Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy (0805.3614v1)

Abstract: We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach constant equilibrium state in the Lp-norm at a rate O(t^-m/2(1-1/p)), as t tends to $\infty$, for p in [min (m,2),+ \infty]. Moreover, we can show that we can approximate, with a faster order of convergence, theconservative part of the solution in terms of the linearized hyperbolic operator for m >= 2, and by a parabolic equation in the spirit of Chapman-Enskog expansion. The main tool is given by a detailed analysis of the Green function for the linearized problem.

Asymptotic Behavior of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy

The paper investigates the asymptotic time behavior of global, smooth solutions to entropy dissipative hyperbolic systems under the Shizuta-Kawashima (SK) condition. The core focus is to establish how these solutions converge towards a constant equilibrium state in the L-norm at a specific rate as time $t \to \infty$. The authors specifically highlight the decay rate $O(t^{1-p})$, where $p \in [\min\{m, 2\}, 1]$, with $m$ being the spatial dimension. More importantly, they approximate the conservative part of the solution at a faster convergence rate compared to the dissipative part.

Analysis and Results

The paper hinges on meticulous analysis of the Green function for the linearized problem. In the one-dimensional case, the solution is decomposed into a diffusive component, reminiscent of heat kernels moving along characteristic directions, and a singular component, which decays exponentially. The authors show that the conservative variable behaves like the heat kernel and decays over time at the same rate as the diffusive Green function. Meanwhile, the dissipative variable, influenced by the dissipative mechanisms present in the system, demonstrates faster decay, notably $t^{-2}$ faster compared to the conservative variable.

For multidimensional cases, direct decomposition of the Green function presents significant difficulty, hence leading the authors to employ Fourier analysis to achieve separation at the level of solution operators. The separation involves conservative and dissipative projections, reinforcing the decomposition structure established for the one-dimensional case.

Implications and Theoretical Contributions

The work presents a robust framework for understanding the long-term behavior of solutions to partially dissipative hyperbolic systems—an area crucial for applications in fluid dynamics and other phenomenologically related fields. The findings elucidate conditions under which global smooth solutions avoid the formation of singularities through entropy dissipation mechanisms, further supported by the SK condition. This enhances our theoretical grasp of how dissipative features influence solution stability and the rate at which they stabilize to equilibrium.

Future Directions

This paper sets a precedent for future analyses, particularly for systems where the SK condition might be violated. Additional exploration into nonnormal degeneracies, such as those occurring in the Kerr-Debye model, would provide deeper insight into broader classes of systems. An interesting direction would include understanding how these dynamics play out in real-world applications, such as electromagnetic wave propagation or gas dynamics covered partially in previous related works.

The multifaceted analysis in this paper offers significant contributions toward dissolving complex behaviors in dissipative systems into manageable analytical components, instrumental in advancing the STUDY of PDE systems with entropy considerations.