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On the global well-posedness of 3-D Boussinesq system with partial viscosity and axisymmetric data
Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions
Gran Sasso Science Institute, Department of Mathematics, Viale Francesco Crispi 7, 67100 - L'Aquila, Italy |
$ n,d\ge 1 $ |
$ \mathbf A: = (A^1,\dots,A^d)\in (\mathbb R^{n\times n})^d $ |
$ B\in \mathbb R^{n\times n} $ |
$ n\times n $ |
$ \begin{equation*} \partial_{t}u+\mathbf A\cdot \nabla_{\mathbf x} u+Bu = 0,\qquad (\mathbf x,t)\in \mathbb R^d\times \mathbb R_+. \end{equation*} $ |
$ u $ |
$ u = u^{(1)}+u^{(2)} $ |
$ u^{(1)} $ |
$ U $ |
$ u^{(1)}-U $ |
$ t^{-\frac d2(\frac 1q-\frac 1p)-\frac 12} $ |
$ t\to +\infty $ |
$ L^p $ |
$ u^{(2)} $ |
$ L^2 $ |
$ u(\cdot,0)\in L^q(\mathbb R^d)\cap L^2(\mathbb R^d) $ |
$ 1\le q\le p\le \infty $ |
$ u^{(1)}-U $ |
$ t^{-\frac d2(\frac 1q-\frac 1p)-1} $ |
$ t\to +\infty $ |
References:
[1] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin-Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-Heidelberg, 1976. |
[3] |
S. Bianchini, B. Hanouzet and R. Natalini,
Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622.
doi: 10.1002/cpa.20195. |
[4] |
A. Bressan,
An ill posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova, 110 (2003), 103-117.
|
[5] |
G. Craciun, A. Brown and A. Friedman,
A dynamical system model of neurofilament transport in axons, J. Theoret. Biol., 237 (2005), 316-322.
doi: 10.1016/j.jtbi.2005.04.018. |
[6] |
S. Goldstein,
On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129-156.
doi: 10.1093/qjmam/4.2.129. |
[7] |
T. Hosono and T. Ogawa,
Large time behavior and Lp-Lq estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118.
doi: 10.1016/j.jde.2004.03.034. |
[8] |
M. Kac, A stochastic model related to the telegrapher's equation. Reprinting of an article published in 1956, Papers arising from a Conference on Stochastic Differential Equations (Univ. Alberta, Edmonton, Alta., 1972), Rocky Mountain J. Math., 4 (1974), 497–509.
doi: 10.1216/RMJ-1974-4-3-497. |
[9] |
T. Kato, Perturbation Theory For Linear Operators, Springer-Verlag, Berlin, 1995. |
[10] |
S. Kawashima,
Global existence and stability of solutions for discrete velocity models of the Boltzmann equation, North-Holland Math. Stud., 98 (1984), 59-85.
doi: 10.1016/S0304-0208(08)71492-0. |
[11] |
P. Marcati and K. Nishihara,
The Lp-Lq estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations, 191 (2003), 445-469.
doi: 10.1016/S0022-0396(03)00026-3. |
[12] |
C. Mascia,
Exact representation of the asymptotic drift speed and diffusion matrix for a class of velocity-jump processes, J. Differential Equations, 260 (2016), 401-426.
doi: 10.1016/j.jde.2015.08.043. |
[13] |
C. Mascia and T. T. Nguyen,
Lp-Lq decay estimates for dissipative linear hyperbolic systems in 1D, J. Differential Equations, 263 (2017), 6189-6230.
doi: 10.1016/j.jde.2017.07.011. |
[14] |
T. Narazaki,
Lp-Lq estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626.
doi: 10.2969/jmsj/1191418647. |
[15] |
K. Nishihara,
Lp-Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z., 244 (2003), 631-649.
doi: 10.1007/s00209-003-0516-0. |
[16] |
D. Serre, Matrices. Theory and Applications, Springer-Verlag, New York, 2002. |
[17] |
Y. Shizuta and S. Kawashima,
Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.
doi: 10.14492/hokmj/1381757663. |
[18] |
Y. Ueda, R. Duan and S. Kawashima,
Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal., 205 (2012), 239-266.
doi: 10.1007/s00205-012-0508-5. |
[19] |
T. Umeda, S. Kawashima and Y. Shizuta,
On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457.
doi: 10.1007/BF03167068. |
show all references
References:
[1] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin-Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-Heidelberg, 1976. |
[3] |
S. Bianchini, B. Hanouzet and R. Natalini,
Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622.
doi: 10.1002/cpa.20195. |
[4] |
A. Bressan,
An ill posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova, 110 (2003), 103-117.
|
[5] |
G. Craciun, A. Brown and A. Friedman,
A dynamical system model of neurofilament transport in axons, J. Theoret. Biol., 237 (2005), 316-322.
doi: 10.1016/j.jtbi.2005.04.018. |
[6] |
S. Goldstein,
On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129-156.
doi: 10.1093/qjmam/4.2.129. |
[7] |
T. Hosono and T. Ogawa,
Large time behavior and Lp-Lq estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118.
doi: 10.1016/j.jde.2004.03.034. |
[8] |
M. Kac, A stochastic model related to the telegrapher's equation. Reprinting of an article published in 1956, Papers arising from a Conference on Stochastic Differential Equations (Univ. Alberta, Edmonton, Alta., 1972), Rocky Mountain J. Math., 4 (1974), 497–509.
doi: 10.1216/RMJ-1974-4-3-497. |
[9] |
T. Kato, Perturbation Theory For Linear Operators, Springer-Verlag, Berlin, 1995. |
[10] |
S. Kawashima,
Global existence and stability of solutions for discrete velocity models of the Boltzmann equation, North-Holland Math. Stud., 98 (1984), 59-85.
doi: 10.1016/S0304-0208(08)71492-0. |
[11] |
P. Marcati and K. Nishihara,
The Lp-Lq estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations, 191 (2003), 445-469.
doi: 10.1016/S0022-0396(03)00026-3. |
[12] |
C. Mascia,
Exact representation of the asymptotic drift speed and diffusion matrix for a class of velocity-jump processes, J. Differential Equations, 260 (2016), 401-426.
doi: 10.1016/j.jde.2015.08.043. |
[13] |
C. Mascia and T. T. Nguyen,
Lp-Lq decay estimates for dissipative linear hyperbolic systems in 1D, J. Differential Equations, 263 (2017), 6189-6230.
doi: 10.1016/j.jde.2017.07.011. |
[14] |
T. Narazaki,
Lp-Lq estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626.
doi: 10.2969/jmsj/1191418647. |
[15] |
K. Nishihara,
Lp-Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z., 244 (2003), 631-649.
doi: 10.1007/s00209-003-0516-0. |
[16] |
D. Serre, Matrices. Theory and Applications, Springer-Verlag, New York, 2002. |
[17] |
Y. Shizuta and S. Kawashima,
Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275.
doi: 10.14492/hokmj/1381757663. |
[18] |
Y. Ueda, R. Duan and S. Kawashima,
Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal., 205 (2012), 239-266.
doi: 10.1007/s00205-012-0508-5. |
[19] |
T. Umeda, S. Kawashima and Y. Shizuta,
On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457.
doi: 10.1007/BF03167068. |
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