# zbMATH — the first resource for mathematics

Further properties of a continuum of model equations with globally defined flux. (English) Zbl 0916.35049
Author’s summary: To develop an understanding of singularity formation in vortex sheets, we consider model equations that exhibit shared characteristics with the vortex sheet equation but are slightly easier to analyze. A model equation is obtained by replacing the flux term in Burgers’ equation by alternatives that contain contributions depending globally on the solution. We consider the continuum of partial differential equations $$u_t= \theta(H(u)u)_x+ (1-\theta)H(u) u_x+ \nu u_{xx}$$, $$0\leq \theta\leq 1$$, $$\nu\geq 0$$, where $$H(u)$$ is the Hilbert transform of $$u$$. We show that when $$\theta= 1/2$$, for $$\nu>0$$, the solution of the equation exists for all time and is unique. We also show with a combination of analytical and numerical means that the solution when $$\theta= 1/2$$ and $$\nu>0$$ is analytic. Using a pseudo-spectral method in space and the Adams-Moulton fourth-order predictor-corrector in time, we compute the numerical solution of the equation with $$\theta= 1/2$$ for various viscosities. The results confirm that for $$\nu>0$$, the solution is well behaved and analytic. The numerical results also confirm that for $$\nu>0$$ and $$\theta= 1/2$$, the solution becomes singular in finite time and finite viscosity prevents singularity formation. We also present, for a certain class of initial conditions, solutions of the equation, with $$0<\theta< 1/3$$ and $$\theta= 1$$, that become infinite for $$\nu\geq 0$$ in finite time.

##### MSC:
 35K55 Nonlinear parabolic equations 35Q53 KdV equations (Korteweg-de Vries equations) 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
Full Text:
##### References:
  Baker, G.R.; Li, X.; Morlet, A.C., Analytic structure of 2 1D transport equations with nonlocal fluxes, Phys. D, 91, 349-375, (1996) · Zbl 0899.76104  Ball, J.M., Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. math., 28, 473-486, (1977) · Zbl 0377.35037  Bernoff, A.J.; Bertozzi, A.L., Singularities in a modified kuramoto – sivashinsky equation describing interface motion for phase transition, Phys. D, 85, 375-404, (1995) · Zbl 0899.76190  Budd, C.; Dold, B.; Stuart, A., Blowup in a partial differential equation with conserved first integral principle, SIAM J. appl. math., 53, 718-742, (1993) · Zbl 0784.35009  Doering, C.R.; Titi, E.S., Exponential decay rate of the power spectrum for solutions of the navier – stokes equations, Phys. fluids, 7, 1384-1390, (1995) · Zbl 1023.76513  Foias, C.; Teman, R., Gevrey class regularity for the solutions of the navier – stokes equations, J. funct. anal., 87, 359-369, (1989) · Zbl 0702.35203  Kreiss, H.-O.; Lorenz, J., Initial-boundary value problems and the navier – stokes equations, (1989), Academic Press San Diego  Morlet, A.C., Some further results for a one-dimensional transport equation with nonlocal flux, Comm. appl. anal., 1, 315-336, (1987) · Zbl 0893.35041  Palais, B., Blowup for nonlinear equations using a comparison principle in Fourier space, Comm. pure appl. math., 41, 165-196, (1988) · Zbl 0674.35045  Stein, E.M., Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Princeton · Zbl 0207.13501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
{"ft_lang_label":"__label__en","ft_lang_prob":0.78080595,"math_prob":0.9967073,"size":4196,"snap":"2021-04-2021-17","text_gpt3_token_len":1181,"char_repetition_ratio":0.12523855,"word_repetition_ratio":0.021311475,"special_character_ratio":0.30886558,"punctuation_ratio":0.22041763,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9981172,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-11T10:56:58Z\",\"WARC-Record-ID\":\"<urn:uuid:d795c2fa-9fbb-48cc-bb91-388aeb5c2f20>\",\"Content-Length\":\"51911\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a9ef851d-cfd3-404d-9e27-250fe209f1c3>\",\"WARC-Concurrent-To\":\"<urn:uuid:77217f95-9a5c-4ee8-964b-9f091ece2b39>\",\"WARC-IP-Address\":\"141.66.194.3\",\"WARC-Target-URI\":\"https://www.zbmath.org/?q=an%3A0916.35049\",\"WARC-Payload-Digest\":\"sha1:SISBOWFXTQY36NAHICB2YN33PDCB7BSJ\",\"WARC-Block-Digest\":\"sha1:XY6UUFBRVJXSOWP52BZIYMN3ZEV7LBU6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618038061820.19_warc_CC-MAIN-20210411085610-20210411115610-00112.warc.gz\"}"}