Betsy Stovall

University of California, Los Angeles, Los Angeles, California, United States

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Publications (5)3.88 Total impact

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    Rowan Killip, Betsy Stovall, Monica Visan
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    ABSTRACT: We analyze the blowup behaviour of solutions to the focusing nonlinear Klein--Gordon equation in spatial dimensions $d\geq 2$. We obtain upper bounds on the blowup rate, both globally in space and in light cones. The results are sharp in the conformal and sub-conformal cases. The argument relies on Lyapunov functionals derived from the dilation identity. We also prove that the critical Sobolev norm diverges near the blowup time.
    Mathematische Annalen 03/2012; · 1.38 Impact Factor
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    Spyridon Dendrinos, Betsy Stovall
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    ABSTRACT: We establish near-optimal mixed-norm estimates for the X-ray transform restricted to polynomial curves with a weight that is a power of the affine arclength. The bounds that we establish depend only on the spatial dimension and the degree of the polynomial. Some of our results are new even in the well-curved case.
    Journal of Functional Analysis 11/2011; · 1.25 Impact Factor
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    Rowan Killip, Betsy Stovall, Monica Visan
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    ABSTRACT: We consider both the defocusing and focusing cubic nonlinear Klein--Gordon equations $$ u_{tt} - \Delta u + u \pm u^3 =0 $$ in two space dimensions for real-valued initial data $u(0)\in H^1_x$ and $u_t(0)\in L^2_x$. We show that in the defocusing case, solutions are global and have finite global $L^4_{t,x}$ spacetime bounds. In the focusing case, we characterize the dichotomy between this behaviour and blowup for initial data with energy less than that of the ground state. These results rely on analogous statements for the two-dimensional cubic nonlinear Schr\"odinger equation, which are known in the defocusing case and for spherically-symmetric initial data in the focusing case. Thus, our results are mostly unconditional. It was previously shown by Nakanishi that spacetime bounds for Klein--Gordon equations imply the same for nonlinear Schr\"odinger equations.
    08/2010;
  • Betsy Stovall
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    ABSTRACT: We establish strong-type endpoint Lp(Rd)→Lq(Rd) bounds for the operator given by convolution with affine arclength measure on polynomial curves for d⩾4. The bounds established depend only on the dimension d and the degree of the polynomial.
    Journal of Functional Analysis 01/2010; 259(12):3205-3229. · 1.25 Impact Factor
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    Betsy Stovall
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    ABSTRACT: We prove that convolution with affine arclength measure on the curve parametrized by $h(t) := (t,t^2,...,t^n)$ is a bounded operator from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ for the full conjectured range of exponents, improving on a result due to M. Christ. We also obtain nearly sharp Lorentz space bounds. Comment: 24 pages, The final version of this article will appear in the Journal of the London Math. Soc
    05/2009;

Publication Stats

16 Citations
3.88 Total Impact Points

Institutions

  • 2011
    • University of California, Los Angeles
      • Department of Mathematics
      Los Angeles, California, United States