Samokhin A. Gradient catastrophes for Burgers equation on a finite interval. Numerical and qualitative study, talk Workshop Geom. of PDEs and Integrability, 14-18 Oct 2013, Teplice nad Becvou, Czech Rep. (abstract): Difference between revisions
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<math>u_t(x,t)=u_{xx}(x,t)+2\eta u(x,t)u_x(x,t)</math> | <math>u_t(x,t)=u_{xx}(x,t)+2\eta u(x,t)u_x(x,t)</math> | ||
on a finite interval: | |||
<math>u(x,0) =f(x), \quad u(\alpha,t) = l(t), \quad u(\beta,t) =r(t), \quad x\in[\alpha,\beta]</math>. | <math>u(x,0) =f(x), \quad u(\alpha,t) = l(t), \quad u(\beta,t) =r(t), \quad x\in[\alpha,\beta]</math>. | ||
The case of constant boundary conditions <math>u(\alpha,t) = A, \quad u(\beta,t) =B</math> and its asymptotics is of special interest here. For such a IVBP viscosity usually produces asymptotic stationary solution which is invariant for some subalgebra of the full symmetry algebra of the equation. But the evolution may also result in a stable gradient catastrophe. | The case of constant boundary conditions <math>u(\alpha,t) = A, \quad u(\beta,t) =B</math> and its asymptotics is of special interest here. For such a IVBP viscosity usually produces asymptotic stationary solution which is invariant for some subalgebra of the full symmetry algebra of the equation. But the evolution may also result in a stable gradient catastrophe. | ||
Revision as of 12:19, 9 July 2013
Speaker: Alexey Samokhin
Title: Gradient catastrophes for Burgers equation on a finite interval. Numerical and qualitative study
Abstract:
We consider initial value - boundary problem (IVBP) for the Burgers equation
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_t(x,t)=u_{xx}(x,t)+2\eta u(x,t)u_x(x,t)}
on a finite interval:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(x,0) =f(x), \quad u(\alpha,t) = l(t), \quad u(\beta,t) =r(t), \quad x\in[\alpha,\beta]} .
The case of constant boundary conditions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(\alpha,t) = A, \quad u(\beta,t) =B}
and its asymptotics is of special interest here. For such a IVBP viscosity usually produces asymptotic stationary solution which is invariant for some subalgebra of the full symmetry algebra of the equation. But the evolution may also result in a stable gradient catastrophe.