Bilinear wave estimates: Difference between revisions

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===Bilinear estimates===
===Bilinear estimates===


* Let <math>d>1</math>. If <math>\phi</math>, <math\psi</math> are free <math>\dot H^{s_1}</math> and <math>\dot H^{s_2}</math> solutions respectively, then one can control <math>\phi\psi</math> in \dot X^{s,b} if and only if
* Let <math>d>1</math>. If <math>\phi</math>, <math>\psi</math> are free <math>\dot H^{s_1}</math> and <math>\dot H^{s_2}</math> solutions respectively, then one can control <math>\phi\psi</math> in \dot X^{s,b} if and only if
** (Scaling) <math>s+b = s_1 + s_2 - (d-1)/2</math>
** (Scaling) <math>s+b = s_1 + s_2 - (d-1)/2</math>
** (Parallel interactions) <math>b \geq (3-d)/4</math>
** (Parallel interactions) <math>b \geq (3-d)/4</math>

Revision as of 20:35, 6 November 2006

Bilinear estimates

  • Let . If , are free and solutions respectively, then one can control in \dot X^{s,b} if and only if
    • (Scaling)
    • (Parallel interactions)
    • (Lack of smoothing)
    • (Frequency cancellation)
    • (No double endpoints) .

See FcKl2000. Null forms can also be handled by identities such as

  • Some bilinear Strichartz estimates are also known. For instance, if , , are as in the linear Strichartz estimates , are solutions, then

as long as FcKl2000. Similar estimates for null forms also exist Pl2002; see also TaVa2000b, Ta-p4.