Abstract
We study the impact of the convective terms on the global solvability or finite time blow up of solutions of dissipative PDEs. We consider the model examples of 1D Burger’s type equations, convective Cahn–Hilliard equation, generalized Kuramoto–Sivashinsky equation and KdV type equations. The following common scenario is established: adding sufficiently strong (in comparison with the destabilizing nonlinearity) convective terms to equation prevents the solutions from blowing up in a finite time and makes the considered system globally well-posed and dissipative and for weak enough convective terms the finite time blow up may occur similar to the case, when the equation does not involve convective term. This kind of result has been previously known for the case of Burger’s type equations and has been strongly based on maximum principle. In contrast to this, our results are based on the weighted energy estimates which do not require the maximum principle for the considered problem.
Similar content being viewed by others
References
Alshin, A., Korpusov, M., Sveshnikov, A.: Blow up in Nonlinear Sobolev Type Equations. De Gruyter Series in Nonlinear Analysis and Applications. De Gruyter Incorporated, Walter (2011)
Ball J.: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Quart. J. Math. Oxford Ser. (2) 28(112), 473–486 (1977)
Bellout H., Benachour S., Titi E.: Finite-time singularity versus global regularity for hyper-viscous Hamilton–Jacobi-like equations. Nonlinearity 16, 1967–1989 (2003)
Bona J., Saut J.: Dispersive blow up of solutions of generalized Korteweg-de Vries equations. J. Differ. Equ. 103, 3–57 (1993)
Bona J., Sun S., Zhang B.: A nonhomogeneous boundary-value problem for the Kortewegde Vries equation posed on a finite domain. Commun. PDEs 28(7-8), 1391–1436 (2003)
Chen T.-F., Levine H., Sacks P.: Analysis of a convective reaction-diffusion equation. Nonlinear Anal. 12, 1349–1370 (1988)
Eden A., Kalantarov V.: 3D convective Cahn–Hilliard equation. Commun. Pure Appl. Anal. 6, 1075–1086 (2007)
Eden A., Kalantarov V., Zelik S.: Global solvability and blow up for the convective Cahn–Hilliard equations with concave potentials. J. Math. Phys. 54(041502), 1–12 (2013)
Foias C., Holm D., Titi E.: The Navier–Stokes-alpha model of fluid turbulence. Phys. D 152–153, 505–519 (2001)
Fursikov A.: On the normal semilinear parabolic equations corresponding to 3D Navier–Stokes system. Syst. Model. Optim. IFIP Adv. Commun. Inform. Technol. 391, 338–347 (2013)
Galaktionov V., Mitidieri E., Pohozhaev S.: Blow up for Higher Order Parabolic, Hyperbolic and Schrödinger Equations. CRC Press, Boca Raton (2015)
Hislop P., Sigal I.: Introduction to Spectral Theory. Springer, Berlin (1996)
Kalantarov V., Ladyzhenskaya O.: The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type. J. Soviet Math. 10, 53–70 (1978)
Ladyzhenskaya, O., Solonnikov, V., Uraltseva, N.: Linear and Quasilinear Equations of Parabolic Types. American Mathematical Society, Providence (1968)
Larios, A., Titi, E.: Global regularity vs. finite-time singularities: some paradigms on the effect of boundary conditions and certain perturbations. arXiv:1401.1534v1
Larkin N.: Modified KdV equation with a source term in a bounded domain. Math. Methods Appl. Sci. 29(7), 751–765 (2006)
Levine H.: Some nonexistence and instability theorems for solutions of formally parabolic equations of the form \({Pu_t = Au + \mathcal{F}(u)}\) . Arch. Ration. Mech. Anal. 51, 371–386 (1973)
Levine H.: Stability and instability for solutions of Burgers’ equation with a semilinear boundary condition. SIAM J. Math. Anal. 19, 312–336 (1988)
Levine H., Payne L.: Nonexistence theorems for the heat equations with nonlinear boundary conditions and for porous medium equation backward in time. J. Differ. Equ. 16, 319–334 (1974)
Levine H., Payne L., Sacks P., Straughan B.: Analysis of a convective reaction-diffusion equation II. SIAM J. Math. Anal. 20, 133–147 (1989)
Martel Y., Merle F.: Blow up in finite time and dynamics of blow up solutions for the L 2-critical generalized KdV equation. J. Am. Math. Soc. 15, 617–664 (2002)
Pokhozhaev S.: On the blow-up of solutions of the Kuramoto–Sivashinsky equation. Sb. Math. 199, 1355–1365 (2008)
Pokhozhaev S.: On the nonexistence of global solutions of the Korteweg-de Vries equation. J. Math. Sci. (NY) 190, 147–156 (2013)
Quittner P., Souplet Ph.: Superlinear parabolic problems. Blow up global existence and steady states. Birkhauser Advanced Texts, Basel (2007)
Souplet P., Weissler F.: Poincares inequality and global solutions of a nonlinear parabolic equation. Ann. Inst. H. Poincare Anal. Non Lineaire 16, 335–371 (1999)
Souplet P., Weissler F.: Self-similar subsolutions and blowup for nonlinear parabolic equations. J. Math. Anal. Appl. 212, 60–74 (1997)
Tao, T.: Nonlinear Dispersive Equations. Local and Global Analysis. American Mathematical Society, Providence (2006)
Tersenov A.: The preventive effect of the convection and of the diffusion in the blow-up phenomenon for parabolic equations. Ann. Inst. H. Poincare Anal. Non Lineaire 21, 533–541 (2004)
Tersenov A.: A condition guaranteeing the abscence of the blow-up phenomenon for the generalized Burgers equation. Nonlinear Anal. 75, 5119–5122 (2012)
Yushkov E.: Blowup in Korteweg-de Vries-type systems. Theor. Math. Phys. 173, 1498–1506 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A.V. Fursikov
This work is partially supported by the Grant 14-41-00044 of RSF.
Rights and permissions
About this article
Cite this article
Bilgin, B., Kalantarov, V. & Zelik, S. Preventing Blow up by Convective Terms in Dissipative PDE’s. J. Math. Fluid Mech. 18, 463–479 (2016). https://doi.org/10.1007/s00021-016-0270-9
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00021-016-0270-9