×

Estimates for the maximum modulus of solutions to the Burgers system of equations. (Russian, English) Zbl 1050.35088

Sib. Mat. Zh. 45, No. 3, 653-657 (2004); translation in Sib. Math. J. 45, No. 3, 536-540 (2004).
Let \(\Omega\) be a bounded domain in \(\mathbb R_n\), let \(\Gamma\) be the boundary of \(\Omega\), and let \(u:\Omega\times [0,T]\to\mathbb R_n\) be a vector function. For \(x\in\Omega\), \(t\in [0,T]\), the author studies the system \(u_t-\nu\Delta u+(u\cdot\nabla)u=f+\text{div\, } g\) with initial and boundary conditions \(u(x,0)=u_0(x)\) for \(x\in\Omega\) and \(u(x,t)=u_*(x,t)\) for \((x,t)\in\partial\Omega\times [0,T]\). Here \(g\) is a field of \(n\times n\) matrices and \(\text{div\, } g\) is a vector with components \((\text{div\, } g)_i=\sum_{j=1}^{n}\partial g_{ij}/\partial x_{ij}\), \(i=1,\dots,n\). The author establishes some nonlinear estimates for the maximum modulus of solutions to this problem on using the particular features of convective bounds.

MSC:

35Q35 PDEs in connection with fluid mechanics
35K15 Initial value problems for second-order parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
PDFBibTeX XMLCite
Full Text: EuDML EMIS