On some energy inequalities and supnorm estimates for advection-diffusion equations in \(\mathbb R^n\). (English) Zbl 1284.35091
Summary: We show how some suitable standard energy inequalities can be used for a simple short derivation of the fundamental supnorm estimate
\[
||u(\cdot, t)||_{L^\infty (\mathbb R^n)}\leq K(n,p)||u(\cdot, 0)||_{L^p(\mathbb R^n)}t^{-\frac{n}{2_p}} \;\forall t>0
\]
for solutions of advection-diffusion equations of the form \(u_t+\mathrm{div}\;\mathbf f(u)={\varDelta}u\) (and some generalizations) in the whole space \(\mathbb R^n\), where \(n\geq 1\) is arbitrary. Our approach can be used for other parabolic equations as well, including degenerate cases like porous medium type equations and \(p\)-Laplacian evolution equations. Some open problems and related results of interest are also given.
MSC:
| 35B45 | A priori estimates in context of PDEs |
| 35K58 | Semilinear parabolic equations |
| 35K15 | Initial value problems for second-order parabolic equations |
Keywords:
advection-diffusion equations; energy inequalities; porous medium type equations; \(p\)-Laplacian evolution equationsReferences:
| [1] | Kreiss, H. O.; Lorenz, J., Initial-Boundary Value Problems and the Navier-Stokes Equations (1989), Academic Press: Academic Press Boston · Zbl 0689.35001 |
| [2] | Oleinik, O. A.; Kruzhkov, S. N., Quasilinear second order parabolic equations with many independent variables, Russian Math. Surveys, 16, 105-146 (1961) · Zbl 0112.32604 |
| [3] | Serre, D., Systems of Conservation Laws, Vol. I (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0930.35001 |
| [4] | Cipriani, F.; Grillo, G., Uniform bounds for solutions to quasilinear parabolic equations, J. Differential Equations, 177, 209-234 (2001) · Zbl 1036.35043 |
| [5] | Escobedo, M.; Zuazua, E., Large time behavior for convection-diffusion equations in \(R^N\), J. Funct. Anal., 100, 119-161 (1991) · Zbl 0762.35011 |
| [6] | Porzio, M. M., On decay estimates, J. Evol. Equ., 9, 561-591 (2009) · Zbl 1239.35023 |
| [7] | Schonbek, M. E., Uniform decay rates for parabolic conservation laws, Nonlinear Anal. TMA, 10, 943-956 (1986) · Zbl 0617.35060 |
| [8] | Zingano, P. R., Nonlinear \(L^2\) stability under large disturbances, J. Comput. Appl. Math., 103, 207-219 (1999) · Zbl 0942.35027 |
| [9] | Nash, J., Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80, 931-954 (1958) · Zbl 0096.06902 |
| [10] | Hörmander, L., Lectures on Nonlinear Hyperbolic Differential Equations (1997), Springer: Springer Berlin · Zbl 0881.35001 |
| [11] | Escobedo, M.; Vazquez, J. L.; Zuazua, E., Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Ration. Mech. Anal., 124, 43-65 (1993) · Zbl 0807.35059 |
| [12] | Carlen, E. A.; Loss, M., Sharp constant in Nash’s inequality, Int. Math. Res. Not. IMRN, 213-215 (1993) · Zbl 0822.35018 |
| [13] | Beckner, W., Geometric proof of Nash’s inequality, Int. Math. Res. Not. IMRN, 67-71 (1998) · Zbl 0895.35015 |
| [14] | Braz e. Silva, P.; Schütz, L.; Zingano, P. R., Decay estimates for solutions of quasilinear parabolic equations in heterogeneous media, Adv. Differ. Equ. Control Process., 6, 101-112 (2010) · Zbl 1219.35025 |
| [15] | Evans, L. C., Partial Differential Equations (1998), Amer. Math. Society: Amer. Math. Society Providence · Zbl 0902.35002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
