envelope manifolds of solutions.

  23. 23.

    Let us emphasize also that an integral 4-plane, where the components \(g_{ij,\beta}\), \(0\le|\beta|\le 3\), satisfy conditions in (42), but not all conditions in (48), has the corresponding integral manifold, say \(\widetilde{V}\), that passes for N 0, but it is not contained in \((RF)\). (This is represented by V outside in Fig. 1.) Let us denote the corresponding metric with \(\widetilde{g}\). Therefore, the retraction method imposes also to \(g_{ij,\beta}\), with \(0\le|\beta|\le 3\), to satisfy conditions reported in (48). The relation between \(\widetilde{V}\equiv V_{outside}\subset \textit{JD}^2(E)\), and \(V\subset (RF)\), can be realized with a deformation, \(g_{\lambda}\), connecting the corresponding metrics \(\widetilde{g}\) and g. More precisely, \(g_{\lambda}=\widetilde{g}+\lambda[g-\widetilde{g}]\), \(\lambda\in[0,1]\). The integral manifolds \(V_{\lambda}\subset \textit{JD}^2(E)\), generated by \(g_{\lambda}\), are not contained in \((RF)\) for any \(\lambda\in[0,1]\), but all pass for N 0. In fact, since \((g_{\lambda})_{ij,\beta}=\widetilde{g}_{ij,\beta}+\lambda[g_{ij,\beta}-\widetilde{g}_{ij,\beta}]\), \(|\beta|\ge 0\), we get that \((g_{\lambda})_{ij,\beta}|_{N_0}=\widetilde{g}_{ij,\beta}|_{N_0}=g_{ij,\beta}|_{N_0}\), for \(0\le|\beta|\le 2\), but \((g_{\lambda})_{ij,\beta}|_{N_0}\) do not satisfy conditions in (48) for \(|\beta|=3\). This is just the meaning of the retraction method considered in the proof of Lemma 14.

  24. 24.

    This generalizes a previous result by Hamilton [27], and after separately by De Turk [19] and Chow and Knopp [16], that proved existence and uniqueness of nonsingular solution for Cauchy problem in some Ricci flow equation. Let us also emphasize that our approach to find solutions for Cauchy problems, works also when \(N_0\subset (RF)\) is diffeomorphic to a 3-dimensional spacelike submanifold of W, that is not necessarily representable by a section of \(E_{t=0}\cong\widetilde{S^0_2M}\to M\).

  25. 25.

    The proof of the Poincaré conjecture given here refers to the Ricci flow equation, according to some ideas pioneered by Hamilton [26–30], and followed also by Perelman [43, 44]. However the arguments used here are completely different from ones used by Hamilton and Perelman. (For general information on the relations between Poincaré conjecture and Ricci flow equation, see, e.g., Refs.[5, 17, 18] and papers quoted there.) Here we used our general PDE’s algebraic topological theory, previously developed in some works. Compare also with our previous proof given in [2, 3], where, instead was not yet introduced the relation between PDEs and crystallographic groups. Let us note also that whether M is homeomorphic to S 3, then M is necessarily diffeomorphic to S 3. (See [78, 80] for details. For related subjects see also [77, 81–86].)

  26. 26.

    \(\mathbb{R}[[x^1,\dots,x^n]]\) denotes the algebra of formal series \(\sum_{i_1,\dots,i_n}a_{i_1\cdots i_n}(x^1)^{i_1}\cdots(x^n)^{i_n}\), with \(a_{i_1\cdots i_n}\in\mathbb{R}\). Real analytic functions in the indeterminates \((x^1,\dots,x^n)\), are identified with above formal series having nonzero converging radius. Thus real analytic functions belong to a subalgebra of \(\mathbb{R}[[x^1,\dots,x^n]]\). This last can be also called the algebra of real formal analytic functions in the indeterminates \((x^1,\dots,x^n)\).

  27. 27.

    For general information on the geometric theory of PDEs see, e.g., [9, 12, 14, 22– 25, 33, 34, 38, 103–105]. In particular, for singular PDE geometry, see the book [60] and the recent papers [3, 79] where many boundary value problems are explicitly considered. For basic information on differential topology and algebraic topology, see, e.g., [9, 24, 31, 39, 41, 47, 94, 99, 98, 102–105, 107, 108].

  28. 28.

    If \(\mathfrak{a}\) is any ideal of A, the radical of \(\mathfrak{a}\) is the following ideal \(r(\mathfrak{a})\equiv \sqrt{\mathfrak{a}}\equiv\{x\in A | x^n\in\mathfrak{a}for some\) n>0\(\}\equiv rad(\mathfrak{a})\). If \(\sqrt{\mathfrak{a}}=\mathfrak{a}\), then \(\mathfrak{a}\) is called radical ideal or perfect. One has also that \(r(\mathfrak{a})\) is the intersection of all prime ideals \(\mathfrak{p}\subset A\), containing \(\mathfrak{a}\). In particular, the radical of the zero ideal \(<0>\) is the nilradical, nil(A), of A, i.e., the set of all nilpotent elements of A. Therefore nil(A) is the intersection of all prime ideals, (since all ideals must contain 0). One has also \(nil(A)\subset rad(A)\), where rad(A) is the ideal of A defined by intersection of all maximal ideals \(\mathfrak{m}\subset A\). If \(\mathfrak{a}\) is a radical ideal, then \(A/\mathfrak{a}\) is reduced, i.e., the set of its nilpotent elements is reduced to \(\{0\}\). In particular \(A/nil(A)\) is reduced. If \(\pi:A\to A/\mathfrak{a}\) is the canonical projection, then \(\pi^{-1}(nil(A/\mathfrak{a}))=r(\mathfrak{a})\).

  29. 29.

    Let us emphasize that by using Lemma 14 we can identify admissible smooth 1-dimensional integral manifolds in A j , \(j=1,2\). In fact, A j are formally integrable and completely integrable PDEs. So we can use Lemma 14(1), but also Lemma 14(3), since \(\pi_{1,0}(A_j)=F\), and \(A_j\to F\) are affine subbundles of \(\textit{JD}(F)\to F\), with associated vector bundle the symbol g 1.

  30. 30.

    Note that the bifurcation does not necessarily imply that the tangent planes in the points of \(V_{ij}\subset V\) to the components V i and V j , should be different.

  31. 31.

    But, in general, it is \(\Omega_{n-1}^{{}^{(ij)}E_k}\not=0\).

  32. 32.

    Let us emphasize that admissible Cauchy integral manifolds can be found in each component of \(\widehat{\underline{(\textit{MHD})}}\), thanks to Lemma 14. More precisely, the proceeding followed for the Navier–Stokes equation in Example 13, to solve Cauchy problems there, can be applied also to \(\widehat{\underline{(\textit{MHD})}}\). In fact, envelopment solutions can be built also for all the components of this last equation, since they are formally integrable and completely integrable PDEs.

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Acknowledgements

I would like to thank the editors for inviting me to write my contribution to this Springer book. A similar version of this work was earlier put on arXiv:0811.3693v18.

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  1. Department SBAI-Mathematics, University of Roma La Sapienza, Via A. Scarpa, 16-10-00166, Roma, Italy

    Agostino Prástaro

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  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Dept. Industrial & Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

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Prástaro, A. (2014). Extended Crystal PDEs. In: Rassias, T., Pardalos, P. (eds) Mathematics Without Boundaries. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1106-6_18

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