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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