On the topological classification of rarefaction curves in systems of three conservation laws.

*(English)*Zbl 1201.34011Consider the system of \(n\) conservation laws in one space dimension \(x\): \(U_t + H(U)_x = 0\), where \(U(x,t) \in \mathbb{R}^n\) is an unknown smooth function and \(H(\cdot): \mathbb{R}^n \to \mathbb{R}^n\) is a given smooth function (called flux). This system is equivalent to the following system of first-order partial differential equations: \(U_t + DH(U)U_x = 0\), where \(DH(U)\) is the Jacobian matrix of \(H\). Consider solutions of the form \(U(x,t) = \tilde U (\lambda)\), where \(\lambda = x/t\). This leads to the equation

\[ DH(U) \dot U = \lambda \dot U, \tag{1} \]

where \(\dot U = dU/d\lambda\) (from here on, we omit the sign tilde over \(U\)). The variable \(\lambda\) plays a double role. Firstly, it is a parameter which plays the role of time: \(U=U(\lambda)\). Secondly, from (1) it follows that \(\lambda\) is an eigenvalue of the matrix \(DH(U)\) with eigenvector \(\dot U\). Thus, equation (1) defines the field of eigenspaces of \(DH(U)\) which corresponds to real eigenvalues. The integral curves of equation (1), i.e., the curves tangent to one of the eigenspaces of \(DH(U)\) at each point, are called rarefaction curves. For instance, in a neighborhood of a point \(U \in \mathbb{R}^n\) where the matrix \(DH(U)\) has \(n\) real simple eigenvalues, equation (1) defines \(n\) linearly independent directions and there are \(n\) non-singular families of rarefaction curves tangent to these directions. Singularities of rarefaction curves occur at the points where the matrix \(DH(U)\) has multiple eigenvalues (such points of the phase space are called singular). Here, it is necessary to consider two types of multiplicities. Algebraic multiplicity is the standard multiplicity of \(\lambda\) as a root of the characteristic polynomial of \(DH(U)\). Geometric multiplicity of \(\lambda\) is the number of linearly independent eigenvectors with \(\lambda\). Singularities of rarefaction curves are also singularities of the system of \(n-1\) implicit differential equations obtained from (1) by eliminating \(\lambda\). In this paper, the authors consider the case \(n=3\) and a generic flux function \(H\). They study the local structure of rarefaction curves at singular points of two types: when \(DH(U)\) has an eigenvalue with algebraic multiplicity two and geometric multiplicity one; and when \(DH(U)\) has an eigenvalue with algebraic multiplicity three and geometric multiplicity one. Local topological normal forms of the configuration of rarefaction curves are given, structural stability of these configurations (under \(C^3\) Whitney perturbation of the flux function) is proved.

\[ DH(U) \dot U = \lambda \dot U, \tag{1} \]

where \(\dot U = dU/d\lambda\) (from here on, we omit the sign tilde over \(U\)). The variable \(\lambda\) plays a double role. Firstly, it is a parameter which plays the role of time: \(U=U(\lambda)\). Secondly, from (1) it follows that \(\lambda\) is an eigenvalue of the matrix \(DH(U)\) with eigenvector \(\dot U\). Thus, equation (1) defines the field of eigenspaces of \(DH(U)\) which corresponds to real eigenvalues. The integral curves of equation (1), i.e., the curves tangent to one of the eigenspaces of \(DH(U)\) at each point, are called rarefaction curves. For instance, in a neighborhood of a point \(U \in \mathbb{R}^n\) where the matrix \(DH(U)\) has \(n\) real simple eigenvalues, equation (1) defines \(n\) linearly independent directions and there are \(n\) non-singular families of rarefaction curves tangent to these directions. Singularities of rarefaction curves occur at the points where the matrix \(DH(U)\) has multiple eigenvalues (such points of the phase space are called singular). Here, it is necessary to consider two types of multiplicities. Algebraic multiplicity is the standard multiplicity of \(\lambda\) as a root of the characteristic polynomial of \(DH(U)\). Geometric multiplicity of \(\lambda\) is the number of linearly independent eigenvectors with \(\lambda\). Singularities of rarefaction curves are also singularities of the system of \(n-1\) implicit differential equations obtained from (1) by eliminating \(\lambda\). In this paper, the authors consider the case \(n=3\) and a generic flux function \(H\). They study the local structure of rarefaction curves at singular points of two types: when \(DH(U)\) has an eigenvalue with algebraic multiplicity two and geometric multiplicity one; and when \(DH(U)\) has an eigenvalue with algebraic multiplicity three and geometric multiplicity one. Local topological normal forms of the configuration of rarefaction curves are given, structural stability of these configurations (under \(C^3\) Whitney perturbation of the flux function) is proved.

Reviewer: Alexey Remizov (Trieste)

##### MSC:

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |

53A25 | Differential line geometry |

35L65 | Hyperbolic conservation laws |

##### Keywords:

rarefaction curves; systems of three conservation laws; systems of implicit differential equations; singular points
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\textit{M. J. D. Carneiro} et al., Bull. Braz. Math. Soc. (N.S.) 41, No. 1, 139--160 (2010; Zbl 1201.34011)

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