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We prove that vanishing viscosity solutions to smooth non-degenerate systems of balance laws, having small bounded variation, in one space dimension, must be functions of special bounded variation ( SBV ). For more than one equation, this SBV -regularity for non-degenerate fluxes is new also in the case of systems of conservation laws outside the context of genuine nonlinearity. For general smooth strictly hyperbolic systems of balance laws, this regularity fails, as known for systems of conservation laws: in such case we generalize the SBV -like regularity of the eigenvalue functions of the Jacobian matrix of the flux from conservation to balance laws. Proofs are based on extending Oleinink-type balance estimates, with the introduction of new source measure, a localization argument from [ 14 , 40 ], and observations in real analysis.

We consider the Cauchy problem for a strictly hyperbolic $2\\times 2$ system of conservation laws in one space dimension $u_t+[F(u)]_x=0, u(0,x)=\\bar u(x),$ which is neither linearly degenerate nor genuinely non-linear. We make the following assumption on the characteristic fields. If $r_i(u), \\i=1,2,$ denotes the $i$-th right eigenvector of $DF(u)$ and $\\lambda_i(u)$ the corresponding eigenvalue, then the set $\\{u: \\nabla \\lambda_i \\cdot r_i (u) = 0\\}$ is a smooth curve in the $u$-plane that is transversal to the vector field $r_i(u)$. Systems of conservation laws that fulfill such assumptions arise in studying elastodynamics or rigid heat conductors at low temperature.For such systems we prove the existence of a closed domain $\\mathcal{D} \\subset L^1,$ containing all functions with sufficiently small total variation, and of a uniformly Lipschitz continuous semigroup $S:\\mathcal{D} \\times [0,+\\infty)\\rightarrow \\mathcal{D}$ with the following properties. Each trajectory $t \\mapsto S_t \\bar u$ of $S$ is a weak solution of (1). Viceversa, if a piecewise Lipschitz, entropic solution $u= u(t,x)$ of (1) exists for $t \\in [0,T],$ then it coincides with the trajectory of $S$, i.e. $u(t,\\cdot) = S_t \\bar u. This result yields the uniqueness and continuous dependence of weak, entropy-admissible solutions of the Cauchy problem with small initial data, for systems satysfying the above assumption.

Consider a general strictly hyperbolic, quasilinear system, in one space dimesion where , is a smooth matrix-valued map. Given an initial datum u (0, ·) with small total variation, let u ( t , ·) be the corresponding (unique) vanishing viscosity solution of (1) obtained as a limit of solutions to the viscous parabolic approximation u t  +  A ( u ) u x  =  μ u xx , as  μ → 0. For every T  ≥ 0, we prove the a-priori bound for an approximate solution of (1) constructed by the Glimm scheme, with mesh size , and with a suitable choice of the sampling sequence. This result provides for general hyperbolic systems the same type of error estimates valid for Glimm approximate solutions of hyperbolic systems of conservation laws u t  +  F ( u ) x = 0 satisfying the classical Lax or Liu assumptions on the eigenvalues λ k ( u ) and on the eigenvectors r k ( u ) of the Jacobian matrix A ( u ) =  DF ( u ). The estimate (2) is obtained introducing a new wave interaction functional with a cubic term that controls the nonlinear coupling of waves of the same family and at the same time decreases at interactions by a quantity that is of the same order of the product of the wave strength times the change in the wave speeds. This is precisely the type of errors arising in a wave tracing analysis of the Glimm scheme, which is crucial to control in order to achieve an accurate estimate of the convergence rate as (2).

Consider the Cauchy problem for a strictly hyperbolic, N  × N quasilinear system in one space dimension where is a smooth matrix-valued map, and the initial data is assumed to have small total variation. We investigate the rate of convergence of approximate solutions of (1) constructed by the Glimm scheme, under the assumption that, letting λ k ( u ), r k ( u ) denote the k -th eigenvalue and a corresponding eigenvector of A ( u ), respectively, for each k -th characteristic family the linearly degenerate manifold is either the whole space, or it is empty, or it consists of a finite number of smooth, N –1-dimensional, connected, manifolds that are transversal to the characteristic vector field r k . We introduce a Glimm type functional which is the sum of the cubic interaction potential defined in B ianchini (Discrete Contin Dyn Syst 9:133–166, 2003), and of a quadratic term that takes into account interactions of waves of the same family with strength smaller than some fixed threshold parameter. Relying on an adapted wave tracing method, and on the decrease amount of such a functional, we obtain the same type of error estimates valid for Glimm approximate solutions of hyperbolic systems satisfying the classical Lax assumptions of genuine nonlinearity or linear degeneracy of the characteristic families.

We deal with the non characteristic initial and boundary value problem for an n × n strictly hyperbolic system of conservation laws in one space dimension \\[_tu + _xF(u) = 0, u(0, x) = u(x), b( u((t), t)) = g(t). (*)\\]Here F is a smooth vector field defined in an open, convex neighborhood of the origin of \\(R^n, u\\) and g are functions with small total variation, \\(x = (t)\\) is a non characteristic Lipschitz boundary profile, and b a \\(C^1\\) function. We prove that the front tracking solutions to (*) constructed by D. Amadori in [1] are stable for the \\(L^1\\) topology. This implies the existence of a Standard Riemann Semigroup and hence the well-posedness of (*).

We consider the initial value problem with boundary control for a scalar nonlinear conservation law equation* u_t+[f(u)]_x=0, u(0,x)=0, u( 0)= uın U, $\\ast$ equation* on the domain$\\Omega=\\{(t,x)\\in\\real^2: t\\geq 0, x\\geq 0\\}$ . Here$u=u(t,x)$is the state variable,${\\cal U}$is a set of bounded boundary data regarded as controls, and$f$is assumed to be strictly convex. We give a characterization of the set of attainable profiles at a fixed time$T>0$and at a fixed point$\\bar x>0$ : equation* aligned &=\\u(T,): u is a solution of ()\\,\\\ &=\\u( x): u is a solution of ()\\, aligned U= ( R^+). equation* Moreover we prove that$\\rag$and$\\ragx$are compact subsets of$\\elleuno$and$\\elleuno_{loc}$ , respectively, whenever${\\cal U}$is a set of controls which pointwise satisfy closed convex constraints, together with some additional integral inequalities.

We prove that vanishing viscosity solutions to smooth non-degenerate systems of balance laws having small bounded variation, in one space dimension, must be functions of special bounded variation. For more than one equation, this is new also in the case of systems of conservation laws out of the context of genuine nonlinearity. For general smooth strictly hyperbolic systems of balance laws, this regularity fails, as known for systems of balance laws: we generalize the SBV-like regularity of the eigenvalue functions of the Jacobian matrix of flux from conservation to balance laws. Proofs are based on extending Oleinink-type balance estimates, with the introduction of new source measures, classical localization arguments, and observations in real analysis.