Optimal decay estimates of a regularity-loss type system with constraint condition (Q1680446)

The paper is devoted to the Cauchy problem for the following wave-heat coupled system in \([0,\infty) \times \mathbb{R}^n\): \[ \begin{aligned} & u_{tt} - \Delta u + \gamma \theta=0, \quad \theta_t - \gamma u_t - \nu \Delta \theta=0,\\ & u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x),\quad \theta(0,x)=\theta_0(x), \end{aligned} \] where \(\gamma \in \mathbb{R} \setminus \{0\}\) and \(\nu \in \mathbb{R}_+\). Instead of this model the author studies the new model (\(v:=\nabla u\), \(w:=u_t\)) \[ \begin{aligned} & v_{t} - \nabla w=0,\quad w_t - \operatorname{div} v + \gamma \theta=0, \quad \theta_t - \gamma w - \nu \Delta \theta=0,\\ & v(0,x)=v_0(x),\quad w(0,x)=w_0(x),\quad \theta(0,x)=\theta_0(x). \end{aligned} \] It is clear that any distributional solution \(v:=\big(v^1,\cdots,v^n\big)\) satisfies the compatibility condition \(\partial_{x_j} v^k = \partial_{x_k} v^j\) for \(1 \leq j,k\leq n\). For this reason these compatibility conditions appear as constraint conditions in the further approach. After introducing the new vector-function \(U=(v,w,\theta)^T\) the above model is transferred to the new model \[ U_t + \sum_{j=1}^n A^j U_{x_j} - \sum_{j,k=1}^n B^{jk} U_{x_jx_k}+ LU=0, \quad U(0,x)=U_0(x), \] with symmetric matrices \(A^j\) and \(B^{jk}\) and a skew-symmetric matrix \(L\). It turns out this system has a regularity-loss structure and is uniformly dissipative of type (2,3). One feels both effects in the pointwise energy estimate in the phase space and the corresponding energy estimate in the physical space. In the estimate in physical space one has a loss of regularity and a decay depending on additional regularity \(L^m\), \(m \in [1,2)\), for the data. The main tool is the energy method in the Fourier space. The optimality of the estimate is verified by studying the asymptotical behavior of the eigenvalues for small and large frequencies as well. Special considerations for the asymptotic profile and asymptotic stability of the solutions in the one-dimensional case (no any constrain condition) complete the paper. There are no corollaries with results for the starting model.