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<p>The one-dimensional scalar surface growth model, arising from the physical process of molecular epitaxy, shares many striking similarities with the three-dimensional incompressible NavierStokes equations. While significant results exist regarding the existence and regularity of solutions for this model, the large-time behavior of solutions remains unexplored, to the best of our knowledge. This paper aims to establish the first decay result for weak solutions. We prove that there exists a weak solution <i>u</i> satisfying <Equation ID="Equ1"> <EquationSource Format="TEX">\(\Vert{u}\Vert_{2}\lesssim(1+t)^{1\over{8}}.\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo>∥</mo> <mrow> <mi>u</mi> </mrow> <msub> <mo>∥</mo> <mrow> <mn>2</mn> </mrow> </msub> <mo>≲</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> </mrow> </mfrac> </mrow> </msup> <mo>.</mo> </math></EquationSource> </Equation> Furthermore, we demonstrate that this decay rate is sharp by constructing initial data for which the corresponding weak solution <i>u</i> satisfies <Equation ID="Equ2"> <EquationSource Format="TEX">\(\Vert{u}\Vert_{2}\gtrsim(1+t)^{1\over{8}}.\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo>∥</mo> <mrow> <mi>u</mi> </mrow> <msub> <mo>∥</mo> <mrow> <mn>2</mn> </mrow> </msub> <mo>≳</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> </mrow> </mfrac> </mrow> </msup> <mo>.</mo> </math></EquationSource> </Equation></p>

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Large Time Behaviour of Weak Solutions to the Surface Growth Model

  • 期刊论文

  • 发表日期：

    2026年1月29日

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  • Jia-qi Yang,
  • Wen-xuan Zhu

摘要

The one-dimensional scalar surface growth model, arising from the physical process of molecular epitaxy, shares many striking similarities with the three-dimensional incompressible NavierStokes equations. While significant results exist regarding the existence and regularity of solutions for this model, the large-time behavior of solutions remains unexplored, to the best of our knowledge. This paper aims to establish the first decay result for weak solutions. We prove that there exists a weak solution u satisfying \(\Vert{u}\Vert_{2}\lesssim(1+t)^{1\over{8}}.\) ∥ u ∥ 2 ≲ ( 1 + t ) 1 8 . Furthermore, we demonstrate that this decay rate is sharp by constructing initial data for which the corresponding weak solution u satisfies \(\Vert{u}\Vert_{2}\gtrsim(1+t)^{1\over{8}}.\) ∥ u ∥ 2 ≳ ( 1 + t ) 1 8 .