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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.

MSC:
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
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