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Register a new userGlobal existence and smoothness for solutions of viscous Burgers equation. (2) The unbounded case: a characteristic flow study
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Jeremie Unterberger
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We show that the homogeneous viscous Burgers equation $(partial_t-etaDelta) u(t,x)+(ucdot abla)u(t,x)=0, (t,x)in{mathbb{R}}_+times{mathbb{R}}^d$ $(dge 1, eta>0)$ has a globally defined smooth solution if the initial condition $u_0$ is a smooth function growing like $o(|x|)$ at infinity. The proof relies mostly on estimates of the random characteristic flow defined by a Feynman-Kac representation of the solution. Viscosity independent a priori bounds for the solution are derived from these. The regularity of the solution is then proved for fixed $eta>0$ using Schauder estimates. The result extends with few modifications to initial conditions growing abnormally large in regions with small relative volume, separated by well-behaved bulk regions, provided these are stable under the characteristic flow with high probability. We provide a large family of examples for which this loose criterion may be verified by hand.
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We prove that the viscous Burgers equation has a globally defined smooth solution in all dimensions provided the initial condition and the forcing term are smooth and bounded together with their derivatives. Such solutions may have infinite energy. The proof does not rely on energy estimates, but on a combination of the maximum principle and quantitative Schauder estimates. We obtain precise bounds on the sup norm of the solution and its derivatives, making it plain that there is no exponential increase in time. In particular, these bounds are time-independent if the forcing term is zero. To get a classical solution, it suffices to assume that the initial condition and the forcing term have bounded derivatives up to order two.
We have shown in a recent collaboration that the Cauchy problem for the multi-dimensional Burgers equation is well-posed when the initial data u(0) is taken in the Lebesgue space L 1 (R n), and more generally in L p (R n). We investigate here the situation where u(0) is a bounded measure instead, focusing on the case n = 2. This is motivated by the description of the asymptotic behaviour of solutions with integrable data, as t $rightarrow$ +$infty$. MSC2010: 35F55, 35L65. Notations. We denote $times$ p the norm in Lebesgue L p (R n). The space of bounded measure over R m is M (R m) and its norm is denoted $times$ M. The Dirac mass at X $in$ R n is $delta$ X or $delta$ x=X. If $ u$ $in$ M (R m) and $mu$ $in$ M (R q), then $ u$ $otimes$ $mu$ is the measure over R m+q uniquely defined by $ u$ $otimes$ $mu$, $psi$ = $ u$, f $mu$, g whenever $psi$(x, y) $ otequiv$ f (x)g(y). The closed halves of the real line are denoted R + and R --. * U.M.P.A., UMR CNRS-ENSL # 5669. 46 all{e}e dItalie,
In this paper, we consider the advective Cahn-Hilliard equation in 2D with shear flow: $$ begin{cases} u_t+v_1(y) partial_x u+gamma Delta^2 u=gamma Delta(u^3-u) quad & quad textrm{on} quad mathbb T^2; u textrm{periodic} quad & quad textrm{on} quad partial mathbb T^2, end{cases} $$ where $mathbb T^2$ is the two-dimensional torus. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we show the global existence of solutions with arbitrary initial $H^2$ data. The main difficulty of this paper is to handle the high-regularity and non-linearity underlying the term $Delta(u^3)$ in a proper way. For such a purpose, we modify the methods by Iyer, Xu, and Zlatov{s} in 2021 under a shear flow setting.
By studying the linearization of contour dynamics equation and using implicit function theorem, we prove the existence of co-rotating and travelling global solutions for the gSQG equation, which extends the result of Hmidi and Mateu cite{HM} to $alphain[1,2)$. Moreover, we prove the $C^infty$ regularity of vortices boundary, and show the convexity of each vortices component.
Analytic solutions for Burgers equations with source terms, possibly stiff, represent an important element to assess numerical schemes. Here we present a procedure, based on the characteristic technique to obtain analytic solutions for these equations with smooth initial conditions.
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