Search results for "35K15"

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Existence of dynamical low-rank approximations to parabolic problems

2021

The existence and uniqueness of weak solutions to dynamical low-rank evolution problems for parabolic partial differential equations in two spatial dimensions is shown, covering also non-diagonal diffusion in the elliptic part. The proof is based on a variational time-stepping scheme on the low-rank manifold. Moreover, this scheme is shown to be closely related to practical methods for computing such low-rank evolutions.

Algebra and Number TheoryPartial differential equationRank (linear algebra)Applied MathematicsNumerical Analysis (math.NA)010103 numerical & computational mathematics01 natural sciencesManifold010101 applied mathematics35K15 35R01 (Primary) 15A69 65L05 (Secondary)Computational MathematicsMathematics - Analysis of PDEsScheme (mathematics)FOS: MathematicsApplied mathematicsUniquenessMathematics - Numerical Analysisddc:5100101 mathematicsDiffusion (business)Analysis of PDEs (math.AP)Mathematics
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Stability of degenerate parabolic Cauchy problems

2015

We prove that solutions to Cauchy problems related to the $p$-parabolic equations are stable with respect to the nonlinearity exponent $p$. More specifically, solutions with a fixed initial trace converge in an $L^q$-space to a solution of the limit problem as $p>2$ varies.

Trace (linear algebra)Applied MathematicsDegenerate energy levelsMathematical analysista111nonlinear parabolic equationsCauchy distribution35K55 35K15stabilityStability (probability)Nonlinear systemMathematics - Analysis of PDEsBarenblatt solutionsExponentFOS: MathematicsInitial value problemLimit (mathematics)initial value problemsCauchy problemsAnalysisMathematicsAnalysis of PDEs (math.AP)Communications on pure and applied analysis
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