Parabolic Systems with Polynomial Growth and Regularity

2011/03/10 by Frank Duzaar, Giuseppe Mingione, Klaus Steffen · 13 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows

paper · doi:10.1090/s0065-9266-2011-00614-3

Abstract

We establish a series of optimal regularity results for solutions to general non-linear parabolic systems <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline minus normal d normal i normal v a left-parenthesis x comma t comma u comma upper D u right-parenthesis plus upper H equals 0 comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo> − </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">v</mml:mi> </mml:mrow> <mml:mtext> </mml:mtext> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width="thinmathspace"/> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u_t- \mathrm div a(x,t,u,Du)+H=0 ,</mml:annotation> </mml:semantics> </mml:math> </disp-formula> under the main assumption of polynomial growth at rate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> i.e. <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue a left-parenthesis x comma t comma u comma upper D u right-parenthesis EndAbsoluteValue less-than-or-equal-to upper L left-parenthesis 1 plus StartAbsoluteValue upper D u EndAbsoluteValue Superscript p minus 1 Baseline right-parenthesis comma p greater-than-or-equal-to 2 period"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo> ≤ </mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mo>,</mml:mo> <mml:mspace width="2em"/> <mml:mi>p</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> <mml:mspace width="thickmathspace"/> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">|a(x,t,u,Du)|≤ L(1+|Du|^p-1) , p ≥ 2 .</mml:annotation> </mml:semantics> </mml:math> </disp-formula> We give a unified treatment of various interconnected aspects of the regularity theory: optimal partial regularity results for the spatial gradient of solutions, the first estimates on the (parabolic) Hausdorff dimension of the related singular set, and the first Calderón-Zygmund estimates for non-homogeneous problems are here achieved.

Citations

• Harmonic type approximation lemmas

Cited by

• Parabolic systems beyond standard growth
• Partial regularity for parabolic systems with VMO-coefficients
• On the Global Existence of a Class of Strongly Coupled Parabolic Systems
• Gradient regularity for strongly singular or degenerate elliptic and parabolic equations
• Partial regularity for parabolic systems of double phase type
• The boundedness-by-entropy method for cross-diffusion systems
• Regularity of weak solutions to a certain class of parabolic system
• A conditional regularity result for p-harmonic flows
• Parabolic Lipschitz truncation and Caloric Approximation
• The parabolic p-Laplacian with fractional differentiability
• Global gradient estimates for nonlinear obstacle problems with nonstandard growth
• Space-time approximation of stochastic p-Laplace systems
• Irregular double-phase evolution problem: existence and global regularity

Related

• A tree-free approach to regularity structures: The regular case for quasi-linear equations
• Error Bounds for Parametric Polynomial Systems with Applications to Higher-Order Stability Analysis and Convergence Rates
• A-caloric approximation and partial regularity for parabolic systems with Orlicz growth
• 𝐿²-solvability of the Dirichlet, Neumann and regularity problems for parabolic equations with time-independent Hölder-continuous coefficients
• A Quantitative Regularity Estimate for Nonnegative Supersolutions of Fully Nonlinear Uniformly Parabolic Equations