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On the system of conservation laws and its perturbation in the Besov spaces. (English) Zbl 1103.35071
The main object of the present paper is to study the system of conservation laws \[ A_0(u) \frac{\partial u}{\partial t} + \sum_{k=1}^N A_k(u) \frac{\partial u}{\partial x_k} = 0,\quad x\in\mathbb{R}^N, \;t>0, \] with initial data \(u(x,0)=u_0(x)\), \(x\in\mathbb{R}^N\), where \(u=(u_1, \dots, u_N)\), \(A_0(u)\) is a positive-definite symmetric matrix, and \(A_k(u)\)’s are symmetric matrices.
The main aim is to prove local unique existence and the extension principle in Besov spaces \(B^{\frac{N}{2}+1}_{2,1}(\mathbb{R}^N)\) and, respectively, global existence for small data in \(B^{\frac{N}{2}-1}_{2,1}(\mathbb{R}^N)\) with an added dissipation term.
The first main result gives the local in time existence, i.e., that for \(u_0\in B^{\frac{N}{2}+1}_{2,1}(\mathbb{R}^N)\) there is a \(T>0\) such that the unique solution \(u(t)\) of the above system belongs to \[ L^\infty\left([0,T], B^{\frac{N}{2}+1}_{2,1}(\mathbb{R}^N)\right) \cap C\left([0,T], B^s_{2,1}(\mathbb{R}^N)\right) \cap \mathrm{Lip}\left([0,T], B^{\frac{N}{2}}_{2,1}(\mathbb{R}^N)\right) \] with \(0\leq s<\frac{N}{2}+1\). Secondly, it provides a blow-up criterion for the local in time solution in \(B^{\frac{N}{2}+1}_{2,1}(\mathbb{R}^N)\) for \(T_\ast>T\). Concerning the second assertion the setting is modified by \[ \frac{\partial u}{\partial t} + \sum_{k=1}^N A_k(u) \frac{\partial u}{\partial x_k} = \mu \Delta u,\quad x\in\mathbb{R}^N, \;t>0. \] For \(u_0\in B^{\frac{N}{2}}_{2,1}(\mathbb{R}^N)\) with \(\| u_0\| _{\dot{B}^{\frac{N}{2}-1}_{2,1}(\mathbb{R}^N)} < c \mu\), the existence of a unique solution \[ u\in C\left([0,\infty), B^{\frac{N}{2}}_{2,1}(\mathbb{R}^N)\right) \cap L^1\left([0,\infty), \dot{B}^{\frac{N}{2}+1}_{2,1}(\mathbb{R}^N)\right) \] is shown. Here \(B^s_{p,q}\) and \(\dot{B}^s_{p,q}\) denote the inhomogeneous and homogeneous Besov spaces, as usual.
In Section 2 all definitions and some preparatory assertions for equivalent norms and multiplication and interpolation estimates, respectively, are collected. Section 3 contains the proof of the first result, whereas Section 4 is devoted to the proof of the second outcome. Littlewood-Paley decompositions and energy type estimates are essentially used.

35L65 Hyperbolic conservation laws
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35L45 Initial value problems for first-order hyperbolic systems
35F25 Initial value problems for nonlinear first-order PDEs